Theorem dvdadd1 ≪ | index | src | ≫ 

theorem dvdadd1 (a b n: nat): $ n || a -> (n || b <-> n || a + b) $;
StepHypRefExpression
1 idvd 
(x + y) * n = a + b -> n || a + b
2 addmul 
(x + y) * n = x * n + y * n
3 addeq 
x * n = a -> y * n = b -> x * n + y * n = a + b
4 3 imp 
x * n = a /\ y * n = b -> x * n + y * n = a + b
5 2, 4 syl5eq 
x * n = a /\ y * n = b -> (x + y) * n = a + b
6 1, 5 syl 
x * n = a /\ y * n = b -> n || a + b
7 6 eexda 
x * n = a -> E. y y * n = b -> n || a + b
8 7 conv dvd 
x * n = a -> n || b -> n || a + b
9 idvd 
(z - x) * n = b -> n || b
10 addcan2 
a + (z - x) * n = a + b <-> (z - x) * n = b
11 addeq1 
x * n = a -> x * n + (z - x) * n = a + (z - x) * n
12 11 anwl 
x * n = a /\ z * n = a + b -> x * n + (z - x) * n = a + (z - x) * n
13 addmul 
(x + (z - x)) * n = x * n + (z - x) * n
14 mul0 
(x + (z - x)) * 0 = 0
15 muleq2 
n = 0 -> (x + (z - x)) * n = (x + (z - x)) * 0
16 14, 15 syl6eq 
n = 0 -> (x + (z - x)) * n = 0
17 mul0 
z * 0 = 0
18 muleq2 
n = 0 -> z * n = z * 0
19 17, 18 syl6eq 
n = 0 -> z * n = 0
20 16, 19 eqtr4d 
n = 0 -> (x + (z - x)) * n = z * n
21 20 anwr 
x * n = a /\ z * n = a + b /\ n = 0 -> (x + (z - x)) * n = z * n
22 pncan3 
x <= z -> x + (z - x) = z
23 leaddid1 
a <= a + b
24 lt01 
0 < n <-> n != 0
25 24 conv ne 
0 < n <-> ~n = 0
26 lemul1 
0 < n -> (x <= z <-> x * n <= z * n)
27 25, 26 sylbir 
~n = 0 -> (x <= z <-> x * n <= z * n)
28 27 anwr 
x * n = a /\ z * n = a + b /\ ~n = 0 -> (x <= z <-> x * n <= z * n)
29 anll 
x * n = a /\ z * n = a + b /\ ~n = 0 -> x * n = a
30 anlr 
x * n = a /\ z * n = a + b /\ ~n = 0 -> z * n = a + b
31 29, 30 leeqd 
x * n = a /\ z * n = a + b /\ ~n = 0 -> (x * n <= z * n <-> a <= a + b)
32 28, 31 bitrd 
x * n = a /\ z * n = a + b /\ ~n = 0 -> (x <= z <-> a <= a + b)
33 23, 32 mpbiri 
x * n = a /\ z * n = a + b /\ ~n = 0 -> x <= z
34 22, 33 syl 
x * n = a /\ z * n = a + b /\ ~n = 0 -> x + (z - x) = z
35 34 muleq1d 
x * n = a /\ z * n = a + b /\ ~n = 0 -> (x + (z - x)) * n = z * n
36 21, 35 casesda 
x * n = a /\ z * n = a + b -> (x + (z - x)) * n = z * n
37 anr 
x * n = a /\ z * n = a + b -> z * n = a + b
38 36, 37 eqtrd 
x * n = a /\ z * n = a + b -> (x + (z - x)) * n = a + b
39 13, 38 syl5eqr 
x * n = a /\ z * n = a + b -> x * n + (z - x) * n = a + b
40 12, 39 eqtr3d 
x * n = a /\ z * n = a + b -> a + (z - x) * n = a + b
41 10, 40 sylib 
x * n = a /\ z * n = a + b -> (z - x) * n = b
42 9, 41 syl 
x * n = a /\ z * n = a + b -> n || b
43 42 eexda 
x * n = a -> E. z z * n = a + b -> n || b
44 43 conv dvd 
x * n = a -> n || a + b -> n || b
45 8, 44 ibid 
x * n = a -> (n || b <-> n || a + b)
46 45 eex 
E. x x * n = a -> (n || b <-> n || a + b)
47 46 conv dvd 
n || a -> (n || b <-> n || a + b)

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)