Theorem leupto ≪ | index | src | ≫

theorem leupto (a b: nat): $ a <= b <-> upto a <= upto b $;

Step	Hyp	Ref	Expression
1		bitr4	(a <= b <-> 2 ^ a <= 2 ^ b) -> (upto a <= upto b <-> 2 ^ a <= 2 ^ b) -> (a <= b <-> upto a <= upto b)
2		lepow2	1 < 2 -> (a <= b <-> 2 ^ a <= 2 ^ b)
3		d1lt2	1 < 2
4	2, 3	ax_mp	a <= b <-> 2 ^ a <= 2 ^ b
5	1, 4	ax_mp	(upto a <= upto b <-> 2 ^ a <= 2 ^ b) -> (a <= b <-> upto a <= upto b)
6		bitr	(upto a <= upto b <-> upto a + 1 <= upto b + 1) -> (upto a + 1 <= upto b + 1 <-> 2 ^ a <= 2 ^ b) -> (upto a <= upto b <-> 2 ^ a <= 2 ^ b)
7		leadd1	upto a <= upto b <-> upto a + 1 <= upto b + 1
8	6, 7	ax_mp	(upto a + 1 <= upto b + 1 <-> 2 ^ a <= 2 ^ b) -> (upto a <= upto b <-> 2 ^ a <= 2 ^ b)
9		leeq	upto a + 1 = 2 ^ a -> upto b + 1 = 2 ^ b -> (upto a + 1 <= upto b + 1 <-> 2 ^ a <= 2 ^ b)
10		uptoadd1	upto a + 1 = 2 ^ a
11	9, 10	ax_mp	upto b + 1 = 2 ^ b -> (upto a + 1 <= upto b + 1 <-> 2 ^ a <= 2 ^ b)
12		uptoadd1	upto b + 1 = 2 ^ b
13	11, 12	ax_mp	upto a + 1 <= upto b + 1 <-> 2 ^ a <= 2 ^ b
14	8, 13	ax_mp	upto a <= upto b <-> 2 ^ a <= 2 ^ b
15	5, 14	ax_mp	a <= b <-> upto a <= upto 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)