Theorem Sumr ≪ | index | src | ≫

pub theorem Sumr (A B: set) (n: nat): $ b1 n e. Sum A B <-> n e. B $;

Step	Hyp	Ref	Expression
1		ifppos	odd i -> (ifp (odd i) (i // 2 e. B) (i // 2 e. A) <-> i // 2 e. B)
2		oddeq	i = b1 n -> (odd i <-> odd (b1 n))
3		b1odd	odd (b1 n)
4	3	a1i	i = b1 n -> odd (b1 n)
5	2, 4	mpbird	i = b1 n -> odd i
6	1, 5	syl	i = b1 n -> (ifp (odd i) (i // 2 e. B) (i // 2 e. A) <-> i // 2 e. B)
7		b1div2	b1 n // 2 = n
8		diveq1	i = b1 n -> i // 2 = b1 n // 2
9	7, 8	syl6eq	i = b1 n -> i // 2 = n
10	9	eleq1d	i = b1 n -> (i // 2 e. B <-> n e. B)
11	6, 10	bitrd	i = b1 n -> (ifp (odd i) (i // 2 e. B) (i // 2 e. A) <-> n e. B)
12	11	elabe	b1 n e. {i \| ifp (odd i) (i // 2 e. B) (i // 2 e. A)} <-> n e. B
13	12	conv Sum	b1 n e. Sum A B <-> n e. 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)