Theorem caser ≪ | index | src | ≫

pub theorem caser (A B: set) (n: nat): $ case A B @ b1 n = B @ n $;

Step	Hyp	Ref	Expression
1		ifpos	odd i -> if (odd i) (B @ (i // 2)) (A @ (i // 2)) = B @ (i // 2)
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 -> if (odd i) (B @ (i // 2)) (A @ (i // 2)) = B @ (i // 2)
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	appeq2d	i = b1 n -> B @ (i // 2) = B @ n
11	6, 10	eqtrd	i = b1 n -> if (odd i) (B @ (i // 2)) (A @ (i // 2)) = B @ n
12	11	applame	(\ i, if (odd i) (B @ (i // 2)) (A @ (i // 2))) @ b1 n = B @ n
13	12	conv case	case A B @ b1 n = B @ n

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)