Theorem cardb0 ≪ | index | src | ≫

theorem cardb0 (n: nat): $ card (b0 n) = card n $;

Step	Hyp	Ref	Expression
1		b00	b0 0 = 0
2		b0eq	n = 0 -> b0 n = b0 0
3		id	n = 0 -> n = 0
4	2, 3	eqeqd	n = 0 -> (b0 n = n <-> b0 0 = 0)
5	1, 4	mpbiri	n = 0 -> b0 n = n
6	5	cardeqd	n = 0 -> card (b0 n) = card n
7		add0	card n + 0 = card n
8		addeq	card (b0 n // 2) = card n -> b0 n % 2 = 0 -> card (b0 n // 2) + b0 n % 2 = card n + 0
9		cardeq	b0 n // 2 = n -> card (b0 n // 2) = card n
10		b0div2	b0 n // 2 = n
11	9, 10	ax_mp	card (b0 n // 2) = card n
12	8, 11	ax_mp	b0 n % 2 = 0 -> card (b0 n // 2) + b0 n % 2 = card n + 0
13		b0mod2	b0 n % 2 = 0
14	12, 13	ax_mp	card (b0 n // 2) + b0 n % 2 = card n + 0
15		b0ne0	b0 n != 0 <-> n != 0
16	15	conv ne	b0 n != 0 <-> ~n = 0
17		cardval	b0 n != 0 -> card (b0 n) = card (b0 n // 2) + b0 n % 2
18	16, 17	sylbir	~n = 0 -> card (b0 n) = card (b0 n // 2) + b0 n % 2
19	14, 18	syl6eq	~n = 0 -> card (b0 n) = card n + 0
20	7, 19	syl6eq	~n = 0 -> card (b0 n) = card n
21	6, 20	cases	card (b0 n) = card 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)