h1de2i - Hilbert Space Explorer

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Theorem h1de2i 32167

Description: Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 17-Jul-2001.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
h1de2.1	⊢ 𝐴 ∈ ℋ
h1de2.2	⊢ 𝐵 ∈ ℋ

**Assertion**
Ref	Expression
h1de2i	⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))

Proof of Theorem h1de2i Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		h1de2.2	. . . . . . . . 9 ⊢ 𝐵 ∈ ℋ
2	1, 1	hicli 31695	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
3		h1de2.1	. . . . . . . 8 ⊢ 𝐴 ∈ ℋ
4	2, 3	hvmulcli 31628	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ∈ ℋ
5	3, 1	hicli 31695	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
6	5, 1	hvmulcli 31628	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ∈ ℋ
7		his2sub 31706	. . . . . . 7 ⊢ ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ∈ ℋ ∧ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐴) = ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐴) − (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐴)))
8	4, 6, 3, 7	mp3an 1490	. . . . . 6 ⊢ ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐴) = ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐴) − (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐴))
9		ax-his3 31698	. . . . . . . . 9 ⊢ (((𝐵 ·_ih 𝐵) ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐴) = ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐴)))
10	2, 3, 3, 9	mp3an 1490	. . . . . . . 8 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐴) = ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐴))
11	3, 3	hicli 31695	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐴) ∈ ℂ
12	2, 11	mulcomi 11327	. . . . . . . 8 ⊢ ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐴)) = ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵))
13	10, 12	eqtri 2786	. . . . . . 7 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐴) = ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵))
14		ax-his3 31698	. . . . . . . 8 ⊢ (((𝐴 ·_ih 𝐵) ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐴) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)))
15	5, 1, 3, 14	mp3an 1490	. . . . . . 7 ⊢ (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐴) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴))
16	13, 15	oveq12i 7439	. . . . . 6 ⊢ ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐴) − (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐴)) = (((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)))
17	8, 16	eqtr2i 2787	. . . . 5 ⊢ (((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴))) = ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐴)
18		his2sub 31706	. . . . . . . 8 ⊢ ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ∈ ℋ ∧ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐵) = ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) − (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵)))
19	4, 6, 1, 18	mp3an 1490	. . . . . . 7 ⊢ ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐵) = ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) − (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵))
20	2, 5	mulcomi 11327	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐵))
21		ax-his3 31698	. . . . . . . . . 10 ⊢ (((𝐵 ·_ih 𝐵) ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) = ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐵)))
22	2, 3, 1, 21	mp3an 1490	. . . . . . . . 9 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) = ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐵))
23		ax-his3 31698	. . . . . . . . . 10 ⊢ (((𝐴 ·_ih 𝐵) ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐵)))
24	5, 1, 1, 23	mp3an 1490	. . . . . . . . 9 ⊢ (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐵))
25	20, 22, 24	3eqtr4i 2796	. . . . . . . 8 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) = (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵)
26	4, 1	hicli 31695	. . . . . . . . 9 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) ∈ ℂ
27	6, 1	hicli 31695	. . . . . . . . 9 ⊢ (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵) ∈ ℂ
28	26, 27	subeq0i 11648	. . . . . . . 8 ⊢ (((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) − (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵)) = 0 ↔ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) = (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵))
29	25, 28	mpbir 234	. . . . . . 7 ⊢ ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) − (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵)) = 0
30	19, 29	eqtri 2786	. . . . . 6 ⊢ ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐵) = 0
31	1	h1dei 32164	. . . . . . . . 9 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ ℋ ((𝐵 ·_ih 𝑥) = 0 → (𝐴 ·_ih 𝑥) = 0)))
32	3, 31	mpbiran 722	. . . . . . . 8 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ ∀𝑥 ∈ ℋ ((𝐵 ·_ih 𝑥) = 0 → (𝐴 ·_ih 𝑥) = 0))
33	4, 6	hvsubcli 31635	. . . . . . . . 9 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ∈ ℋ
34		oveq2 7435	. . . . . . . . . . . 12 ⊢ (𝑥 = (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) → (𝐵 ·_ih 𝑥) = (𝐵 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))))
35	34	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑥 = (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) → ((𝐵 ·_ih 𝑥) = 0 ↔ (𝐵 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))) = 0))
36		oveq2 7435	. . . . . . . . . . . 12 ⊢ (𝑥 = (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) → (𝐴 ·_ih 𝑥) = (𝐴 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))))
37	36	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑥 = (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) → ((𝐴 ·_ih 𝑥) = 0 ↔ (𝐴 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))) = 0))
38	35, 37	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑥 = (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) → (((𝐵 ·_ih 𝑥) = 0 → (𝐴 ·_ih 𝑥) = 0) ↔ ((𝐵 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))) = 0 → (𝐴 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))) = 0)))
39	38	rspcv 3575	. . . . . . . . 9 ⊢ ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ∈ ℋ → (∀𝑥 ∈ ℋ ((𝐵 ·_ih 𝑥) = 0 → (𝐴 ·_ih 𝑥) = 0) → ((𝐵 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))) = 0 → (𝐴 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))) = 0)))
40	33, 39	ax-mp 5	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℋ ((𝐵 ·_ih 𝑥) = 0 → (𝐴 ·_ih 𝑥) = 0) → ((𝐵 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))) = 0 → (𝐴 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))) = 0))
41	32, 40	sylbi 220	. . . . . . 7 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐵 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))) = 0 → (𝐴 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))) = 0))
42		orthcom 31722	. . . . . . . 8 ⊢ (((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐵) = 0 ↔ (𝐵 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))) = 0))
43	33, 1, 42	mp2an 705	. . . . . . 7 ⊢ (((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐵) = 0 ↔ (𝐵 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))) = 0)
44		orthcom 31722	. . . . . . . 8 ⊢ (((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ∈ ℋ ∧ 𝐴 ∈ ℋ) → (((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐴) = 0 ↔ (𝐴 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))) = 0))
45	33, 3, 44	mp2an 705	. . . . . . 7 ⊢ (((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐴) = 0 ↔ (𝐴 ·_ih (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))) = 0)
46	41, 43, 45	3imtr4g 299	. . . . . 6 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → (((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐵) = 0 → ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐴) = 0))
47	30, 46	mpi 21	. . . . 5 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐴) = 0)
48	17, 47	eqtrid 2810	. . . 4 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → (((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴))) = 0)
49	11, 2	mulcli 11326	. . . . 5 ⊢ ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) ∈ ℂ
50	1, 3	hicli 31695	. . . . . 6 ⊢ (𝐵 ·_ih 𝐴) ∈ ℂ
51	5, 50	mulcli 11326	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)) ∈ ℂ
52	49, 51	subeq0i 11648	. . . 4 ⊢ ((((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴))) = 0 ↔ ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)))
53	48, 52	sylib 221	. . 3 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)))
54	53	eqcomd 2769	. 2 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)) = ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)))
55	3, 1	bcseqi 31734	. 2 ⊢ (((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)) = ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) ↔ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))
56	54, 55	sylib 221	1 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 = wceq 1569 ∈ wcel 2146 ∀wral 3079 {csn 4587 ‘cfv 6546 (class class class)co 7427 ℂcc 11208 0cc0 11210 · cmul 11215 − cmin 11551 ℋchba 31533 ·_ℎ csm 31535 ·_ih csp 31536 −_ℎ cmv 31539 ⊥cort 31544

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-inf2 9652 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288 ax-addf 11289 ax-mulf 11290 ax-hilex 31613 ax-hfvadd 31614 ax-hvcom 31615 ax-hvass 31616 ax-hv0cl 31617 ax-hvaddid 31618 ax-hfvmul 31619 ax-hvmulid 31620 ax-hvmulass 31621 ax-hvdistr1 31622 ax-hvdistr2 31623 ax-hvmul0 31624 ax-hfi 31693 ax-his1 31696 ax-his2 31697 ax-his3 31698 ax-his4 31699 ax-hcompl 31816

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-iin 4956 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-se 5608 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-of 7700 df-om 7885 df-1st 8008 df-2nd 8009 df-supp 8188 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-2o 8488 df-er 8728 df-map 8860 df-pm 8861 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9365 df-fi 9413 df-sup 9444 df-inf 9445 df-oi 9514 df-card 10030 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-3 12416 df-4 12417 df-5 12418 df-6 12419 df-7 12420 df-8 12421 df-9 12422 df-n0 12617 df-z 12704 df-dec 12825 df-uz 12976 df-q 13086 df-rp 13131 df-xneg 13251 df-xadd 13252 df-xmul 13253 df-ioo 13490 df-icc 13493 df-fz 13650 df-fzo 13799 df-seq 14155 df-exp 14215 df-hash 14485 df-cj 15278 df-re 15279 df-im 15280 df-sqrt 15414 df-abs 15415 df-clim 15667 df-sum 15866 df-struct 17337 df-sets 17354 df-slot 17372 df-ndx 17384 df-base 17400 df-ress 17421 df-plusg 17453 df-mulr 17454 df-starv 17455 df-sca 17456 df-vsca 17457 df-ip 17458 df-tset 17459 df-ple 17460 df-ds 17462 df-unif 17463 df-hom 17464 df-cco 17465 df-rest 17605 df-topn 17606 df-0g 17624 df-gsum 17625 df-topgen 17626 df-pt 17627 df-prds 17630 df-xrs 17686 df-qtop 17691 df-imas 17692 df-xps 17694 df-mre 17768 df-mrc 17769 df-acs 17771 df-mgm 18828 df-sgrp 18920 df-mnd 18936 df-submnd 18991 df-mulg 19290 df-cntz 19543 df-cmn 20008 df-psmet 21682 df-xmet 21683 df-met 21684 df-bl 21685 df-mopn 21686 df-cnfld 21691 df-top 23224 df-topon 23241 df-topsp 23263 df-bases 23276 df-cn 23557 df-cnp 23558 df-lm 23559 df-haus 23645 df-tx 23893 df-hmeo 24086 df-xms 24651 df-ms 24652 df-tms 24653 df-cau 25589 df-grpo 31107 df-gid 31108 df-ginv 31109 df-gdiv 31110 df-ablo 31159 df-vc 31173 df-nv 31206 df-va 31209 df-ba 31210 df-sm 31211 df-0v 31212 df-vs 31213 df-nmcv 31214 df-ims 31215 df-dip 31315 df-hnorm 31582 df-hvsub 31585 df-hlim 31586 df-hcau 31587 df-sh 31821 df-ch 31835 df-oc 31866

This theorem is used by: h1de2bi 32168

W3C validator