h1de2i - Hilbert Space Explorer

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Theorem h1de2i 31489

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 31017	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
3		h1de2.1	. . . . . . . 8 ⊢ 𝐴 ∈ ℋ
4	2, 3	hvmulcli 30950	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ∈ ℋ
5	3, 1	hicli 31017	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
6	5, 1	hvmulcli 30950	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ∈ ℋ
7		his2sub 31028	. . . . . . 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 1463	. . . . . 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 31020	. . . . . . . . 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 1463	. . . . . . . 8 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐴) = ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐴))
11	3, 3	hicli 31017	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐴) ∈ ℂ
12	2, 11	mulcomi 11189	. . . . . . . 8 ⊢ ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐴)) = ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵))
13	10, 12	eqtri 2753	. . . . . . 7 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐴) = ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵))
14		ax-his3 31020	. . . . . . . 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 1463	. . . . . . 7 ⊢ (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐴) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴))
16	13, 15	oveq12i 7402	. . . . . 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 2754	. . . . 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 31028	. . . . . . . 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 1463	. . . . . . 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 11189	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐵))
21		ax-his3 31020	. . . . . . . . . 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 1463	. . . . . . . . 9 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) = ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐵))
23		ax-his3 31020	. . . . . . . . . 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 1463	. . . . . . . . 9 ⊢ (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐵))
25	20, 22, 24	3eqtr4i 2763	. . . . . . . 8 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) = (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵)
26	4, 1	hicli 31017	. . . . . . . . 9 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) ∈ ℂ
27	6, 1	hicli 31017	. . . . . . . . 9 ⊢ (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵) ∈ ℂ
28	26, 27	subeq0i 11509	. . . . . . . 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 231	. . . . . . 7 ⊢ ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) − (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵)) = 0
30	19, 29	eqtri 2753	. . . . . 6 ⊢ ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐵) = 0
31	1	h1dei 31486	. . . . . . . . 9 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ ℋ ((𝐵 ·_ih 𝑥) = 0 → (𝐴 ·_ih 𝑥) = 0)))
32	3, 31	mpbiran 709	. . . . . . . 8 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ ∀𝑥 ∈ ℋ ((𝐵 ·_ih 𝑥) = 0 → (𝐴 ·_ih 𝑥) = 0))
33	4, 6	hvsubcli 30957	. . . . . . . . 9 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ∈ ℋ
34		oveq2 7398	. . . . . . . . . . . 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 2732	. . . . . . . . . . 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 7398	. . . . . . . . . . . 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 2732	. . . . . . . . . . 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 344	. . . . . . . . . 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 3587	. . . . . . . . 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 217	. . . . . . 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 31044	. . . . . . . 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 692	. . . . . . 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 31044	. . . . . . . 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 692	. . . . . . 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 296	. . . . . 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 20	. . . . 5 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐴) = 0)
48	17, 47	eqtrid 2777	. . . 4 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → (((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴))) = 0)
49	11, 2	mulcli 11188	. . . . 5 ⊢ ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) ∈ ℂ
50	1, 3	hicli 31017	. . . . . 6 ⊢ (𝐵 ·_ih 𝐴) ∈ ℂ
51	5, 50	mulcli 11188	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)) ∈ ℂ
52	49, 51	subeq0i 11509	. . . 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 218	. . 3 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)))
54	53	eqcomd 2736	. 2 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)) = ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)))
55	3, 1	bcseqi 31056	. 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 218	1 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2109 ∀wral 3045 {csn 4592 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 0cc0 11075 · cmul 11080 − cmin 11412 ℋchba 30855 ·_ℎ csm 30857 ·_ih csp 30858 −_ℎ cmv 30861 ⊥cort 30866

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155 ax-hilex 30935 ax-hfvadd 30936 ax-hvcom 30937 ax-hvass 30938 ax-hv0cl 30939 ax-hvaddid 30940 ax-hfvmul 30941 ax-hvmulid 30942 ax-hvmulass 30943 ax-hvdistr1 30944 ax-hvdistr2 30945 ax-hvmul0 30946 ax-hfi 31015 ax-his1 31018 ax-his2 31019 ax-his3 31020 ax-his4 31021 ax-hcompl 31138

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-icc 13320 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-sum 15660 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cn 23121 df-cnp 23122 df-lm 23123 df-haus 23209 df-tx 23456 df-hmeo 23649 df-xms 24215 df-ms 24216 df-tms 24217 df-cau 25163 df-grpo 30429 df-gid 30430 df-ginv 30431 df-gdiv 30432 df-ablo 30481 df-vc 30495 df-nv 30528 df-va 30531 df-ba 30532 df-sm 30533 df-0v 30534 df-vs 30535 df-nmcv 30536 df-ims 30537 df-dip 30637 df-hnorm 30904 df-hvsub 30907 df-hlim 30908 df-hcau 30909 df-sh 31143 df-ch 31157 df-oc 31188

This theorem is referenced by: h1de2bi 31490

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