h1de2i - Hilbert Space Explorer

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Theorem h1de2i 31455

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 30983	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
3		h1de2.1	. . . . . . . 8 ⊢ 𝐴 ∈ ℋ
4	2, 3	hvmulcli 30916	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ∈ ℋ
5	3, 1	hicli 30983	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
6	5, 1	hvmulcli 30916	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ∈ ℋ
7		his2sub 30994	. . . . . . 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 30986	. . . . . . . . 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 30983	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐴) ∈ ℂ
12	2, 11	mulcomi 11158	. . . . . . . 8 ⊢ ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐴)) = ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵))
13	10, 12	eqtri 2752	. . . . . . 7 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐴) = ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵))
14		ax-his3 30986	. . . . . . . 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 7381	. . . . . 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 2753	. . . . 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 30994	. . . . . . . 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 11158	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐵))
21		ax-his3 30986	. . . . . . . . . 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 30986	. . . . . . . . . 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 2762	. . . . . . . 8 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) = (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵)
26	4, 1	hicli 30983	. . . . . . . . 9 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) ∈ ℂ
27	6, 1	hicli 30983	. . . . . . . . 9 ⊢ (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵) ∈ ℂ
28	26, 27	subeq0i 11478	. . . . . . . 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 2752	. . . . . 6 ⊢ ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐵) = 0
31	1	h1dei 31452	. . . . . . . . 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 30923	. . . . . . . . 9 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ∈ ℋ
34		oveq2 7377	. . . . . . . . . . . 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 2731	. . . . . . . . . . 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 7377	. . . . . . . . . . . 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 2731	. . . . . . . . . . 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 3581	. . . . . . . . 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 31010	. . . . . . . 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 31010	. . . . . . . 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 2776	. . . 4 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → (((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴))) = 0)
49	11, 2	mulcli 11157	. . . . 5 ⊢ ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) ∈ ℂ
50	1, 3	hicli 30983	. . . . . 6 ⊢ (𝐵 ·_ih 𝐴) ∈ ℂ
51	5, 50	mulcli 11157	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)) ∈ ℂ
52	49, 51	subeq0i 11478	. . . 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 2735	. 2 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)) = ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)))
55	3, 1	bcseqi 31022	. 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 3044 {csn 4585 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 0cc0 11044 · cmul 11049 − cmin 11381 ℋchba 30821 ·_ℎ csm 30823 ·_ih csp 30824 −_ℎ cmv 30827 ⊥cort 30832

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 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124 ax-hilex 30901 ax-hfvadd 30902 ax-hvcom 30903 ax-hvass 30904 ax-hv0cl 30905 ax-hvaddid 30906 ax-hfvmul 30907 ax-hvmulid 30908 ax-hvmulass 30909 ax-hvdistr1 30910 ax-hvdistr2 30911 ax-hvmul0 30912 ax-hfi 30981 ax-his1 30984 ax-his2 30985 ax-his3 30986 ax-his4 30987 ax-hcompl 31104

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-sum 15629 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cn 23090 df-cnp 23091 df-lm 23092 df-haus 23178 df-tx 23425 df-hmeo 23618 df-xms 24184 df-ms 24185 df-tms 24186 df-cau 25132 df-grpo 30395 df-gid 30396 df-ginv 30397 df-gdiv 30398 df-ablo 30447 df-vc 30461 df-nv 30494 df-va 30497 df-ba 30498 df-sm 30499 df-0v 30500 df-vs 30501 df-nmcv 30502 df-ims 30503 df-dip 30603 df-hnorm 30870 df-hvsub 30873 df-hlim 30874 df-hcau 30875 df-sh 31109 df-ch 31123 df-oc 31154

This theorem is referenced by: h1de2bi 31456

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