h1de2i - Hilbert Space Explorer

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Theorem h1de2i 30806

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 30334	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
3		h1de2.1	. . . . . . . 8 ⊢ 𝐴 ∈ ℋ
4	2, 3	hvmulcli 30267	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ∈ ℋ
5	3, 1	hicli 30334	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
6	5, 1	hvmulcli 30267	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ∈ ℋ
7		his2sub 30345	. . . . . . 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 1462	. . . . . 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 30337	. . . . . . . . 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 1462	. . . . . . . 8 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐴) = ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐴))
11	3, 3	hicli 30334	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐴) ∈ ℂ
12	2, 11	mulcomi 11222	. . . . . . . 8 ⊢ ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐴)) = ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵))
13	10, 12	eqtri 2761	. . . . . . 7 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐴) = ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵))
14		ax-his3 30337	. . . . . . . 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 1462	. . . . . . 7 ⊢ (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐴) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴))
16	13, 15	oveq12i 7421	. . . . . 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 2762	. . . . 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 30345	. . . . . . . 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 1462	. . . . . . 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 11222	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐵))
21		ax-his3 30337	. . . . . . . . . 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 1462	. . . . . . . . 9 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) = ((𝐵 ·_ih 𝐵) · (𝐴 ·_ih 𝐵))
23		ax-his3 30337	. . . . . . . . . 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 1462	. . . . . . . . 9 ⊢ (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐵))
25	20, 22, 24	3eqtr4i 2771	. . . . . . . 8 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) = (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵)
26	4, 1	hicli 30334	. . . . . . . . 9 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) ∈ ℂ
27	6, 1	hicli 30334	. . . . . . . . 9 ⊢ (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵) ∈ ℂ
28	26, 27	subeq0i 11540	. . . . . . . 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 230	. . . . . . 7 ⊢ ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) ·_ih 𝐵) − (((𝐴 ·_ih 𝐵) ·_ℎ 𝐵) ·_ih 𝐵)) = 0
30	19, 29	eqtri 2761	. . . . . 6 ⊢ ((((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ·_ih 𝐵) = 0
31	1	h1dei 30803	. . . . . . . . 9 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ ℋ ((𝐵 ·_ih 𝑥) = 0 → (𝐴 ·_ih 𝑥) = 0)))
32	3, 31	mpbiran 708	. . . . . . . 8 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ ∀𝑥 ∈ ℋ ((𝐵 ·_ih 𝑥) = 0 → (𝐴 ·_ih 𝑥) = 0))
33	4, 6	hvsubcli 30274	. . . . . . . . 9 ⊢ (((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) −_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) ∈ ℋ
34		oveq2 7417	. . . . . . . . . . . 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 2735	. . . . . . . . . . 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 7417	. . . . . . . . . . . 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 2735	. . . . . . . . . . 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 345	. . . . . . . . . 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 3609	. . . . . . . . 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 216	. . . . . . 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 30361	. . . . . . . 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 691	. . . . . . 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 30361	. . . . . . . 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 691	. . . . . . 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 2785	. . . 4 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → (((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴))) = 0)
49	11, 2	mulcli 11221	. . . . 5 ⊢ ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) ∈ ℂ
50	1, 3	hicli 30334	. . . . . 6 ⊢ (𝐵 ·_ih 𝐴) ∈ ℂ
51	5, 50	mulcli 11221	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)) ∈ ℂ
52	49, 51	subeq0i 11540	. . . 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 217	. . 3 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)))
54	53	eqcomd 2739	. 2 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐴 ·_ih 𝐵) · (𝐵 ·_ih 𝐴)) = ((𝐴 ·_ih 𝐴) · (𝐵 ·_ih 𝐵)))
55	3, 1	bcseqi 30373	. 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 217	1 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {csn 4629 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 0cc0 11110 · cmul 11115 − cmin 11444 ℋchba 30172 ·_ℎ csm 30174 ·_ih csp 30175 −_ℎ cmv 30178 ⊥cort 30183

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 ax-hilex 30252 ax-hfvadd 30253 ax-hvcom 30254 ax-hvass 30255 ax-hv0cl 30256 ax-hvaddid 30257 ax-hfvmul 30258 ax-hvmulid 30259 ax-hvmulass 30260 ax-hvdistr1 30261 ax-hvdistr2 30262 ax-hvmul0 30263 ax-hfi 30332 ax-his1 30335 ax-his2 30336 ax-his3 30337 ax-his4 30338 ax-hcompl 30455

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-icc 13331 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cn 22731 df-cnp 22732 df-lm 22733 df-haus 22819 df-tx 23066 df-hmeo 23259 df-xms 23826 df-ms 23827 df-tms 23828 df-cau 24773 df-grpo 29746 df-gid 29747 df-ginv 29748 df-gdiv 29749 df-ablo 29798 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-vs 29852 df-nmcv 29853 df-ims 29854 df-dip 29954 df-hnorm 30221 df-hvsub 30224 df-hlim 30225 df-hcau 30226 df-sh 30460 df-ch 30474 df-oc 30505

This theorem is referenced by: h1de2bi 30807

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