eigorth - Hilbert Space Explorer

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Theorem eigorth 32320

Description: A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for two eigenvectors 𝐴 and 𝐵 to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
eigorth	⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) ∧ (((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))) → ((𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵) ↔ (𝐴 ·_ih 𝐵) = 0))

**Proof of Theorem eigorth**
Step	Hyp	Ref	Expression
1		fveq2 6879	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝑇‘𝐴) = (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)))
2		oveq2 7422	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐶 ·_ℎ 𝐴) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)))
3	1, 2	eqeq12d 2776	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → ((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) ↔ (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ))))
4	3	anbi1d 643	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ↔ ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵))))
5	4	anbi1d 643	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → ((((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) ↔ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))))
6		oveq1 7421	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ·_ih (𝑇‘𝐵)) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)))
7	1	oveq1d 7429	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → ((𝑇‘𝐴) ·_ih 𝐵) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵))
8	6, 7	eqeq12d 2776	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → ((𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵)))
9		oveq1 7421	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ·_ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵))
10	9	eqeq1d 2762	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → ((𝐴 ·_ih 𝐵) = 0 ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = 0))
11	8, 10	bibi12d 348	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (((𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵) ↔ (𝐴 ·_ih 𝐵) = 0) ↔ ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = 0)))
12	5, 11	imbi12d 347	. . 3 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (((((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵) ↔ (𝐴 ·_ih 𝐵) = 0)) ↔ ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = 0))))
13		fveq2 6879	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝑇‘𝐵) = (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
14		oveq2 7422	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝐷 ·_ℎ 𝐵) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
15	13, 14	eqeq12d 2776	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → ((𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵) ↔ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))))
16	15	anbi2d 642	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ↔ ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))))
17	16	anbi1d 643	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) ↔ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ 𝐶 ≠ (∗‘𝐷))))
18	13	oveq2d 7430	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))))
19		oveq2 7422	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
20	18, 19	eqeq12d 2776	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))))
21		oveq2 7422	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
22	21	eqeq1d 2762	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = 0 ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0))
23	20, 22	bibi12d 348	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = 0) ↔ ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0)))
24	17, 23	imbi12d 347	. . 3 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = 0)) ↔ ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ 𝐶 ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0))))
25		oveq1 7421	. . . . . . 7 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)))
26	25	eqeq2d 2771	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ↔ (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ))))
27	26	anbi1d 643	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ↔ ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))))
28		neeq1 3017	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 ≠ (∗‘𝐷) ↔ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)))
29	27, 28	anbi12d 644	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ 𝐶 ≠ (∗‘𝐷)) ↔ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷))))
30	29	imbi1d 344	. . 3 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ 𝐶 ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0)) ↔ ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0))))
31		oveq1 7421	. . . . . . 7 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (if(𝐷 ∈ ℂ, 𝐷, 0) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
32	31	eqeq2d 2771	. . . . . 6 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → ((𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (if(𝐷 ∈ ℂ, 𝐷, 0) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))))
33	32	anbi2d 642	. . . . 5 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ↔ ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (if(𝐷 ∈ ℂ, 𝐷, 0) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))))
34		fveq2 6879	. . . . . 6 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (∗‘𝐷) = (∗‘if(𝐷 ∈ ℂ, 𝐷, 0)))
35	34	neeq2d 3015	. . . . 5 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷) ↔ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘if(𝐷 ∈ ℂ, 𝐷, 0))))
36	33, 35	anbi12d 644	. . . 4 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)) ↔ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (if(𝐷 ∈ ℂ, 𝐷, 0) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘if(𝐷 ∈ ℂ, 𝐷, 0)))))
37	36	imbi1d 344	. . 3 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0)) ↔ ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (if(𝐷 ∈ ℂ, 𝐷, 0) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘if(𝐷 ∈ ℂ, 𝐷, 0))) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0))))
38		ifhvhv0 31504	. . . 4 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ ℋ
39		ifhvhv0 31504	. . . 4 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ∈ ℋ
40		0cn 11223	. . . . 5 ⊢ 0 ∈ ℂ
41	40	elimel 4552	. . . 4 ⊢ if(𝐶 ∈ ℂ, 𝐶, 0) ∈ ℂ
42	40	elimel 4552	. . . 4 ⊢ if(𝐷 ∈ ℂ, 𝐷, 0) ∈ ℂ
43	38, 39, 41, 42	eigorthi 32319	. . 3 ⊢ ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (if(𝐷 ∈ ℂ, 𝐷, 0) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘if(𝐷 ∈ ℂ, 𝐷, 0))) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0))
44	12, 24, 30, 37, 43	dedth4h 4544	. 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵) ↔ (𝐴 ·_ih 𝐵) = 0)))
45	44	imp 412	1 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) ∧ (((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))) → ((𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵) ↔ (𝐴 ·_ih 𝐵) = 0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ifcif 4482 ‘cfv 6533 (class class class)co 7414 ℂcc 11123 0cc0 11125 ∗ccj 15184 ℋchba 31401 ·_ℎ csm 31403 ·_ih csp 31404 0_ℎc0v 31406

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-hv0cl 31485 ax-hfvmul 31487 ax-hfi 31561 ax-his1 31564 ax-his3 31566

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-er 8697 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-cj 15187 df-re 15188 df-im 15189

This theorem is used by: eighmorth 32446

W3C validator