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Theorem eigorth 31870

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 6920	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝑇‘𝐴) = (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)))
2		oveq2 7456	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐶 ·_ℎ 𝐴) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)))
3	1, 2	eqeq12d 2756	. . . . . 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 630	. . . . 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 630	. . . 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 7455	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ·_ih (𝑇‘𝐵)) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)))
7	1	oveq1d 7463	. . . . . 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 2756	. . . . 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 7455	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ·_ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵))
10	9	eqeq1d 2742	. . . . 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 345	. . . 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 344	. . 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 6920	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝑇‘𝐵) = (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
14		oveq2 7456	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝐷 ·_ℎ 𝐵) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
15	13, 14	eqeq12d 2756	. . . . . 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 629	. . . . 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 630	. . . 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 7464	. . . . . 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 7456	. . . . . 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 2756	. . . . 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 7456	. . . . . 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 2742	. . . . 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 345	. . . 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 344	. . 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 7455	. . . . . . 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 2751	. . . . . 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 630	. . . . 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 3009	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 ≠ (∗‘𝐷) ↔ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)))
29	27, 28	anbi12d 631	. . . 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 341	. . 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 7455	. . . . . . 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 2751	. . . . . 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 629	. . . . 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 6920	. . . . . 6 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (∗‘𝐷) = (∗‘if(𝐷 ∈ ℂ, 𝐷, 0)))
35	34	neeq2d 3007	. . . . 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 631	. . . 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 341	. . 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 31054	. . . 4 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ ℋ
39		ifhvhv0 31054	. . . 4 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ∈ ℋ
40		0cn 11282	. . . . 5 ⊢ 0 ∈ ℂ
41	40	elimel 4617	. . . 4 ⊢ if(𝐶 ∈ ℂ, 𝐶, 0) ∈ ℂ
42	40	elimel 4617	. . . 4 ⊢ if(𝐷 ∈ ℂ, 𝐷, 0) ∈ ℂ
43	38, 39, 41, 42	eigorthi 31869	. . 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 4609	. 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 406	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

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ifcif 4548 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 0cc0 11184 ∗ccj 15145 ℋchba 30951 ·_ℎ csm 30953 ·_ih csp 30954 0_ℎc0v 30956

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-hv0cl 31035 ax-hfvmul 31037 ax-hfi 31111 ax-his1 31114 ax-his3 31116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-2 12356 df-cj 15148 df-re 15149 df-im 15150

This theorem is referenced by: eighmorth 31996

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