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Theorem eigorth 29773

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 6674	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝑇‘𝐴) = (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)))
2		oveq2 7178	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐶 ·_ℎ 𝐴) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)))
3	1, 2	eqeq12d 2754	. . . . . 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 633	. . . . 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 633	. . . 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 7177	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ·_ih (𝑇‘𝐵)) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)))
7	1	oveq1d 7185	. . . . . 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 2754	. . . . 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 7177	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ·_ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵))
10	9	eqeq1d 2740	. . . . 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 349	. . . 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 348	. . 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 6674	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝑇‘𝐵) = (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
14		oveq2 7178	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝐷 ·_ℎ 𝐵) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
15	13, 14	eqeq12d 2754	. . . . . 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 632	. . . . 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 633	. . . 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 7186	. . . . . 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 7178	. . . . . 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 2754	. . . . 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 7178	. . . . . 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 2740	. . . . 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 349	. . . 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 348	. . 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 7177	. . . . . . 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 2749	. . . . . 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 633	. . . . 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 2996	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 ≠ (∗‘𝐷) ↔ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)))
29	27, 28	anbi12d 634	. . . 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 345	. . 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 7177	. . . . . . 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 2749	. . . . . 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 632	. . . . 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 6674	. . . . . 6 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (∗‘𝐷) = (∗‘if(𝐷 ∈ ℂ, 𝐷, 0)))
35	34	neeq2d 2994	. . . . 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 634	. . . 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 345	. . 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 28957	. . . 4 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ ℋ
39		ifhvhv0 28957	. . . 4 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ∈ ℋ
40		0cn 10711	. . . . 5 ⊢ 0 ∈ ℂ
41	40	elimel 4483	. . . 4 ⊢ if(𝐶 ∈ ℂ, 𝐶, 0) ∈ ℂ
42	40	elimel 4483	. . . 4 ⊢ if(𝐷 ∈ ℂ, 𝐷, 0) ∈ ℂ
43	38, 39, 41, 42	eigorthi 29772	. . 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 4475	. 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 410	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 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ifcif 4414 ‘cfv 6339 (class class class)co 7170 ℂcc 10613 0cc0 10615 ∗ccj 14545 ℋchba 28854 ·_ℎ csm 28856 ·_ih csp 28857 0_ℎc0v 28859

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-hv0cl 28938 ax-hfvmul 28940 ax-hfi 29014 ax-his1 29017 ax-his3 29019

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-2 11779 df-cj 14548 df-re 14549 df-im 14550

This theorem is referenced by: eighmorth 29899

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