eigorth - Hilbert Space Explorer

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Theorem eigorth 31824

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 6881	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝑇‘𝐴) = (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)))
2		oveq2 7418	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐶 ·_ℎ 𝐴) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)))
3	1, 2	eqeq12d 2752	. . . . . 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 631	. . . . 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 631	. . . 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 7417	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ·_ih (𝑇‘𝐵)) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)))
7	1	oveq1d 7425	. . . . . 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 2752	. . . . 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 7417	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ·_ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵))
10	9	eqeq1d 2738	. . . . 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 6881	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝑇‘𝐵) = (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
14		oveq2 7418	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝐷 ·_ℎ 𝐵) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
15	13, 14	eqeq12d 2752	. . . . . 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 630	. . . . 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 631	. . . 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 7426	. . . . . 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 7418	. . . . . 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 2752	. . . . 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 7418	. . . . . 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 2738	. . . . 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 7417	. . . . . . 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 2747	. . . . . 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 631	. . . . 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 2995	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 ≠ (∗‘𝐷) ↔ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)))
29	27, 28	anbi12d 632	. . . 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 7417	. . . . . . 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 2747	. . . . . 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 630	. . . . 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 6881	. . . . . 6 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (∗‘𝐷) = (∗‘if(𝐷 ∈ ℂ, 𝐷, 0)))
35	34	neeq2d 2993	. . . . 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 632	. . . 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 31008	. . . 4 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ ℋ
39		ifhvhv0 31008	. . . 4 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ∈ ℋ
40		0cn 11232	. . . . 5 ⊢ 0 ∈ ℂ
41	40	elimel 4575	. . . 4 ⊢ if(𝐶 ∈ ℂ, 𝐶, 0) ∈ ℂ
42	40	elimel 4575	. . . 4 ⊢ if(𝐷 ∈ ℂ, 𝐷, 0) ∈ ℂ
43	38, 39, 41, 42	eigorthi 31823	. . 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 4567	. 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 1540 ∈ wcel 2109 ≠ wne 2933 ifcif 4505 ‘cfv 6536 (class class class)co 7410 ℂcc 11132 0cc0 11134 ∗ccj 15120 ℋchba 30905 ·_ℎ csm 30907 ·_ih csp 30908 0_ℎc0v 30910

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 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-hv0cl 30989 ax-hfvmul 30991 ax-hfi 31065 ax-his1 31068 ax-his3 31070

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-cj 15123 df-re 15124 df-im 15125

This theorem is referenced by: eighmorth 31950

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