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Theorem eigorth 31560
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)) ∧ (((𝑇𝐴) = (𝐶 · 𝐴) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))) → ((𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0))

Proof of Theorem eigorth
StepHypRef Expression
1 fveq2 6881 . . . . . . 7 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝑇𝐴) = (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)))
2 oveq2 7409 . . . . . . 7 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐶 · 𝐴) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)))
31, 2eqeq12d 2740 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((𝑇𝐴) = (𝐶 · 𝐴) ↔ (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0))))
43anbi1d 629 . . . . 5 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (((𝑇𝐴) = (𝐶 · 𝐴) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ↔ ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇𝐵) = (𝐷 · 𝐵))))
54anbi1d 629 . . . 4 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((((𝑇𝐴) = (𝐶 · 𝐴) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) ↔ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))))
6 oveq1 7408 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐴 ·ih (𝑇𝐵)) = (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇𝐵)))
71oveq1d 7416 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((𝑇𝐴) ·ih 𝐵) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih 𝐵))
86, 7eqeq12d 2740 . . . . 5 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih 𝐵)))
9 oveq1 7408 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐴 ·ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵))
109eqeq1d 2726 . . . . 5 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((𝐴 ·ih 𝐵) = 0 ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵) = 0))
118, 10bibi12d 345 . . . 4 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (((𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0) ↔ ((if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵) = 0)))
125, 11imbi12d 344 . . 3 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (((((𝑇𝐴) = (𝐶 · 𝐴) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0)) ↔ ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵) = 0))))
13 fveq2 6881 . . . . . . 7 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (𝑇𝐵) = (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)))
14 oveq2 7409 . . . . . . 7 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (𝐷 · 𝐵) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0)))
1513, 14eqeq12d 2740 . . . . . 6 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → ((𝑇𝐵) = (𝐷 · 𝐵) ↔ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0))))
1615anbi2d 628 . . . . 5 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ↔ ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0)))))
1716anbi1d 629 . . . 4 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) ↔ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0))) ∧ 𝐶 ≠ (∗‘𝐷))))
1813oveq2d 7417 . . . . . 6 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇𝐵)) = (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0))))
19 oveq2 7409 . . . . . 6 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih 𝐵) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)))
2018, 19eqeq12d 2740 . . . . 5 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → ((if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih if(𝐵 ∈ ℋ, 𝐵, 0))))
21 oveq2 7409 . . . . . 6 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)))
2221eqeq1d 2726 . . . . 5 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → ((if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵) = 0 ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) = 0))
2320, 22bibi12d 345 . . . 4 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (((if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵) = 0) ↔ ((if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) = 0)))
2417, 23imbi12d 344 . . 3 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵) = 0)) ↔ ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0))) ∧ 𝐶 ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) = 0))))
25 oveq1 7408 . . . . . . 7 (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)))
2625eqeq2d 2735 . . . . . 6 (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) ↔ (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0))))
2726anbi1d 629 . . . . 5 (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0))) ↔ ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0)))))
28 neeq1 2995 . . . . 5 (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 ≠ (∗‘𝐷) ↔ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)))
2927, 28anbi12d 630 . . . 4 (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0))) ∧ 𝐶 ≠ (∗‘𝐷)) ↔ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷))))
3029imbi1d 341 . . 3 (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0))) ∧ 𝐶 ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) = 0)) ↔ ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) = 0))))
31 oveq1 7408 . . . . . . 7 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0)) = (if(𝐷 ∈ ℂ, 𝐷, 0) · if(𝐵 ∈ ℋ, 𝐵, 0)))
3231eqeq2d 2735 . . . . . 6 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → ((𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0)) ↔ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (if(𝐷 ∈ ℂ, 𝐷, 0) · if(𝐵 ∈ ℋ, 𝐵, 0))))
3332anbi2d 628 . . . . 5 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0))) ↔ ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (if(𝐷 ∈ ℂ, 𝐷, 0) · if(𝐵 ∈ ℋ, 𝐵, 0)))))
34 fveq2 6881 . . . . . 6 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (∗‘𝐷) = (∗‘if(𝐷 ∈ ℂ, 𝐷, 0)))
3534neeq2d 2993 . . . . 5 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷) ↔ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘if(𝐷 ∈ ℂ, 𝐷, 0))))
3633, 35anbi12d 630 . . . 4 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)) ↔ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (if(𝐷 ∈ ℂ, 𝐷, 0) · if(𝐵 ∈ ℋ, 𝐵, 0))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘if(𝐷 ∈ ℂ, 𝐷, 0)))))
3736imbi1d 341 . . 3 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) = 0)) ↔ ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (if(𝐷 ∈ ℂ, 𝐷, 0) · if(𝐵 ∈ ℋ, 𝐵, 0))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘if(𝐷 ∈ ℂ, 𝐷, 0))) → ((if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) = 0))))
38 ifhvhv0 30744 . . . 4 if(𝐴 ∈ ℋ, 𝐴, 0) ∈ ℋ
39 ifhvhv0 30744 . . . 4 if(𝐵 ∈ ℋ, 𝐵, 0) ∈ ℋ
40 0cn 11203 . . . . 5 0 ∈ ℂ
4140elimel 4589 . . . 4 if(𝐶 ∈ ℂ, 𝐶, 0) ∈ ℂ
4240elimel 4589 . . . 4 if(𝐷 ∈ ℂ, 𝐷, 0) ∈ ℂ
4338, 39, 41, 42eigorthi 31559 . . 3 ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (if(𝐷 ∈ ℂ, 𝐷, 0) · if(𝐵 ∈ ℋ, 𝐵, 0))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘if(𝐷 ∈ ℂ, 𝐷, 0))) → ((if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih if(𝐵 ∈ ℋ, 𝐵, 0)) = 0))
4412, 24, 30, 37, 43dedth4h 4581 . 2 (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((((𝑇𝐴) = (𝐶 · 𝐴) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0)))
4544imp 406 1 ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) ∧ (((𝑇𝐴) = (𝐶 · 𝐴) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))) → ((𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wne 2932  ifcif 4520  cfv 6533  (class class class)co 7401  cc 11104  0cc0 11106  ccj 15040  chba 30641   · csm 30643   ·ih csp 30644  0c0v 30646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-hv0cl 30725  ax-hfvmul 30727  ax-hfi 30801  ax-his1 30804  ax-his3 30806
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-po 5578  df-so 5579  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-2 12272  df-cj 15043  df-re 15044  df-im 15045
This theorem is referenced by:  eighmorth  31686
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