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Theorem eigorth 31867
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 6907 . . . . . . 7 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝑇𝐴) = (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)))
2 oveq2 7439 . . . . . . 7 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐶 · 𝐴) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)))
31, 2eqeq12d 2751 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((𝑇𝐴) = (𝐶 · 𝐴) ↔ (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0))))
43anbi1d 631 . . . . 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 631 . . . 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 7438 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐴 ·ih (𝑇𝐵)) = (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇𝐵)))
71oveq1d 7446 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((𝑇𝐴) ·ih 𝐵) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih 𝐵))
86, 7eqeq12d 2751 . . . . 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 7438 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐴 ·ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵))
109eqeq1d 2737 . . . . 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 6907 . . . . . . 7 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (𝑇𝐵) = (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)))
14 oveq2 7439 . . . . . . 7 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (𝐷 · 𝐵) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0)))
1513, 14eqeq12d 2751 . . . . . 6 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → ((𝑇𝐵) = (𝐷 · 𝐵) ↔ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0))))
1615anbi2d 630 . . . . 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 631 . . . 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 7447 . . . . . 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 7439 . . . . . 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 2751 . . . . 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 7439 . . . . . 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 2737 . . . . 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 7438 . . . . . . 7 (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)))
2625eqeq2d 2746 . . . . . 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 631 . . . . 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 3001 . . . . 5 (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 ≠ (∗‘𝐷) ↔ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)))
2927, 28anbi12d 632 . . . 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 7438 . . . . . . 7 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0)) = (if(𝐷 ∈ ℂ, 𝐷, 0) · if(𝐵 ∈ ℋ, 𝐵, 0)))
3231eqeq2d 2746 . . . . . 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 630 . . . . 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 6907 . . . . . 6 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (∗‘𝐷) = (∗‘if(𝐷 ∈ ℂ, 𝐷, 0)))
3534neeq2d 2999 . . . . 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 632 . . . 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 31051 . . . 4 if(𝐴 ∈ ℋ, 𝐴, 0) ∈ ℋ
39 ifhvhv0 31051 . . . 4 if(𝐵 ∈ ℋ, 𝐵, 0) ∈ ℋ
40 0cn 11251 . . . . 5 0 ∈ ℂ
4140elimel 4600 . . . 4 if(𝐶 ∈ ℂ, 𝐶, 0) ∈ ℂ
4240elimel 4600 . . . 4 if(𝐷 ∈ ℂ, 𝐷, 0) ∈ ℂ
4338, 39, 41, 42eigorthi 31866 . . 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 4592 . 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 206  wa 395   = wceq 1537  wcel 2106  wne 2938  ifcif 4531  cfv 6563  (class class class)co 7431  cc 11151  0cc0 11153  ccj 15132  chba 30948   · csm 30950   ·ih csp 30951  0c0v 30953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-hv0cl 31032  ax-hfvmul 31034  ax-hfi 31108  ax-his1 31111  ax-his3 31113
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-2 12327  df-cj 15135  df-re 15136  df-im 15137
This theorem is referenced by:  eighmorth  31993
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