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Theorem eigorth 30200
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 6774 . . . . . . 7 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝑇𝐴) = (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)))
2 oveq2 7283 . . . . . . 7 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐶 · 𝐴) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)))
31, 2eqeq12d 2754 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((𝑇𝐴) = (𝐶 · 𝐴) ↔ (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0))))
43anbi1d 630 . . . . 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 630 . . . 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 7282 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐴 ·ih (𝑇𝐵)) = (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇𝐵)))
71oveq1d 7290 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((𝑇𝐴) ·ih 𝐵) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih 𝐵))
86, 7eqeq12d 2754 . . . . 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 7282 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐴 ·ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵))
109eqeq1d 2740 . . . . 5 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((𝐴 ·ih 𝐵) = 0 ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵) = 0))
118, 10bibi12d 346 . . . 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 345 . . 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 6774 . . . . . . 7 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (𝑇𝐵) = (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)))
14 oveq2 7283 . . . . . . 7 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (𝐷 · 𝐵) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0)))
1513, 14eqeq12d 2754 . . . . . 6 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → ((𝑇𝐵) = (𝐷 · 𝐵) ↔ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0))))
1615anbi2d 629 . . . . 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 630 . . . 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 7291 . . . . . 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 7283 . . . . . 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 2754 . . . . 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 7283 . . . . . 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 2740 . . . . 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 346 . . . 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 345 . . 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 7282 . . . . . . 7 (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)))
2625eqeq2d 2749 . . . . . 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 630 . . . . 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 3006 . . . . 5 (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 ≠ (∗‘𝐷) ↔ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)))
2927, 28anbi12d 631 . . . 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 342 . . 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 7282 . . . . . . 7 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0)) = (if(𝐷 ∈ ℂ, 𝐷, 0) · if(𝐵 ∈ ℋ, 𝐵, 0)))
3231eqeq2d 2749 . . . . . 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 629 . . . . 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 6774 . . . . . 6 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (∗‘𝐷) = (∗‘if(𝐷 ∈ ℂ, 𝐷, 0)))
3534neeq2d 3004 . . . . 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 631 . . . 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 342 . . 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 29384 . . . 4 if(𝐴 ∈ ℋ, 𝐴, 0) ∈ ℋ
39 ifhvhv0 29384 . . . 4 if(𝐵 ∈ ℋ, 𝐵, 0) ∈ ℋ
40 0cn 10967 . . . . 5 0 ∈ ℂ
4140elimel 4528 . . . 4 if(𝐶 ∈ ℂ, 𝐶, 0) ∈ ℂ
4240elimel 4528 . . . 4 if(𝐷 ∈ ℂ, 𝐷, 0) ∈ ℂ
4338, 39, 41, 42eigorthi 30199 . . 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 4520 . 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 407 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 396   = wceq 1539  wcel 2106  wne 2943  ifcif 4459  cfv 6433  (class class class)co 7275  cc 10869  0cc0 10871  ccj 14807  chba 29281   · csm 29283   ·ih csp 29284  0c0v 29286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-hv0cl 29365  ax-hfvmul 29367  ax-hfi 29441  ax-his1 29444  ax-his3 29446
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-2 12036  df-cj 14810  df-re 14811  df-im 14812
This theorem is referenced by:  eighmorth  30326
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