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Theorem eigorth 29607
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 6663 . . . . . . 7 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝑇𝐴) = (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)))
2 oveq2 7156 . . . . . . 7 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐶 · 𝐴) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)))
31, 2eqeq12d 2835 . . . . . 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 7155 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐴 ·ih (𝑇𝐵)) = (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇𝐵)))
71oveq1d 7163 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((𝑇𝐴) ·ih 𝐵) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih 𝐵))
86, 7eqeq12d 2835 . . . . 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 7155 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐴 ·ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵))
109eqeq1d 2821 . . . . 5 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((𝐴 ·ih 𝐵) = 0 ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵) = 0))
118, 10bibi12d 348 . . . 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 347 . . 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 6663 . . . . . . 7 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (𝑇𝐵) = (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)))
14 oveq2 7156 . . . . . . 7 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (𝐷 · 𝐵) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0)))
1513, 14eqeq12d 2835 . . . . . 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 7164 . . . . . 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 7156 . . . . . 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 2835 . . . . 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 7156 . . . . . 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 2821 . . . . 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 348 . . . 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 347 . . 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 7155 . . . . . . 7 (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)))
2625eqeq2d 2830 . . . . . 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 3076 . . . . 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 344 . . 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 7155 . . . . . . 7 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0)) = (if(𝐷 ∈ ℂ, 𝐷, 0) · if(𝐵 ∈ ℋ, 𝐵, 0)))
3231eqeq2d 2830 . . . . . 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 6663 . . . . . 6 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (∗‘𝐷) = (∗‘if(𝐷 ∈ ℂ, 𝐷, 0)))
3534neeq2d 3074 . . . . 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 344 . . 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 28791 . . . 4 if(𝐴 ∈ ℋ, 𝐴, 0) ∈ ℋ
39 ifhvhv0 28791 . . . 4 if(𝐵 ∈ ℋ, 𝐵, 0) ∈ ℋ
40 0cn 10625 . . . . 5 0 ∈ ℂ
4140elimel 4532 . . . 4 if(𝐶 ∈ ℂ, 𝐶, 0) ∈ ℂ
4240elimel 4532 . . . 4 if(𝐷 ∈ ℂ, 𝐷, 0) ∈ ℂ
4338, 39, 41, 42eigorthi 29606 . . 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 4524 . 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 409 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 208  wa 398   = wceq 1531  wcel 2108  wne 3014  ifcif 4465  cfv 6348  (class class class)co 7148  cc 10527  0cc0 10529  ccj 14447  chba 28688   · csm 28690   ·ih csp 28691  0c0v 28693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-hv0cl 28772  ax-hfvmul 28774  ax-hfi 28848  ax-his1 28851  ax-his3 28853
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-2 11692  df-cj 14450  df-re 14451  df-im 14452
This theorem is referenced by:  eighmorth  29733
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