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Theorem eigorth 29038
 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 6331 . . . . . . 7 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝑇𝐴) = (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)))
2 oveq2 6799 . . . . . . 7 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐶 · 𝐴) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)))
31, 2eqeq12d 2784 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((𝑇𝐴) = (𝐶 · 𝐴) ↔ (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) = (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0))))
43anbi1d 740 . . . . 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 740 . . . 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 6798 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐴 ·ih (𝑇𝐵)) = (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih (𝑇𝐵)))
71oveq1d 6806 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((𝑇𝐴) ·ih 𝐵) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0)) ·ih 𝐵))
86, 7eqeq12d 2784 . . . . 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 6798 . . . . . 6 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → (𝐴 ·ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵))
109eqeq1d 2771 . . . . 5 (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0) → ((𝐴 ·ih 𝐵) = 0 ↔ (if(𝐴 ∈ ℋ, 𝐴, 0) ·ih 𝐵) = 0))
118, 10bibi12d 334 . . . 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 333 . . 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 6331 . . . . . . 7 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (𝑇𝐵) = (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)))
14 oveq2 6799 . . . . . . 7 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → (𝐷 · 𝐵) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0)))
1513, 14eqeq12d 2784 . . . . . 6 (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0) → ((𝑇𝐵) = (𝐷 · 𝐵) ↔ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0)) = (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0))))
1615anbi2d 739 . . . . 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 740 . . . 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 6807 . . . . . 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 6799 . . . . . 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 2784 . . . . 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 6799 . . . . . 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 2771 . . . . 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 334 . . . 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 333 . . 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 6798 . . . . . . 7 (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 · if(𝐴 ∈ ℋ, 𝐴, 0)) = (if(𝐶 ∈ ℂ, 𝐶, 0) · if(𝐴 ∈ ℋ, 𝐴, 0)))
2625eqeq2d 2779 . . . . . 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 740 . . . . 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 3003 . . . . 5 (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 ≠ (∗‘𝐷) ↔ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)))
2927, 28anbi12d 746 . . . 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 330 . . 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 6798 . . . . . . 7 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (𝐷 · if(𝐵 ∈ ℋ, 𝐵, 0)) = (if(𝐷 ∈ ℂ, 𝐷, 0) · if(𝐵 ∈ ℋ, 𝐵, 0)))
3231eqeq2d 2779 . . . . . 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 739 . . . . 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 6331 . . . . . 6 (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (∗‘𝐷) = (∗‘if(𝐷 ∈ ℂ, 𝐷, 0)))
3534neeq2d 3001 . . . . 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 746 . . . 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 330 . . 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 28220 . . . 4 if(𝐴 ∈ ℋ, 𝐴, 0) ∈ ℋ
39 ifhvhv0 28220 . . . 4 if(𝐵 ∈ ℋ, 𝐵, 0) ∈ ℋ
40 0cn 10232 . . . . 5 0 ∈ ℂ
4140elimel 4286 . . . 4 if(𝐶 ∈ ℂ, 𝐶, 0) ∈ ℂ
4240elimel 4286 . . . 4 if(𝐷 ∈ ℂ, 𝐷, 0) ∈ ℂ
4338, 39, 41, 42eigorthi 29037 . . 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 4278 . 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 443 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 196   ∧ wa 383   = wceq 1629   ∈ wcel 2143   ≠ wne 2941  ifcif 4222  ‘cfv 6030  (class class class)co 6791  ℂcc 10134  0cc0 10136  ∗ccj 14047   ℋchil 28117   ·ℎ csm 28119   ·ih csp 28120  0ℎc0v 28122 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  ax-un 7094  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213  ax-hv0cl 28201  ax-hfvmul 28203  ax-hfi 28277  ax-his1 28280  ax-his3 28282 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1070  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-nel 3045  df-ral 3064  df-rex 3065  df-reu 3066  df-rmo 3067  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-id 5156  df-po 5169  df-so 5170  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-er 7894  df-en 8108  df-dom 8109  df-sdom 8110  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-div 10885  df-2 11279  df-cj 14050  df-re 14051  df-im 14052 This theorem is referenced by:  eighmorth  29164
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