eigorth - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > eigorth		Structured version Visualization version GIF version

Theorem eigorth 31857

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)) ∧ (((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))) → ((𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵) ↔ (𝐴 ·_ih 𝐵) = 0))

**Proof of Theorem eigorth**
Step	Hyp	Ref	Expression
1		fveq2 6906	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝑇‘𝐴) = (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)))
2		oveq2 7439	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐶 ·_ℎ 𝐴) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)))
3	1, 2	eqeq12d 2753	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → ((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) ↔ (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ))))
4	3	anbi1d 631	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ↔ ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵))))
5	4	anbi1d 631	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → ((((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) ↔ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))))
6		oveq1 7438	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ·_ih (𝑇‘𝐵)) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)))
7	1	oveq1d 7446	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → ((𝑇‘𝐴) ·_ih 𝐵) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵))
8	6, 7	eqeq12d 2753	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → ((𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵)))
9		oveq1 7438	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ·_ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵))
10	9	eqeq1d 2739	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → ((𝐴 ·_ih 𝐵) = 0 ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = 0))
11	8, 10	bibi12d 345	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (((𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵) ↔ (𝐴 ·_ih 𝐵) = 0) ↔ ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = 0)))
12	5, 11	imbi12d 344	. . 3 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (((((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵) ↔ (𝐴 ·_ih 𝐵) = 0)) ↔ ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = 0))))
13		fveq2 6906	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝑇‘𝐵) = (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
14		oveq2 7439	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝐷 ·_ℎ 𝐵) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
15	13, 14	eqeq12d 2753	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → ((𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵) ↔ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))))
16	15	anbi2d 630	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ↔ ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))))
17	16	anbi1d 631	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) ↔ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ 𝐶 ≠ (∗‘𝐷))))
18	13	oveq2d 7447	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))))
19		oveq2 7439	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
20	18, 19	eqeq12d 2753	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))))
21		oveq2 7439	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
22	21	eqeq1d 2739	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = 0 ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0))
23	20, 22	bibi12d 345	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = 0) ↔ ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0)))
24	17, 23	imbi12d 344	. . 3 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘𝐵)) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih 𝐵) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = 0)) ↔ ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ 𝐶 ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0))))
25		oveq1 7438	. . . . . . 7 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)))
26	25	eqeq2d 2748	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ↔ (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ))))
27	26	anbi1d 631	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ↔ ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))))
28		neeq1 3003	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (𝐶 ≠ (∗‘𝐷) ↔ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)))
29	27, 28	anbi12d 632	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ 𝐶 ≠ (∗‘𝐷)) ↔ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷))))
30	29	imbi1d 341	. . 3 ⊢ (𝐶 = if(𝐶 ∈ ℂ, 𝐶, 0) → (((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (𝐶 ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ 𝐶 ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0)) ↔ ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0))))
31		oveq1 7438	. . . . . . 7 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (if(𝐷 ∈ ℂ, 𝐷, 0) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
32	31	eqeq2d 2748	. . . . . 6 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → ((𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (if(𝐷 ∈ ℂ, 𝐷, 0) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))))
33	32	anbi2d 630	. . . . 5 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ↔ ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (if(𝐷 ∈ ℂ, 𝐷, 0) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))))
34		fveq2 6906	. . . . . 6 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (∗‘𝐷) = (∗‘if(𝐷 ∈ ℂ, 𝐷, 0)))
35	34	neeq2d 3001	. . . . 5 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷) ↔ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘if(𝐷 ∈ ℂ, 𝐷, 0))))
36	33, 35	anbi12d 632	. . . 4 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)) ↔ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (if(𝐷 ∈ ℂ, 𝐷, 0) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘if(𝐷 ∈ ℂ, 𝐷, 0)))))
37	36	imbi1d 341	. . 3 ⊢ (𝐷 = if(𝐷 ∈ ℂ, 𝐷, 0) → (((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (𝐷 ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘𝐷)) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0)) ↔ ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (if(𝐷 ∈ ℂ, 𝐷, 0) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘if(𝐷 ∈ ℂ, 𝐷, 0))) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0))))
38		ifhvhv0 31041	. . . 4 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ ℋ
39		ifhvhv0 31041	. . . 4 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ∈ ℋ
40		0cn 11253	. . . . 5 ⊢ 0 ∈ ℂ
41	40	elimel 4595	. . . 4 ⊢ if(𝐶 ∈ ℂ, 𝐶, 0) ∈ ℂ
42	40	elimel 4595	. . . 4 ⊢ if(𝐷 ∈ ℂ, 𝐷, 0) ∈ ℂ
43	38, 39, 41, 42	eigorthi 31856	. . 3 ⊢ ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) = (if(𝐶 ∈ ℂ, 𝐶, 0) ·_ℎ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ∧ (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = (if(𝐷 ∈ ℂ, 𝐷, 0) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ∧ if(𝐶 ∈ ℂ, 𝐶, 0) ≠ (∗‘if(𝐷 ∈ ℂ, 𝐷, 0))) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih (𝑇‘if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0_ℎ)) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) = 0))
44	12, 24, 30, 37, 43	dedth4h 4587	. 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵) ↔ (𝐴 ·_ih 𝐵) = 0)))
45	44	imp 406	1 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) ∧ (((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))) → ((𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵) ↔ (𝐴 ·_ih 𝐵) = 0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ifcif 4525 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 ∗ccj 15135 ℋchba 30938 ·_ℎ csm 30940 ·_ih csp 30941 0_ℎc0v 30943

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-hv0cl 31022 ax-hfvmul 31024 ax-hfi 31098 ax-his1 31101 ax-his3 31103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-2 12329 df-cj 15138 df-re 15139 df-im 15140

This theorem is referenced by: eighmorth 31983

W3C validator