Theorem hmop 31726

Description: Basic inner product property of a Hermitian operator. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
hmop	⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵))

Proof of Theorem hmop

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elhmop 31677	. . . 4 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦)))
2	1	simprbi 496	. . 3 ⊢ (𝑇 ∈ HrmOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))
3	2	3ad2ant1 1131	. 2 ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))
4		oveq1 7422	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝑦)))
5		fveq2 6892	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
6	5	oveq1d 7430	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) ·ih 𝑦) = ((𝑇‘𝐴) ·ih 𝑦))
7	4, 6	eqeq12d 2744	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·ih 𝑦)))
8		fveq2 6892	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
9	8	oveq2d 7431	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝐵)))
10		oveq2 7423	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑇‘𝐴) ·ih 𝑦) = ((𝑇‘𝐴) ·ih 𝐵))
11	9, 10	eqeq12d 2744	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ·ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵)))
12	7, 11	rspc2v 3619	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦) → (𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵)))
13	12	3adant1 1128	. 2 ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦) → (𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵)))
14	3, 13	mpd 15	1 ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵))

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3057  ⟶wf 6539  ‘cfv 6543  (class class class)co 7415   ℋchba 30723   ·_ih csp 30726  HrmOpcho 30754

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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-hilex 30803

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-map 8841  df-hmop 31648

This theorem is referenced by:  hmopre  31727  hmopadj  31743  hmoplin  31746  eighmre  31767  eighmorth  31768  hmopbdoptHIL  31792  hmops  31824  hmopm  31825  hmopco  31827  leopsq  31933  hmopidmpji  31956