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Theorem adj1 31756

Description: Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
adj1	⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝑇‘𝐵)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵))

Proof of Theorem adj1 Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funadj 31709	. . . . . . 7 ⊢ Fun adj_ℎ
2		funfvop 7059	. . . . . . 7 ⊢ ((Fun adj_ℎ ∧ 𝑇 ∈ dom adj_ℎ) → ⟨𝑇, (adj_ℎ‘𝑇)⟩ ∈ adj_ℎ)
3	1, 2	mpan 689	. . . . . 6 ⊢ (𝑇 ∈ dom adj_ℎ → ⟨𝑇, (adj_ℎ‘𝑇)⟩ ∈ adj_ℎ)
4		dfadj2 31708	. . . . . 6 ⊢ adj_ℎ = {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦))}
5	3, 4	eleqtrdi 2839	. . . . 5 ⊢ (𝑇 ∈ dom adj_ℎ → ⟨𝑇, (adj_ℎ‘𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦))})
6		fvex 6910	. . . . . 6 ⊢ (adj_ℎ‘𝑇) ∈ V
7		feq1 6703	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8		fveq1 6896	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑇 → (𝑧‘𝑦) = (𝑇‘𝑦))
9	8	oveq2d 7436	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → (𝑥 ·_ih (𝑧‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)))
10	9	eqeq1d 2730	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → ((𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦)))
11	10	2ralbidv 3215	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦)))
12	7, 11	3anbi13d 1435	. . . . . . 7 ⊢ (𝑧 = 𝑇 → ((𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦))))
13		feq1 6703	. . . . . . . 8 ⊢ (𝑤 = (adj_ℎ‘𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adj_ℎ‘𝑇): ℋ⟶ ℋ))
14		fveq1 6896	. . . . . . . . . . 11 ⊢ (𝑤 = (adj_ℎ‘𝑇) → (𝑤‘𝑥) = ((adj_ℎ‘𝑇)‘𝑥))
15	14	oveq1d 7435	. . . . . . . . . 10 ⊢ (𝑤 = (adj_ℎ‘𝑇) → ((𝑤‘𝑥) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))
16	15	eqeq2d 2739	. . . . . . . . 9 ⊢ (𝑤 = (adj_ℎ‘𝑇) → ((𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦)))
17	16	2ralbidv 3215	. . . . . . . 8 ⊢ (𝑤 = (adj_ℎ‘𝑇) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦)))
18	13, 17	3anbi23d 1436	. . . . . . 7 ⊢ (𝑤 = (adj_ℎ‘𝑇) → ((𝑇: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ (adj_ℎ‘𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))))
19	12, 18	opelopabg 5540	. . . . . 6 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ (adj_ℎ‘𝑇) ∈ V) → (⟨𝑇, (adj_ℎ‘𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦))} ↔ (𝑇: ℋ⟶ ℋ ∧ (adj_ℎ‘𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))))
20	6, 19	mpan2 690	. . . . 5 ⊢ (𝑇 ∈ dom adj_ℎ → (⟨𝑇, (adj_ℎ‘𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦))} ↔ (𝑇: ℋ⟶ ℋ ∧ (adj_ℎ‘𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))))
21	5, 20	mpbid 231	. . . 4 ⊢ (𝑇 ∈ dom adj_ℎ → (𝑇: ℋ⟶ ℋ ∧ (adj_ℎ‘𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦)))
22	21	simp3d 1142	. . 3 ⊢ (𝑇 ∈ dom adj_ℎ → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))
23		oveq1 7427	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih (𝑇‘𝑦)))
24		fveq2 6897	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((adj_ℎ‘𝑇)‘𝑥) = ((adj_ℎ‘𝑇)‘𝐴))
25	24	oveq1d 7435	. . . . 5 ⊢ (𝑥 = 𝐴 → (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝑦))
26	23, 25	eqeq12d 2744	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦) ↔ (𝐴 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝑦)))
27		fveq2 6897	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
28	27	oveq2d 7436	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih (𝑇‘𝐵)))
29		oveq2 7428	. . . . 5 ⊢ (𝑦 = 𝐵 → (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵))
30	28, 29	eqeq12d 2744	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝑦) ↔ (𝐴 ·_ih (𝑇‘𝐵)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵)))
31	26, 30	rspc2v 3620	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦) → (𝐴 ·_ih (𝑇‘𝐵)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵)))
32	22, 31	syl5com 31	. 2 ⊢ (𝑇 ∈ dom adj_ℎ → ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝑇‘𝐵)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵)))
33	32	3impib 1114	1 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝑇‘𝐵)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3058 Vcvv 3471 ⟨cop 4635 {copab 5210 dom cdm 5678 Fun wfun 6542 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ℋchba 30742 ·_ih csp 30745 adj_ℎcado 30778

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 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-hfvadd 30823 ax-hvcom 30824 ax-hvass 30825 ax-hv0cl 30826 ax-hvaddid 30827 ax-hfvmul 30828 ax-hvmulid 30829 ax-hvdistr2 30832 ax-hvmul0 30833 ax-hfi 30902 ax-his1 30905 ax-his2 30906 ax-his3 30907 ax-his4 30908

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 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-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-2 12306 df-cj 15079 df-re 15080 df-im 15081 df-hvsub 30794 df-adjh 31672

This theorem is referenced by: adj2 31757 adjadj 31759 hmopadj2 31764

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