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Theorem adj1 32285

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 32238	. . . . . . 7 ⊢ Fun adj_ℎ
2		funfvop 7045	. . . . . . 7 ⊢ ((Fun adj_ℎ ∧ 𝑇 ∈ dom adj_ℎ) → ⟨𝑇, (adj_ℎ‘𝑇)⟩ ∈ adj_ℎ)
3	1, 2	mpan 702	. . . . . 6 ⊢ (𝑇 ∈ dom adj_ℎ → ⟨𝑇, (adj_ℎ‘𝑇)⟩ ∈ adj_ℎ)
4		dfadj2 32237	. . . . . 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 2873	. . . . 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 6894	. . . . . 6 ⊢ (adj_ℎ‘𝑇) ∈ V
7		feq1 6683	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8		fveq1 6880	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑇 → (𝑧‘𝑦) = (𝑇‘𝑦))
9	8	oveq2d 7426	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → (𝑥 ·_ih (𝑧‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)))
10	9	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → ((𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦)))
11	10	2ralbidv 3229	. . . . . . . 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 1466	. . . . . . 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 6683	. . . . . . . 8 ⊢ (𝑤 = (adj_ℎ‘𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adj_ℎ‘𝑇): ℋ⟶ ℋ))
14		fveq1 6880	. . . . . . . . . . 11 ⊢ (𝑤 = (adj_ℎ‘𝑇) → (𝑤‘𝑥) = ((adj_ℎ‘𝑇)‘𝑥))
15	14	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝑤 = (adj_ℎ‘𝑇) → ((𝑤‘𝑥) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))
16	15	eqeq2d 2774	. . . . . . . . 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 3229	. . . . . . . 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 1467	. . . . . . 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 5523	. . . . . 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 703	. . . . 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 235	. . . 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 1162	. . 3 ⊢ (𝑇 ∈ dom adj_ℎ → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))
23		oveq1 7417	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih (𝑇‘𝑦)))
24		fveq2 6881	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((adj_ℎ‘𝑇)‘𝑥) = ((adj_ℎ‘𝑇)‘𝐴))
25	24	oveq1d 7425	. . . . 5 ⊢ (𝑥 = 𝐴 → (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝑦))
26	23, 25	eqeq12d 2779	. . . 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 6881	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
28	27	oveq2d 7426	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih (𝑇‘𝐵)))
29		oveq2 7418	. . . . 5 ⊢ (𝑦 = 𝐵 → (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵))
30	28, 29	eqeq12d 2779	. . . 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 3592	. . 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 32	. 2 ⊢ (𝑇 ∈ dom adj_ℎ → ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝑇‘𝐵)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵)))
33	32	3impib 1134	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 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∀wral 3079 Vcvv 3455 ⟨cop 4595 {copab 5173 dom cdm 5661 Fun wfun 6530 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℋchba 31271 ·_ih csp 31274 adj_ℎcado 31307

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-hfvadd 31352 ax-hvcom 31353 ax-hvass 31354 ax-hv0cl 31355 ax-hvaddid 31356 ax-hfvmul 31357 ax-hvmulid 31358 ax-hvdistr2 31361 ax-hvmul0 31362 ax-hfi 31431 ax-his1 31434 ax-his2 31435 ax-his3 31436 ax-his4 31437

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-cj 15146 df-re 15147 df-im 15148 df-hvsub 31323 df-adjh 32201

This theorem is referenced by: adj2 32286 adjadj 32288 hmopadj2 32293

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