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Theorem adj1 31860

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 31813	. . . . . . 7 ⊢ Fun adj_ℎ
2		funfvop 7039	. . . . . . 7 ⊢ ((Fun adj_ℎ ∧ 𝑇 ∈ dom adj_ℎ) → ⟨𝑇, (adj_ℎ‘𝑇)⟩ ∈ adj_ℎ)
3	1, 2	mpan 690	. . . . . 6 ⊢ (𝑇 ∈ dom adj_ℎ → ⟨𝑇, (adj_ℎ‘𝑇)⟩ ∈ adj_ℎ)
4		dfadj2 31812	. . . . . 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 2844	. . . . 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 6888	. . . . . 6 ⊢ (adj_ℎ‘𝑇) ∈ V
7		feq1 6685	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8		fveq1 6874	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑇 → (𝑧‘𝑦) = (𝑇‘𝑦))
9	8	oveq2d 7419	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → (𝑥 ·_ih (𝑧‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)))
10	9	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → ((𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦)))
11	10	2ralbidv 3205	. . . . . . . 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 1440	. . . . . . 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 6685	. . . . . . . 8 ⊢ (𝑤 = (adj_ℎ‘𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adj_ℎ‘𝑇): ℋ⟶ ℋ))
14		fveq1 6874	. . . . . . . . . . 11 ⊢ (𝑤 = (adj_ℎ‘𝑇) → (𝑤‘𝑥) = ((adj_ℎ‘𝑇)‘𝑥))
15	14	oveq1d 7418	. . . . . . . . . 10 ⊢ (𝑤 = (adj_ℎ‘𝑇) → ((𝑤‘𝑥) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))
16	15	eqeq2d 2746	. . . . . . . . 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 3205	. . . . . . . 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 1441	. . . . . . 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 5513	. . . . . 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 691	. . . . 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 232	. . . 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 1144	. . 3 ⊢ (𝑇 ∈ dom adj_ℎ → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))
23		oveq1 7410	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih (𝑇‘𝑦)))
24		fveq2 6875	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((adj_ℎ‘𝑇)‘𝑥) = ((adj_ℎ‘𝑇)‘𝐴))
25	24	oveq1d 7418	. . . . 5 ⊢ (𝑥 = 𝐴 → (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝑦))
26	23, 25	eqeq12d 2751	. . . 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 6875	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
28	27	oveq2d 7419	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih (𝑇‘𝐵)))
29		oveq2 7411	. . . . 5 ⊢ (𝑦 = 𝐵 → (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵))
30	28, 29	eqeq12d 2751	. . . 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 3612	. . 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 1116	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3051 Vcvv 3459 ⟨cop 4607 {copab 5181 dom cdm 5654 Fun wfun 6524 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 ℋchba 30846 ·_ih csp 30849 adj_ℎcado 30882

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 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-hfvadd 30927 ax-hvcom 30928 ax-hvass 30929 ax-hv0cl 30930 ax-hvaddid 30931 ax-hfvmul 30932 ax-hvmulid 30933 ax-hvdistr2 30936 ax-hvmul0 30937 ax-hfi 31006 ax-his1 31009 ax-his2 31010 ax-his3 31011 ax-his4 31012

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8717 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-cj 15116 df-re 15117 df-im 15118 df-hvsub 30898 df-adjh 31776

This theorem is referenced by: adj2 31861 adjadj 31863 hmopadj2 31868

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