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Theorem adj1 31905

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 31858	. . . . . . 7 ⊢ Fun adj_ℎ
2		funfvop 6978	. . . . . . 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 31857	. . . . . 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 2841	. . . . 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 6830	. . . . . 6 ⊢ (adj_ℎ‘𝑇) ∈ V
7		feq1 6624	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8		fveq1 6816	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑇 → (𝑧‘𝑦) = (𝑇‘𝑦))
9	8	oveq2d 7357	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → (𝑥 ·_ih (𝑧‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)))
10	9	eqeq1d 2733	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → ((𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦)))
11	10	2ralbidv 3196	. . . . . . . 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 6624	. . . . . . . 8 ⊢ (𝑤 = (adj_ℎ‘𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adj_ℎ‘𝑇): ℋ⟶ ℋ))
14		fveq1 6816	. . . . . . . . . . 11 ⊢ (𝑤 = (adj_ℎ‘𝑇) → (𝑤‘𝑥) = ((adj_ℎ‘𝑇)‘𝑥))
15	14	oveq1d 7356	. . . . . . . . . 10 ⊢ (𝑤 = (adj_ℎ‘𝑇) → ((𝑤‘𝑥) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))
16	15	eqeq2d 2742	. . . . . . . . 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 3196	. . . . . . . 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 5473	. . . . . 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 7348	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih (𝑇‘𝑦)))
24		fveq2 6817	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((adj_ℎ‘𝑇)‘𝑥) = ((adj_ℎ‘𝑇)‘𝐴))
25	24	oveq1d 7356	. . . . 5 ⊢ (𝑥 = 𝐴 → (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝑦))
26	23, 25	eqeq12d 2747	. . . 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 6817	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
28	27	oveq2d 7357	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih (𝑇‘𝐵)))
29		oveq2 7349	. . . . 5 ⊢ (𝑦 = 𝐵 → (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵))
30	28, 29	eqeq12d 2747	. . . 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 3583	. . 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 1541 ∈ wcel 2111 ∀wral 3047 Vcvv 3436 ⟨cop 4577 {copab 5148 dom cdm 5611 Fun wfun 6470 ⟶wf 6472 ‘cfv 6476 (class class class)co 7341 ℋchba 30891 ·_ih csp 30894 adj_ℎcado 30927

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-hfvadd 30972 ax-hvcom 30973 ax-hvass 30974 ax-hv0cl 30975 ax-hvaddid 30976 ax-hfvmul 30977 ax-hvmulid 30978 ax-hvdistr2 30981 ax-hvmul0 30982 ax-hfi 31051 ax-his1 31054 ax-his2 31055 ax-his3 31056 ax-his4 31057

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-cj 15001 df-re 15002 df-im 15003 df-hvsub 30943 df-adjh 31821

This theorem is referenced by: adj2 31906 adjadj 31908 hmopadj2 31913

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