Theorem adj1 30196

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

Assertion

Ref	Expression
adj1	⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = (((adjℎ‘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 30149	. . . . . . 7 ⊢ Fun adjℎ
2		funfvop 6909	. . . . . . 7 ⊢ ((Fun adjℎ ∧ 𝑇 ∈ dom adjℎ) → ⟨𝑇, (adjℎ‘𝑇)⟩ ∈ adjℎ)
3	1, 2	mpan 686	. . . . . 6 ⊢ (𝑇 ∈ dom adjℎ → ⟨𝑇, (adjℎ‘𝑇)⟩ ∈ adjℎ)
4		dfadj2 30148	. . . . . 6 ⊢ adjℎ = {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦))}
5	3, 4	eleqtrdi 2849	. . . . 5 ⊢ (𝑇 ∈ dom adjℎ → ⟨𝑇, (adjℎ‘𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦))})
6		fvex 6769	. . . . . 6 ⊢ (adjℎ‘𝑇) ∈ V
7		feq1 6565	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8		fveq1 6755	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑇 → (𝑧‘𝑦) = (𝑇‘𝑦))
9	8	oveq2d 7271	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → (𝑥 ·ih (𝑧‘𝑦)) = (𝑥 ·ih (𝑇‘𝑦)))
10	9	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → ((𝑥 ·ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦)))
11	10	2ralbidv 3122	. . . . . . . 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 1436	. . . . . . 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 6565	. . . . . . . 8 ⊢ (𝑤 = (adjℎ‘𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adjℎ‘𝑇): ℋ⟶ ℋ))
14		fveq1 6755	. . . . . . . . . . 11 ⊢ (𝑤 = (adjℎ‘𝑇) → (𝑤‘𝑥) = ((adjℎ‘𝑇)‘𝑥))
15	14	oveq1d 7270	. . . . . . . . . 10 ⊢ (𝑤 = (adjℎ‘𝑇) → ((𝑤‘𝑥) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦))
16	15	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑤 = (adjℎ‘𝑇) → ((𝑥 ·ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦)))
17	16	2ralbidv 3122	. . . . . . . 8 ⊢ (𝑤 = (adjℎ‘𝑇) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦)))
18	13, 17	3anbi23d 1437	. . . . . . 7 ⊢ (𝑤 = (adjℎ‘𝑇) → ((𝑇: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦))))
19	12, 18	opelopabg 5444	. . . . . 6 ⊢ ((𝑇 ∈ dom adjℎ ∧ (adjℎ‘𝑇) ∈ V) → (⟨𝑇, (adjℎ‘𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦))} ↔ (𝑇: ℋ⟶ ℋ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦))))
20	6, 19	mpan2 687	. . . . 5 ⊢ (𝑇 ∈ dom adjℎ → (⟨𝑇, (adjℎ‘𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦))} ↔ (𝑇: ℋ⟶ ℋ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦))))
21	5, 20	mpbid 231	. . . 4 ⊢ (𝑇 ∈ dom adjℎ → (𝑇: ℋ⟶ ℋ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦)))
22	21	simp3d 1142	. . 3 ⊢ (𝑇 ∈ dom adjℎ → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦))
23		oveq1 7262	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝑦)))
24		fveq2 6756	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((adjℎ‘𝑇)‘𝑥) = ((adjℎ‘𝑇)‘𝐴))
25	24	oveq1d 7270	. . . . 5 ⊢ (𝑥 = 𝐴 → (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦))
26	23, 25	eqeq12d 2754	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦)))
27		fveq2 6756	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
28	27	oveq2d 7271	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝐵)))
29		oveq2 7263	. . . . 5 ⊢ (𝑦 = 𝐵 → (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵))
30	28, 29	eqeq12d 2754	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇‘𝐵)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵)))
31	26, 30	rspc2v 3562	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦) → (𝐴 ·ih (𝑇‘𝐵)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵)))
32	22, 31	syl5com 31	. 2 ⊢ (𝑇 ∈ dom adjℎ → ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵)))
33	32	3impib 1114	1 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422  ⟨cop 4564  {copab 5132  dom cdm 5580  Fun wfun 6412  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ℋchba 29182   ·_ih csp 29185  adj_ℎcado 29218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-hfvadd 29263  ax-hvcom 29264  ax-hvass 29265  ax-hv0cl 29266  ax-hvaddid 29267  ax-hfvmul 29268  ax-hvmulid 29269  ax-hvdistr2 29272  ax-hvmul0 29273  ax-hfi 29342  ax-his1 29345  ax-his2 29346  ax-his3 29347  ax-his4 29348

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-2 11966  df-cj 14738  df-re 14739  df-im 14740  df-hvsub 29234  df-adjh 30112

This theorem is referenced by:  adj2  30197  adjadj  30199  hmopadj2  30204