Theorem adj1 31113

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 31066	. . . . . . 7 ⊢ Fun adjℎ
2		funfvop 7037	. . . . . . 7 ⊢ ((Fun adjℎ ∧ 𝑇 ∈ dom adjℎ) → ⟨𝑇, (adjℎ‘𝑇)⟩ ∈ adjℎ)
3	1, 2	mpan 688	. . . . . 6 ⊢ (𝑇 ∈ dom adjℎ → ⟨𝑇, (adjℎ‘𝑇)⟩ ∈ adjℎ)
4		dfadj2 31065	. . . . . 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 2843	. . . . 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 6892	. . . . . 6 ⊢ (adjℎ‘𝑇) ∈ V
7		feq1 6686	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8		fveq1 6878	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑇 → (𝑧‘𝑦) = (𝑇‘𝑦))
9	8	oveq2d 7410	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → (𝑥 ·ih (𝑧‘𝑦)) = (𝑥 ·ih (𝑇‘𝑦)))
10	9	eqeq1d 2734	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → ((𝑥 ·ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦)))
11	10	2ralbidv 3218	. . . . . . . 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 1438	. . . . . . 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 6686	. . . . . . . 8 ⊢ (𝑤 = (adjℎ‘𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adjℎ‘𝑇): ℋ⟶ ℋ))
14		fveq1 6878	. . . . . . . . . . 11 ⊢ (𝑤 = (adjℎ‘𝑇) → (𝑤‘𝑥) = ((adjℎ‘𝑇)‘𝑥))
15	14	oveq1d 7409	. . . . . . . . . 10 ⊢ (𝑤 = (adjℎ‘𝑇) → ((𝑤‘𝑥) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦))
16	15	eqeq2d 2743	. . . . . . . . 9 ⊢ (𝑤 = (adjℎ‘𝑇) → ((𝑥 ·ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦)))
17	16	2ralbidv 3218	. . . . . . . 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 1439	. . . . . . 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 5532	. . . . . 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 689	. . . . 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 1144	. . 3 ⊢ (𝑇 ∈ dom adjℎ → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦))
23		oveq1 7401	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝑦)))
24		fveq2 6879	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((adjℎ‘𝑇)‘𝑥) = ((adjℎ‘𝑇)‘𝐴))
25	24	oveq1d 7409	. . . . 5 ⊢ (𝑥 = 𝐴 → (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦))
26	23, 25	eqeq12d 2748	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦)))
27		fveq2 6879	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
28	27	oveq2d 7410	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝐵)))
29		oveq2 7402	. . . . 5 ⊢ (𝑦 = 𝐵 → (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵))
30	28, 29	eqeq12d 2748	. . . 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 3619	. . 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 1116	1 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474  ⟨cop 4629  {copab 5204  dom cdm 5670  Fun wfun 6527  ⟶wf 6529  ‘cfv 6533  (class class class)co 7394   ℋchba 30099   ·_ih csp 30102  adj_ℎcado 30135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-resscn 11151  ax-1cn 11152  ax-icn 11153  ax-addcl 11154  ax-addrcl 11155  ax-mulcl 11156  ax-mulrcl 11157  ax-mulcom 11158  ax-addass 11159  ax-mulass 11160  ax-distr 11161  ax-i2m1 11162  ax-1ne0 11163  ax-1rid 11164  ax-rnegex 11165  ax-rrecex 11166  ax-cnre 11167  ax-pre-lttri 11168  ax-pre-lttrn 11169  ax-pre-ltadd 11170  ax-pre-mulgt0 11171  ax-hfvadd 30180  ax-hvcom 30181  ax-hvass 30182  ax-hv0cl 30183  ax-hvaddid 30184  ax-hfvmul 30185  ax-hvmulid 30186  ax-hvdistr2 30189  ax-hvmul0 30190  ax-hfi 30259  ax-his1 30262  ax-his2 30263  ax-his3 30264  ax-his4 30265

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-po 5582  df-so 5583  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-er 8688  df-en 8925  df-dom 8926  df-sdom 8927  df-pnf 11234  df-mnf 11235  df-xr 11236  df-ltxr 11237  df-le 11238  df-sub 11430  df-neg 11431  df-div 11856  df-2 12259  df-cj 15030  df-re 15031  df-im 15032  df-hvsub 30151  df-adjh 31029

This theorem is referenced by:  adj2  31114  adjadj  31116  hmopadj2  31121