Theorem adj1 29710

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 29663	. . . . . . 7 ⊢ Fun adjℎ
2		funfvop 6820	. . . . . . 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 29662	. . . . . 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 2923	. . . . 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 6683	. . . . . 6 ⊢ (adjℎ‘𝑇) ∈ V
7		feq1 6495	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8		fveq1 6669	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑇 → (𝑧‘𝑦) = (𝑇‘𝑦))
9	8	oveq2d 7172	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → (𝑥 ·ih (𝑧‘𝑦)) = (𝑥 ·ih (𝑇‘𝑦)))
10	9	eqeq1d 2823	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → ((𝑥 ·ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦)))
11	10	2ralbidv 3199	. . . . . . . 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 1434	. . . . . . 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 6495	. . . . . . . 8 ⊢ (𝑤 = (adjℎ‘𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adjℎ‘𝑇): ℋ⟶ ℋ))
14		fveq1 6669	. . . . . . . . . . 11 ⊢ (𝑤 = (adjℎ‘𝑇) → (𝑤‘𝑥) = ((adjℎ‘𝑇)‘𝑥))
15	14	oveq1d 7171	. . . . . . . . . 10 ⊢ (𝑤 = (adjℎ‘𝑇) → ((𝑤‘𝑥) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦))
16	15	eqeq2d 2832	. . . . . . . . 9 ⊢ (𝑤 = (adjℎ‘𝑇) → ((𝑥 ·ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦)))
17	16	2ralbidv 3199	. . . . . . . 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 1435	. . . . . . 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 5425	. . . . . 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 234	. . . 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 1140	. . 3 ⊢ (𝑇 ∈ dom adjℎ → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦))
23		oveq1 7163	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝑦)))
24		fveq2 6670	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((adjℎ‘𝑇)‘𝑥) = ((adjℎ‘𝑇)‘𝐴))
25	24	oveq1d 7171	. . . . 5 ⊢ (𝑥 = 𝐴 → (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦))
26	23, 25	eqeq12d 2837	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦)))
27		fveq2 6670	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
28	27	oveq2d 7172	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝐵)))
29		oveq2 7164	. . . . 5 ⊢ (𝑦 = 𝐵 → (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵))
30	28, 29	eqeq12d 2837	. . . 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 3633	. . 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 1112	1 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138  Vcvv 3494  ⟨cop 4573  {copab 5128  dom cdm 5555  Fun wfun 6349  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156   ℋchba 28696   ·_ih csp 28699  adj_ℎcado 28732

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-hfvadd 28777  ax-hvcom 28778  ax-hvass 28779  ax-hv0cl 28780  ax-hvaddid 28781  ax-hfvmul 28782  ax-hvmulid 28783  ax-hvdistr2 28786  ax-hvmul0 28787  ax-hfi 28856  ax-his1 28859  ax-his2 28860  ax-his3 28861  ax-his4 28862

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-po 5474  df-so 5475  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-2 11701  df-cj 14458  df-re 14459  df-im 14460  df-hvsub 28748  df-adjh 29626

This theorem is referenced by:  adj2  29711  adjadj  29713  hmopadj2  29718