Theorem adj1 32592

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 32545	. . . . . . 7 ⊢ Fun adjℎ
2		funfvop 7080	. . . . . . 7 ⊢ ((Fun adjℎ ∧ 𝑇 ∈ dom adjℎ) → ⟨𝑇, (adjℎ‘𝑇)⟩ ∈ adjℎ)
3	1, 2	mpan 703	. . . . . 6 ⊢ (𝑇 ∈ dom adjℎ → ⟨𝑇, (adjℎ‘𝑇)⟩ ∈ adjℎ)
4		dfadj2 32544	. . . . . 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 2874	. . . . 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 6929	. . . . . 6 ⊢ (adjℎ‘𝑇) ∈ V
7		feq1 6717	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8		fveq1 6915	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑇 → (𝑧‘𝑦) = (𝑇‘𝑦))
9	8	oveq2d 7467	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → (𝑥 ·ih (𝑧‘𝑦)) = (𝑥 ·ih (𝑇‘𝑦)))
10	9	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → ((𝑥 ·ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦)))
11	10	2ralbidv 3230	. . . . . . . 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 1467	. . . . . . 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 6717	. . . . . . . 8 ⊢ (𝑤 = (adjℎ‘𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adjℎ‘𝑇): ℋ⟶ ℋ))
14		fveq1 6915	. . . . . . . . . . 11 ⊢ (𝑤 = (adjℎ‘𝑇) → (𝑤‘𝑥) = ((adjℎ‘𝑇)‘𝑥))
15	14	oveq1d 7466	. . . . . . . . . 10 ⊢ (𝑤 = (adjℎ‘𝑇) → ((𝑤‘𝑥) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦))
16	15	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑤 = (adjℎ‘𝑇) → ((𝑥 ·ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦)))
17	16	2ralbidv 3230	. . . . . . . 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 1468	. . . . . . 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 5520	. . . . . 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 704	. . . . 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 235	. . . 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 1162	. . 3 ⊢ (𝑇 ∈ dom adjℎ → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦))
23		oveq1 7458	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝑦)))
24		fveq2 6916	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((adjℎ‘𝑇)‘𝑥) = ((adjℎ‘𝑇)‘𝐴))
25	24	oveq1d 7466	. . . . 5 ⊢ (𝑥 = 𝐴 → (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦))
26	23, 25	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦)))
27		fveq2 6916	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
28	27	oveq2d 7467	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝐵)))
29		oveq2 7459	. . . . 5 ⊢ (𝑦 = 𝐵 → (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵))
30	28, 29	eqeq12d 2779	. . . 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 3590	. . 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 32	. 2 ⊢ (𝑇 ∈ dom adjℎ → ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵)))
33	32	3impib 1134	1 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454  ⟨cop 4840  {copab 5173  dom cdm 5658  Fun wfun 6560  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ℋchba 31578   ·_ih csp 31581  adj_ℎcado 31614

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-hfvadd 31659  ax-hvcom 31660  ax-hvass 31661  ax-hv0cl 31662  ax-hvaddid 31663  ax-hfvmul 31664  ax-hvmulid 31665  ax-hvdistr2 31668  ax-hvmul0 31669  ax-hfi 31738  ax-his1 31741  ax-his2 31742  ax-his3 31743  ax-his4 31744

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-cj 15309  df-re 15310  df-im 15311  df-hvsub 31630  df-adjh 32508

This theorem is used by:  adj2  32593  adjadj  32595  hmopadj2  32600