Theorem adj1 32414

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 32367	. . . . . . 7 ⊢ Fun adjℎ
2		funfvop 7042	. . . . . . 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 32366	. . . . . 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 2870	. . . . 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 6891	. . . . . 6 ⊢ (adjℎ‘𝑇) ∈ V
7		feq1 6680	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8		fveq1 6877	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑇 → (𝑧‘𝑦) = (𝑇‘𝑦))
9	8	oveq2d 7429	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → (𝑥 ·ih (𝑧‘𝑦)) = (𝑥 ·ih (𝑇‘𝑦)))
10	9	eqeq1d 2762	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → ((𝑥 ·ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦)))
11	10	2ralbidv 3226	. . . . . . . 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 1466	. . . . . . 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 6680	. . . . . . . 8 ⊢ (𝑤 = (adjℎ‘𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adjℎ‘𝑇): ℋ⟶ ℋ))
14		fveq1 6877	. . . . . . . . . . 11 ⊢ (𝑤 = (adjℎ‘𝑇) → (𝑤‘𝑥) = ((adjℎ‘𝑇)‘𝑥))
15	14	oveq1d 7428	. . . . . . . . . 10 ⊢ (𝑤 = (adjℎ‘𝑇) → ((𝑤‘𝑥) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦))
16	15	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑤 = (adjℎ‘𝑇) → ((𝑥 ·ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦)))
17	16	2ralbidv 3226	. . . . . . . 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 1467	. . . . . . 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 5517	. . . . . 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 7420	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝑦)))
24		fveq2 6878	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((adjℎ‘𝑇)‘𝑥) = ((adjℎ‘𝑇)‘𝐴))
25	24	oveq1d 7428	. . . . 5 ⊢ (𝑥 = 𝐴 → (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦))
26	23, 25	eqeq12d 2776	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇‘𝑦)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦)))
27		fveq2 6878	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
28	27	oveq2d 7429	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝐵)))
29		oveq2 7421	. . . . 5 ⊢ (𝑦 = 𝐵 → (((adjℎ‘𝑇)‘𝐴) ·ih 𝑦) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵))
30	28, 29	eqeq12d 2776	. . . 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 3587	. . 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 2145  ∀wral 3076  Vcvv 3450  ⟨cop 4590  {copab 5167  dom cdm 5655  Fun wfun 6527  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413   ℋchba 31400   ·_ih csp 31403  adj_ℎcado 31436

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-hfvadd 31481  ax-hvcom 31482  ax-hvass 31483  ax-hv0cl 31484  ax-hvaddid 31485  ax-hfvmul 31486  ax-hvmulid 31487  ax-hvdistr2 31490  ax-hvmul0 31491  ax-hfi 31560  ax-his1 31563  ax-his2 31564  ax-his3 31565  ax-his4 31566

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-cj 15186  df-re 15187  df-im 15188  df-hvsub 31452  df-adjh 32330

This theorem is used by:  adj2  32415  adjadj  32417  hmopadj2  32422