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Theorem adj2 31225

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

**Assertion**
Ref	Expression
adj2	⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·_ih 𝐵) = (𝐴 ·_ih ((adj_ℎ‘𝑇)‘𝐵)))

**Proof of Theorem adj2**
Step	Hyp	Ref	Expression
1		adj1 31224	. . . 4 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐵 ·_ih (𝑇‘𝐴)) = (((adj_ℎ‘𝑇)‘𝐵) ·_ih 𝐴))
2		simp2 1137	. . . . 5 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → 𝐵 ∈ ℋ)
3		dmadjop 31179	. . . . . . 7 ⊢ (𝑇 ∈ dom adj_ℎ → 𝑇: ℋ⟶ ℋ)
4	3	ffvelcdmda 7086	. . . . . 6 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) → (𝑇‘𝐴) ∈ ℋ)
5	4	3adant2 1131	. . . . 5 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑇‘𝐴) ∈ ℋ)
6		ax-his1 30373	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝑇‘𝐴) ∈ ℋ) → (𝐵 ·_ih (𝑇‘𝐴)) = (∗‘((𝑇‘𝐴) ·_ih 𝐵)))
7	2, 5, 6	syl2anc 584	. . . 4 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐵 ·_ih (𝑇‘𝐴)) = (∗‘((𝑇‘𝐴) ·_ih 𝐵)))
8		adjcl 31223	. . . . . 6 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐵 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝐵) ∈ ℋ)
9	8	3adant3 1132	. . . . 5 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝐵) ∈ ℋ)
10		simp3 1138	. . . . 5 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → 𝐴 ∈ ℋ)
11		ax-his1 30373	. . . . 5 ⊢ ((((adj_ℎ‘𝑇)‘𝐵) ∈ ℋ ∧ 𝐴 ∈ ℋ) → (((adj_ℎ‘𝑇)‘𝐵) ·_ih 𝐴) = (∗‘(𝐴 ·_ih ((adj_ℎ‘𝑇)‘𝐵))))
12	9, 10, 11	syl2anc 584	. . . 4 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (((adj_ℎ‘𝑇)‘𝐵) ·_ih 𝐴) = (∗‘(𝐴 ·_ih ((adj_ℎ‘𝑇)‘𝐵))))
13	1, 7, 12	3eqtr3d 2780	. . 3 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝑇‘𝐴) ·_ih 𝐵)) = (∗‘(𝐴 ·_ih ((adj_ℎ‘𝑇)‘𝐵))))
14		hicl 30371	. . . . 5 ⊢ (((𝑇‘𝐴) ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·_ih 𝐵) ∈ ℂ)
15	5, 2, 14	syl2anc 584	. . . 4 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑇‘𝐴) ·_ih 𝐵) ∈ ℂ)
16		hicl 30371	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ((adj_ℎ‘𝑇)‘𝐵) ∈ ℋ) → (𝐴 ·_ih ((adj_ℎ‘𝑇)‘𝐵)) ∈ ℂ)
17	10, 9, 16	syl2anc 584	. . . 4 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐴 ·_ih ((adj_ℎ‘𝑇)‘𝐵)) ∈ ℂ)
18		cj11 15111	. . . 4 ⊢ ((((𝑇‘𝐴) ·_ih 𝐵) ∈ ℂ ∧ (𝐴 ·_ih ((adj_ℎ‘𝑇)‘𝐵)) ∈ ℂ) → ((∗‘((𝑇‘𝐴) ·_ih 𝐵)) = (∗‘(𝐴 ·_ih ((adj_ℎ‘𝑇)‘𝐵))) ↔ ((𝑇‘𝐴) ·_ih 𝐵) = (𝐴 ·_ih ((adj_ℎ‘𝑇)‘𝐵))))
19	15, 17, 18	syl2anc 584	. . 3 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((∗‘((𝑇‘𝐴) ·_ih 𝐵)) = (∗‘(𝐴 ·_ih ((adj_ℎ‘𝑇)‘𝐵))) ↔ ((𝑇‘𝐴) ·_ih 𝐵) = (𝐴 ·_ih ((adj_ℎ‘𝑇)‘𝐵))))
20	13, 19	mpbid 231	. 2 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑇‘𝐴) ·_ih 𝐵) = (𝐴 ·_ih ((adj_ℎ‘𝑇)‘𝐵)))
21	20	3com23 1126	1 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·_ih 𝐵) = (𝐴 ·_ih ((adj_ℎ‘𝑇)‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 dom cdm 5676 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 ∗ccj 15045 ℋchba 30210 ·_ih csp 30213 adj_ℎcado 30246

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 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-hilex 30290 ax-hfvadd 30291 ax-hvcom 30292 ax-hvass 30293 ax-hv0cl 30294 ax-hvaddid 30295 ax-hfvmul 30296 ax-hvmulid 30297 ax-hvdistr2 30300 ax-hvmul0 30301 ax-hfi 30370 ax-his1 30373 ax-his2 30374 ax-his3 30375 ax-his4 30376

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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-2 12277 df-cj 15048 df-re 15049 df-im 15050 df-hvsub 30262 df-adjh 31140

This theorem is referenced by: adjadj 31227 adjvalval 31228 adjlnop 31377 adjmul 31383 adjadd 31384 adjcoi 31391 nmopcoadji 31392

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