Theorem adjmul 31600

Description: The adjoint of the scalar product of an operator. Theorem 3.11(ii) of [Beran] p. 106. (Contributed by NM, 21-Feb-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
adjmul	⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝐴 ·op 𝑇)) = ((∗‘𝐴) ·op (adjℎ‘𝑇)))

Proof of Theorem adjmul

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmadjop 31396	. . 3 ⊢ (𝑇 ∈ dom adjℎ → 𝑇: ℋ⟶ ℋ)
2		homulcl 31267	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
3	1, 2	sylan2 593	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
4		cjcl 15056	. . 3 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ)
5		dmadjrn 31403	. . . 4 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ)
6		dmadjop 31396	. . . 4 ⊢ ((adjℎ‘𝑇) ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
7	5, 6	syl 17	. . 3 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
8		homulcl 31267	. . 3 ⊢ (((∗‘𝐴) ∈ ℂ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ) → ((∗‘𝐴) ·op (adjℎ‘𝑇)): ℋ⟶ ℋ)
9	4, 7, 8	syl2an 596	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) → ((∗‘𝐴) ·op (adjℎ‘𝑇)): ℋ⟶ ℋ)
10		adj2 31442	. . . . . . . 8 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
11	10	3expb 1120	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
12	11	adantll 712	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
13	12	oveq2d 7427	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)) = (𝐴 · (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦))))
14	1	ffvelcdmda 7086	. . . . . . . . 9 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
15		ax-his3 30592	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)))
16	14, 15	syl3an2 1164	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)))
17	16	3exp 1119	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → (𝑦 ∈ ℋ → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)))))
18	17	expd 416	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑇 ∈ dom adjℎ → (𝑥 ∈ ℋ → (𝑦 ∈ ℋ → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦))))))
19	18	imp43 428	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)))
20		simpll 765	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝐴 ∈ ℂ)
21		simprl 769	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
22		adjcl 31440	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ) → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ)
23	22	ad2ant2l 744	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ)
24		his52 30595	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ ((adjℎ‘𝑇)‘𝑦) ∈ ℋ) → (𝑥 ·ih ((∗‘𝐴) ·ℎ ((adjℎ‘𝑇)‘𝑦))) = (𝐴 · (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦))))
25	20, 21, 23, 24	syl3anc 1371	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((∗‘𝐴) ·ℎ ((adjℎ‘𝑇)‘𝑦))) = (𝐴 · (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦))))
26	13, 19, 25	3eqtr4d 2782	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝑥 ·ih ((∗‘𝐴) ·ℎ ((adjℎ‘𝑇)‘𝑦))))
27		homval 31249	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
28	1, 27	syl3an2 1164	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
29	28	3expa 1118	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
30	29	adantrr 715	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
31	30	oveq1d 7426	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑦) = ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦))
32		id 22	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → 𝑦 ∈ ℋ)
33		homval 31249	. . . . . . . 8 ⊢ (((∗‘𝐴) ∈ ℂ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (((∗‘𝐴) ·op (adjℎ‘𝑇))‘𝑦) = ((∗‘𝐴) ·ℎ ((adjℎ‘𝑇)‘𝑦)))
34	4, 7, 32, 33	syl3an 1160	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ) → (((∗‘𝐴) ·op (adjℎ‘𝑇))‘𝑦) = ((∗‘𝐴) ·ℎ ((adjℎ‘𝑇)‘𝑦)))
35	34	3expa 1118	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ 𝑦 ∈ ℋ) → (((∗‘𝐴) ·op (adjℎ‘𝑇))‘𝑦) = ((∗‘𝐴) ·ℎ ((adjℎ‘𝑇)‘𝑦)))
36	35	adantrl 714	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((∗‘𝐴) ·op (adjℎ‘𝑇))‘𝑦) = ((∗‘𝐴) ·ℎ ((adjℎ‘𝑇)‘𝑦)))
37	36	oveq2d 7427	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((∗‘𝐴) ·op (adjℎ‘𝑇))‘𝑦)) = (𝑥 ·ih ((∗‘𝐴) ·ℎ ((adjℎ‘𝑇)‘𝑦))))
38	26, 31, 37	3eqtr4d 2782	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((∗‘𝐴) ·op (adjℎ‘𝑇))‘𝑦)))
39	38	ralrimivva 3200	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((∗‘𝐴) ·op (adjℎ‘𝑇))‘𝑦)))
40		adjeq 31443	. 2 ⊢ (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ ((∗‘𝐴) ·op (adjℎ‘𝑇)): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((∗‘𝐴) ·op (adjℎ‘𝑇))‘𝑦))) → (adjℎ‘(𝐴 ·op 𝑇)) = ((∗‘𝐴) ·op (adjℎ‘𝑇)))
41	3, 9, 39, 40	syl3anc 1371	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝐴 ·op 𝑇)) = ((∗‘𝐴) ·op (adjℎ‘𝑇)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  dom cdm 5676  ⟶wf 6539  ‘cfv 6543  (class class class)co 7411  ℂcc 11110   · cmul 11117  ∗ccj 15047   ℋchba 30427   ·_ℎ csm 30429   ·_ih csp 30430   ·_op chot 30447  adj_ℎcado 30463

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-rep 5285  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 30507  ax-hfvadd 30508  ax-hvcom 30509  ax-hvass 30510  ax-hv0cl 30511  ax-hvaddid 30512  ax-hfvmul 30513  ax-hvmulid 30514  ax-hvdistr2 30517  ax-hvmul0 30518  ax-hfi 30587  ax-his1 30590  ax-his2 30591  ax-his3 30592  ax-his4 30593

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 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-2 12279  df-cj 15050  df-re 15051  df-im 15052  df-hvsub 30479  df-homul 31239  df-adjh 31357

This theorem is referenced by: (None)