Theorem adjmul 31097

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 30893	. . 3 ⊢ (𝑇 ∈ dom adjℎ → 𝑇: ℋ⟶ ℋ)
2		homulcl 30764	. . 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 15002	. . 3 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ)
5		dmadjrn 30900	. . . 4 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ)
6		dmadjop 30893	. . . 4 ⊢ ((adjℎ‘𝑇) ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
7	5, 6	syl 17	. . 3 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
8		homulcl 30764	. . 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 30939	. . . . . . . 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 7378	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)) = (𝐴 · (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦))))
14	1	ffvelcdmda 7040	. . . . . . . . 9 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
15		ax-his3 30089	. . . . . . . . 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 30937	. . . . . . 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 30092	. . . . . 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 2781	. . . 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 30746	. . . . . . . 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 7377	. . . 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 30746	. . . . . . . 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 7378	. . . 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 2781	. . 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 3193	. 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 30940	. 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 3060  dom cdm 5638  ⟶wf 6497  ‘cfv 6501  (class class class)co 7362  ℂcc 11058   · cmul 11065  ∗ccj 14993   ℋchba 29924   ·_ℎ csm 29926   ·_ih csp 29927   ·_op chot 29944  adj_ℎcado 29960

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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137  ax-hilex 30004  ax-hfvadd 30005  ax-hvcom 30006  ax-hvass 30007  ax-hv0cl 30008  ax-hvaddid 30009  ax-hfvmul 30010  ax-hvmulid 30011  ax-hvdistr2 30014  ax-hvmul0 30015  ax-hfi 30084  ax-his1 30087  ax-his2 30088  ax-his3 30089  ax-his4 30090

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-po 5550  df-so 5551  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-div 11822  df-2 12225  df-cj 14996  df-re 14997  df-im 14998  df-hvsub 29976  df-homul 30736  df-adjh 30854

This theorem is referenced by: (None)