Theorem adjmul 32353

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 32149	. . 3 ⊢ (𝑇 ∈ dom adjℎ → 𝑇: ℋ⟶ ℋ)
2		homulcl 32020	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
3	1, 2	sylan2 604	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
4		cjcl 15146	. . 3 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ)
5		dmadjrn 32156	. . . 4 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ)
6		dmadjop 32149	. . . 4 ⊢ ((adjℎ‘𝑇) ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
7	5, 6	syl 18	. . 3 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
8		homulcl 32020	. . 3 ⊢ (((∗‘𝐴) ∈ ℂ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ) → ((∗‘𝐴) ·op (adjℎ‘𝑇)): ℋ⟶ ℋ)
9	4, 7, 8	syl2an 607	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) → ((∗‘𝐴) ·op (adjℎ‘𝑇)): ℋ⟶ ℋ)
10		adj2 32195	. . . . . . . 8 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
11	10	3expb 1136	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
12	11	adantll 726	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
13	12	oveq2d 7416	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)) = (𝐴 · (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦))))
14	1	ffvelcdmda 7069	. . . . . . . . 9 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
15		ax-his3 31345	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)))
16	14, 15	syl3an2 1180	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)))
17	16	3exp 1135	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → (𝑦 ∈ ℋ → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)))))
18	17	expd 420	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑇 ∈ dom adjℎ → (𝑥 ∈ ℋ → (𝑦 ∈ ℋ → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦))))))
19	18	imp43 432	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)))
20		simpll 778	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝐴 ∈ ℂ)
21		simprl 782	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
22		adjcl 32193	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ) → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ)
23	22	ad2ant2l 758	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ)
24		his52 31348	. . . . . 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 1394	. . . . 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 2810	. . . 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 32002	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
28	1, 27	syl3an2 1180	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
29	28	3expa 1134	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
30	29	adantrr 729	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
31	30	oveq1d 7415	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑦) = ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦))
32		id 23	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → 𝑦 ∈ ℋ)
33		homval 32002	. . . . . . . 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 1176	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ) → (((∗‘𝐴) ·op (adjℎ‘𝑇))‘𝑦) = ((∗‘𝐴) ·ℎ ((adjℎ‘𝑇)‘𝑦)))
35	34	3expa 1134	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ 𝑦 ∈ ℋ) → (((∗‘𝐴) ·op (adjℎ‘𝑇))‘𝑦) = ((∗‘𝐴) ·ℎ ((adjℎ‘𝑇)‘𝑦)))
36	35	adantrl 728	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((∗‘𝐴) ·op (adjℎ‘𝑇))‘𝑦) = ((∗‘𝐴) ·ℎ ((adjℎ‘𝑇)‘𝑦)))
37	36	oveq2d 7416	. . . 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 2810	. . 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 3208	. 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 32196	. 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 1394	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝐴 ·op 𝑇)) = ((∗‘𝐴) ·op (adjℎ‘𝑇)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  dom cdm 5652  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  ℂcc 11086   · cmul 11093  ∗ccj 15137   ℋchba 31180   ·_ℎ csm 31182   ·_ih csp 31183   ·_op chot 31200  adj_ℎcado 31216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-hilex 31260  ax-hfvadd 31261  ax-hvcom 31262  ax-hvass 31263  ax-hv0cl 31264  ax-hvaddid 31265  ax-hfvmul 31266  ax-hvmulid 31267  ax-hvdistr2 31270  ax-hvmul0 31271  ax-hfi 31340  ax-his1 31343  ax-his2 31344  ax-his3 31345  ax-his4 31346

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-cj 15140  df-re 15141  df-im 15142  df-hvsub 31232  df-homul 31992  df-adjh 32110

This theorem is referenced by: (None)