Theorem adjmul 32163

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 31959	. . 3 ⊢ (𝑇 ∈ dom adjℎ → 𝑇: ℋ⟶ ℋ)
2		homulcl 31830	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
3	1, 2	sylan2 594	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
4		cjcl 15067	. . 3 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ)
5		dmadjrn 31966	. . . 4 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ)
6		dmadjop 31959	. . . 4 ⊢ ((adjℎ‘𝑇) ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
7	5, 6	syl 17	. . 3 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
8		homulcl 31830	. . 3 ⊢ (((∗‘𝐴) ∈ ℂ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ) → ((∗‘𝐴) ·op (adjℎ‘𝑇)): ℋ⟶ ℋ)
9	4, 7, 8	syl2an 597	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) → ((∗‘𝐴) ·op (adjℎ‘𝑇)): ℋ⟶ ℋ)
10		adj2 32005	. . . . . . . 8 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
11	10	3expb 1121	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
12	11	adantll 715	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
13	12	oveq2d 7383	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)) = (𝐴 · (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦))))
14	1	ffvelcdmda 7036	. . . . . . . . 9 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
15		ax-his3 31155	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)))
16	14, 15	syl3an2 1165	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)))
17	16	3exp 1120	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → (𝑦 ∈ ℋ → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)))))
18	17	expd 415	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑇 ∈ dom adjℎ → (𝑥 ∈ ℋ → (𝑦 ∈ ℋ → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦))))))
19	18	imp43 427	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑦) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑦)))
20		simpll 767	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝐴 ∈ ℂ)
21		simprl 771	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
22		adjcl 32003	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ) → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ)
23	22	ad2ant2l 747	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ)
24		his52 31158	. . . . . 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 1374	. . . . 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 31812	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
28	1, 27	syl3an2 1165	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
29	28	3expa 1119	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
30	29	adantrr 718	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
31	30	oveq1d 7382	. . . 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 31812	. . . . . . . 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 1161	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ) → (((∗‘𝐴) ·op (adjℎ‘𝑇))‘𝑦) = ((∗‘𝐴) ·ℎ ((adjℎ‘𝑇)‘𝑦)))
35	34	3expa 1119	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ 𝑦 ∈ ℋ) → (((∗‘𝐴) ·op (adjℎ‘𝑇))‘𝑦) = ((∗‘𝐴) ·ℎ ((adjℎ‘𝑇)‘𝑦)))
36	35	adantrl 717	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((∗‘𝐴) ·op (adjℎ‘𝑇))‘𝑦) = ((∗‘𝐴) ·ℎ ((adjℎ‘𝑇)‘𝑦)))
37	36	oveq2d 7383	. . . 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 3180	. 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 32006	. 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 1374	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝐴 ·op 𝑇)) = ((∗‘𝐴) ·op (adjℎ‘𝑇)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  dom cdm 5631  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  ℂcc 11036   · cmul 11043  ∗ccj 15058   ℋchba 30990   ·_ℎ csm 30992   ·_ih csp 30993   ·_op chot 31010  adj_ℎcado 31026

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-hilex 31070  ax-hfvadd 31071  ax-hvcom 31072  ax-hvass 31073  ax-hv0cl 31074  ax-hvaddid 31075  ax-hfvmul 31076  ax-hvmulid 31077  ax-hvdistr2 31080  ax-hvmul0 31081  ax-hfi 31150  ax-his1 31153  ax-his2 31154  ax-his3 31155  ax-his4 31156

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-cj 15061  df-re 15062  df-im 15063  df-hvsub 31042  df-homul 31802  df-adjh 31920

This theorem is referenced by: (None)