Theorem adjmul 28951

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 e. CC /\ T e. 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 28747	. . 3 |- ( T e. dom adjh -> T : ~H --> ~H )
2		homulcl 28618	. . 3 |- ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) : ~H --> ~H )
3	1, 2	sylan2 491	. 2 |- ( ( A e. CC /\ T e. dom adjh ) -> ( A .op T ) : ~H --> ~H )
4		cjcl 13845	. . 3 |- ( A e. CC -> ( * ` A ) e. CC )
5		dmadjrn 28754	. . . 4 |- ( T e. dom adjh -> ( adjh ` T ) e. dom adjh )
6		dmadjop 28747	. . . 4 |- ( ( adjh ` T ) e. dom adjh -> ( adjh ` T ) : ~H --> ~H )
7	5, 6	syl 17	. . 3 |- ( T e. dom adjh -> ( adjh ` T ) : ~H --> ~H )
8		homulcl 28618	. . 3 |- ( ( ( * ` A ) e. CC /\ ( adjh ` T ) : ~H --> ~H ) -> ( ( * ` A ) .op ( adjh ` T ) ) : ~H --> ~H )
9	4, 7, 8	syl2an 494	. 2 |- ( ( A e. CC /\ T e. dom adjh ) -> ( ( * ` A ) .op ( adjh ` T ) ) : ~H --> ~H )
10		adj2 28793	. . . . . . . 8 |- ( ( T e. dom adjh /\ x e. ~H /\ y e. ~H ) -> ( ( T ` x ) .ih y ) = ( x .ih ( ( adjh ` T ) ` y ) ) )
11	10	3expb 1266	. . . . . . 7 |- ( ( T e. dom adjh /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( T ` x ) .ih y ) = ( x .ih ( ( adjh ` T ) ` y ) ) )
12	11	adantll 750	. . . . . 6 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( T ` x ) .ih y ) = ( x .ih ( ( adjh ` T ) ` y ) ) )
13	12	oveq2d 6666	. . . . 5 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( A x. ( ( T ` x ) .ih y ) ) = ( A x. ( x .ih ( ( adjh ` T ) ` y ) ) ) )
14	1	ffvelrnda 6359	. . . . . . . . 9 |- ( ( T e. dom adjh /\ x e. ~H ) -> ( T ` x ) e. ~H )
15		ax-his3 27941	. . . . . . . . 9 |- ( ( A e. CC /\ ( T ` x ) e. ~H /\ y e. ~H ) -> ( ( A .h ( T ` x ) ) .ih y ) = ( A x. ( ( T ` x ) .ih y ) ) )
16	14, 15	syl3an2 1360	. . . . . . . 8 |- ( ( A e. CC /\ ( T e. dom adjh /\ x e. ~H ) /\ y e. ~H ) -> ( ( A .h ( T ` x ) ) .ih y ) = ( A x. ( ( T ` x ) .ih y ) ) )
17	16	3exp 1264	. . . . . . 7 |- ( A e. CC -> ( ( T e. dom adjh /\ x e. ~H ) -> ( y e. ~H -> ( ( A .h ( T ` x ) ) .ih y ) = ( A x. ( ( T ` x ) .ih y ) ) ) ) )
18	17	expd 452	. . . . . 6 |- ( A e. CC -> ( T e. dom adjh -> ( x e. ~H -> ( y e. ~H -> ( ( A .h ( T ` x ) ) .ih y ) = ( A x. ( ( T ` x ) .ih y ) ) ) ) ) )
19	18	imp43 621	. . . . 5 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( A .h ( T ` x ) ) .ih y ) = ( A x. ( ( T ` x ) .ih y ) ) )
20		simpll 790	. . . . . 6 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ ( x e. ~H /\ y e. ~H ) ) -> A e. CC )
21		simprl 794	. . . . . 6 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ ( x e. ~H /\ y e. ~H ) ) -> x e. ~H )
22		adjcl 28791	. . . . . . 7 |- ( ( T e. dom adjh /\ y e. ~H ) -> ( ( adjh ` T ) ` y ) e. ~H )
23	22	ad2ant2l 782	. . . . . 6 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( adjh ` T ) ` y ) e. ~H )
24		his52 27944	. . . . . 6 |- ( ( A e. CC /\ x e. ~H /\ ( ( adjh ` T ) ` y ) e. ~H ) -> ( x .ih ( ( * ` A ) .h ( ( adjh ` T ) ` y ) ) ) = ( A x. ( x .ih ( ( adjh ` T ) ` y ) ) ) )
25	20, 21, 23, 24	syl3anc 1326	. . . . 5 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( ( * ` A ) .h ( ( adjh ` T ) ` y ) ) ) = ( A x. ( x .ih ( ( adjh ` T ) ` y ) ) ) )
26	13, 19, 25	3eqtr4d 2666	. . . 4 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( A .h ( T ` x ) ) .ih y ) = ( x .ih ( ( * ` A ) .h ( ( adjh ` T ) ` y ) ) ) )
27		homval 28600	. . . . . . . 8 |- ( ( A e. CC /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( A .op T ) ` x ) = ( A .h ( T ` x ) ) )
28	1, 27	syl3an2 1360	. . . . . . 7 |- ( ( A e. CC /\ T e. dom adjh /\ x e. ~H ) -> ( ( A .op T ) ` x ) = ( A .h ( T ` x ) ) )
29	28	3expa 1265	. . . . . 6 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ x e. ~H ) -> ( ( A .op T ) ` x ) = ( A .h ( T ` x ) ) )
30	29	adantrr 753	. . . . 5 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( A .op T ) ` x ) = ( A .h ( T ` x ) ) )
31	30	oveq1d 6665	. . . 4 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( ( A .op T ) ` x ) .ih y ) = ( ( A .h ( T ` x ) ) .ih y ) )
32		id 22	. . . . . . . 8 |- ( y e. ~H -> y e. ~H )
33		homval 28600	. . . . . . . 8 |- ( ( ( * ` A ) e. CC /\ ( adjh ` T ) : ~H --> ~H /\ y e. ~H ) -> ( ( ( * ` A ) .op ( adjh ` T ) ) ` y ) = ( ( * ` A ) .h ( ( adjh ` T ) ` y ) ) )
34	4, 7, 32, 33	syl3an 1368	. . . . . . 7 |- ( ( A e. CC /\ T e. dom adjh /\ y e. ~H ) -> ( ( ( * ` A ) .op ( adjh ` T ) ) ` y ) = ( ( * ` A ) .h ( ( adjh ` T ) ` y ) ) )
35	34	3expa 1265	. . . . . 6 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ y e. ~H ) -> ( ( ( * ` A ) .op ( adjh ` T ) ) ` y ) = ( ( * ` A ) .h ( ( adjh ` T ) ` y ) ) )
36	35	adantrl 752	. . . . 5 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( ( * ` A ) .op ( adjh ` T ) ) ` y ) = ( ( * ` A ) .h ( ( adjh ` T ) ` y ) ) )
37	36	oveq2d 6666	. . . 4 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( ( ( * ` A ) .op ( adjh ` T ) ) ` y ) ) = ( x .ih ( ( * ` A ) .h ( ( adjh ` T ) ` y ) ) ) )
38	26, 31, 37	3eqtr4d 2666	. . 3 |- ( ( ( A e. CC /\ T e. dom adjh ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( ( A .op T ) ` x ) .ih y ) = ( x .ih ( ( ( * ` A ) .op ( adjh ` T ) ) ` y ) ) )
39	38	ralrimivva 2971	. 2 |- ( ( A e. CC /\ T e. dom adjh ) -> A. x e. ~H A. y e. ~H ( ( ( A .op T ) ` x ) .ih y ) = ( x .ih ( ( ( * ` A ) .op ( adjh ` T ) ) ` y ) ) )
40		adjeq 28794	. 2 |- ( ( ( A .op T ) : ~H --> ~H /\ ( ( * ` A ) .op ( adjh ` T ) ) : ~H --> ~H /\ A. x e. ~H A. y e. ~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 1326	1 |- ( ( A e. CC /\ T e. dom adjh ) -> ( adjh ` ( A .op T ) ) = ( ( * ` A ) .op ( adjh ` T ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   x. cmul 9941   *ccj 13836   ~Hchil 27776   .h csm 27778   .ih csp 27779   .op chot 27796   adjhcado 27812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-cj 13839  df-re 13840  df-im 13841  df-hvsub 27828  df-homul 28590  df-adjh 28708

This theorem is referenced by: (None)