Theorem adj1 28792

Description: Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
adj1	|- ( ( T e. dom adjh /\ A e. ~H /\ B e. ~H ) -> ( A .ih ( T ` B ) ) = ( ( ( adjh ` T ) ` A ) .ih B ) )

Proof of Theorem adj1

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

Step	Hyp	Ref	Expression
1		funadj 28745	. . . . . . 7 |- Fun adjh
2		funfvop 6329	. . . . . . 7 |- ( ( Fun adjh /\ T e. dom adjh ) -> <. T , ( adjh ` T ) >. e. adjh )
3	1, 2	mpan 706	. . . . . 6 |- ( T e. dom adjh -> <. T , ( adjh ` T ) >. e. adjh )
4		dfadj2 28744	. . . . . 6 |- adjh = { <. z , w >. | ( z : ~H --> ~H /\ w : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( z ` y ) ) = ( ( w ` x ) .ih y ) ) }
5	3, 4	syl6eleq 2711	. . . . 5 |- ( T e. dom adjh -> <. T , ( adjh ` T ) >. e. { <. z , w >. | ( z : ~H --> ~H /\ w : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( z ` y ) ) = ( ( w ` x ) .ih y ) ) } )
6		fvex 6201	. . . . . 6 |- ( adjh ` T ) e. _V
7		feq1 6026	. . . . . . . 8 |- ( z = T -> ( z : ~H --> ~H <-> T : ~H --> ~H ) )
8		fveq1 6190	. . . . . . . . . . 11 |- ( z = T -> ( z ` y ) = ( T ` y ) )
9	8	oveq2d 6666	. . . . . . . . . 10 |- ( z = T -> ( x .ih ( z ` y ) ) = ( x .ih ( T ` y ) ) )
10	9	eqeq1d 2624	. . . . . . . . 9 |- ( z = T -> ( ( x .ih ( z ` y ) ) = ( ( w ` x ) .ih y ) <-> ( x .ih ( T ` y ) ) = ( ( w ` x ) .ih y ) ) )
11	10	2ralbidv 2989	. . . . . . . 8 |- ( z = T -> ( A. x e. ~H A. y e. ~H ( x .ih ( z ` y ) ) = ( ( w ` x ) .ih y ) <-> A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( w ` x ) .ih y ) ) )
12	7, 11	3anbi13d 1401	. . . . . . 7 |- ( z = T -> ( ( z : ~H --> ~H /\ w : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( z ` y ) ) = ( ( w ` x ) .ih y ) ) <-> ( T : ~H --> ~H /\ w : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( w ` x ) .ih y ) ) ) )
13		feq1 6026	. . . . . . . 8 |- ( w = ( adjh ` T ) -> ( w : ~H --> ~H <-> ( adjh ` T ) : ~H --> ~H ) )
14		fveq1 6190	. . . . . . . . . . 11 |- ( w = ( adjh ` T ) -> ( w ` x ) = ( ( adjh ` T ) ` x ) )
15	14	oveq1d 6665	. . . . . . . . . 10 |- ( w = ( adjh ` T ) -> ( ( w ` x ) .ih y ) = ( ( ( adjh ` T ) ` x ) .ih y ) )
16	15	eqeq2d 2632	. . . . . . . . 9 |- ( w = ( adjh ` T ) -> ( ( x .ih ( T ` y ) ) = ( ( w ` x ) .ih y ) <-> ( x .ih ( T ` y ) ) = ( ( ( adjh ` T ) ` x ) .ih y ) ) )
17	16	2ralbidv 2989	. . . . . . . 8 |- ( w = ( adjh ` T ) -> ( A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( w ` x ) .ih y ) <-> A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( ( adjh ` T ) ` x ) .ih y ) ) )
18	13, 17	3anbi23d 1402	. . . . . . 7 |- ( w = ( adjh ` T ) -> ( ( T : ~H --> ~H /\ w : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( w ` x ) .ih y ) ) <-> ( T : ~H --> ~H /\ ( adjh ` T ) : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( ( adjh ` T ) ` x ) .ih y ) ) ) )
19	12, 18	opelopabg 4993	. . . . . 6 |- ( ( T e. dom adjh /\ ( adjh ` T ) e. _V ) -> ( <. T , ( adjh ` T ) >. e. { <. z , w >. | ( z : ~H --> ~H /\ w : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( z ` y ) ) = ( ( w ` x ) .ih y ) ) } <-> ( T : ~H --> ~H /\ ( adjh ` T ) : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( ( adjh ` T ) ` x ) .ih y ) ) ) )
20	6, 19	mpan2 707	. . . . 5 |- ( T e. dom adjh -> ( <. T , ( adjh ` T ) >. e. { <. z , w >. | ( z : ~H --> ~H /\ w : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( z ` y ) ) = ( ( w ` x ) .ih y ) ) } <-> ( T : ~H --> ~H /\ ( adjh ` T ) : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( ( adjh ` T ) ` x ) .ih y ) ) ) )
21	5, 20	mpbid 222	. . . 4 |- ( T e. dom adjh -> ( T : ~H --> ~H /\ ( adjh ` T ) : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( ( adjh ` T ) ` x ) .ih y ) ) )
22	21	simp3d 1075	. . 3 |- ( T e. dom adjh -> A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( ( adjh ` T ) ` x ) .ih y ) )
23		oveq1 6657	. . . . 5 |- ( x = A -> ( x .ih ( T ` y ) ) = ( A .ih ( T ` y ) ) )
24		fveq2 6191	. . . . . 6 |- ( x = A -> ( ( adjh ` T ) ` x ) = ( ( adjh ` T ) ` A ) )
25	24	oveq1d 6665	. . . . 5 |- ( x = A -> ( ( ( adjh ` T ) ` x ) .ih y ) = ( ( ( adjh ` T ) ` A ) .ih y ) )
26	23, 25	eqeq12d 2637	. . . 4 |- ( x = A -> ( ( x .ih ( T ` y ) ) = ( ( ( adjh ` T ) ` x ) .ih y ) <-> ( A .ih ( T ` y ) ) = ( ( ( adjh ` T ) ` A ) .ih y ) ) )
27		fveq2 6191	. . . . . 6 |- ( y = B -> ( T ` y ) = ( T ` B ) )
28	27	oveq2d 6666	. . . . 5 |- ( y = B -> ( A .ih ( T ` y ) ) = ( A .ih ( T ` B ) ) )
29		oveq2 6658	. . . . 5 |- ( y = B -> ( ( ( adjh ` T ) ` A ) .ih y ) = ( ( ( adjh ` T ) ` A ) .ih B ) )
30	28, 29	eqeq12d 2637	. . . 4 |- ( y = B -> ( ( A .ih ( T ` y ) ) = ( ( ( adjh ` T ) ` A ) .ih y ) <-> ( A .ih ( T ` B ) ) = ( ( ( adjh ` T ) ` A ) .ih B ) ) )
31	26, 30	rspc2v 3322	. . 3 |- ( ( A e. ~H /\ B e. ~H ) -> ( A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( ( adjh ` T ) ` x ) .ih y ) -> ( A .ih ( T ` B ) ) = ( ( ( adjh ` T ) ` A ) .ih B ) ) )
32	22, 31	syl5com 31	. 2 |- ( T e. dom adjh -> ( ( A e. ~H /\ B e. ~H ) -> ( A .ih ( T ` B ) ) = ( ( ( adjh ` T ) ` A ) .ih B ) ) )
33	32	3impib 1262	1 |- ( ( T e. dom adjh /\ A e. ~H /\ B e. ~H ) -> ( A .ih ( T ` B ) ) = ( ( ( adjh ` T ) ` A ) .ih B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.cop 4183   {copab 4712   dom cdm 5114   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   ~Hchil 27776   .ih csp 27779   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-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-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-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-adjh 28708

This theorem is referenced by:  adj2  28793  adjadj  28795  hmopadj2  28800