Theorem homulcl 22339

Description: The scalar product of a Hilbert space operator is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
homulcl	|- ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) : ~H --> ~H )

Proof of Theorem homulcl

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffvelrn 5663	. . . . 5 |- ( ( T : ~H --> ~H /\ x e. ~H ) -> ( T ` x ) e. ~H )
2		hvmulcl 21593	. . . . 5 |- ( ( A e. CC /\ ( T ` x ) e. ~H ) -> ( A .h ( T ` x ) ) e. ~H )
3	1, 2	sylan2 460	. . . 4 |- ( ( A e. CC /\ ( T : ~H --> ~H /\ x e. ~H ) ) -> ( A .h ( T ` x ) ) e. ~H )
4	3	anassrs 629	. . 3 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( T ` x ) ) e. ~H )
5		eqid 2283	. . 3 |- ( x e. ~H |-> ( A .h ( T ` x ) ) ) = ( x e. ~H |-> ( A .h ( T ` x ) ) )
6	4, 5	fmptd 5684	. 2 |- ( ( A e. CC /\ T : ~H --> ~H ) -> ( x e. ~H |-> ( A .h ( T ` x ) ) ) : ~H --> ~H )
7		hommval 22316	. . 3 |- ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) = ( x e. ~H |-> ( A .h ( T ` x ) ) ) )
8	7	feq1d 5379	. 2 |- ( ( A e. CC /\ T : ~H --> ~H ) -> ( ( A .op T ) : ~H --> ~H <-> ( x e. ~H |-> ( A .h ( T ` x ) ) ) : ~H --> ~H ) )
9	6, 8	mpbird 223	1 |- ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) : ~H --> ~H )

Syntax hints:   -> wi 4   /\ wa 358   e. wcel 1684   e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   ~Hchil 21499   .h csm 21501   .op chot 21519

This theorem is referenced by:  honegsubi  22376  homulid2  22380  homco1  22381  homulass  22382  hoadddi  22383  hoadddir  22384  hosubneg  22387  hosubdi  22388  honegsubdi  22390  honegsubdi2  22391  hosub4  22393  hosubsub4  22398  hosubeq0i  22406  nmopnegi  22545  homco2  22557  lnopmi  22580  hmopm  22601  nmophmi  22611  adjmul  22672  opsqrlem1  22720  opsqrlem6  22725

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-hilex 21579  ax-hfvmul 21585

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-homul 22311