Theorem homulcl 23219

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 5831	. . . . 5 |- ( ( T : ~H --> ~H /\ x e. ~H ) -> ( T ` x ) e. ~H )
2		hvmulcl 22473	. . . . 5 |- ( ( A e. CC /\ ( T ` x ) e. ~H ) -> ( A .h ( T ` x ) ) e. ~H )
3	1, 2	sylan2 461	. . . 4 |- ( ( A e. CC /\ ( T : ~H --> ~H /\ x e. ~H ) ) -> ( A .h ( T ` x ) ) e. ~H )
4	3	anassrs 630	. . 3 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( T ` x ) ) e. ~H )
5		eqid 2408	. . 3 |- ( x e. ~H |-> ( A .h ( T ` x ) ) ) = ( x e. ~H |-> ( A .h ( T ` x ) ) )
6	4, 5	fmptd 5856	. 2 |- ( ( A e. CC /\ T : ~H --> ~H ) -> ( x e. ~H |-> ( A .h ( T ` x ) ) ) : ~H --> ~H )
7		hommval 23196	. . 3 |- ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) = ( x e. ~H |-> ( A .h ( T ` x ) ) ) )
8	7	feq1d 5543	. 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 224	1 |- ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) : ~H --> ~H )

Syntax hints:   -> wi 4   /\ wa 359   e. wcel 1721   e. cmpt 4230   -->wf 5413   ` cfv 5417  (class class class)co 6044   CCcc 8948   ~Hchil 22379   .h csm 22381   .op chot 22399

This theorem is referenced by:  honegsubi  23256  homulid2  23260  homco1  23261  homulass  23262  hoadddi  23263  hoadddir  23264  hosubneg  23267  hosubdi  23268  honegsubdi  23270  honegsubdi2  23271  hosub4  23273  hosubsub4  23278  hosubeq0i  23286  nmopnegi  23425  homco2  23437  lnopmi  23460  hmopm  23481  nmophmi  23491  adjmul  23552  opsqrlem1  23600  opsqrlem6  23605

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-hilex 22459  ax-hfvmul 22465

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-map 6983  df-homul 23191