Theorem hosval 28599

Description: Value of the sum of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
hosval	|- ( ( S : ~H --> ~H /\ T : ~H --> ~H /\ A e. ~H ) -> ( ( S +op T ) ` A ) = ( ( S ` A ) +h ( T ` A ) ) )

Proof of Theorem hosval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hosmval 28594	. . . 4 |- ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( S +op T ) = ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) )
2	1	fveq1d 6193	. . 3 |- ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( ( S +op T ) ` A ) = ( ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) ` A ) )
3		fveq2 6191	. . . . 5 |- ( x = A -> ( S ` x ) = ( S ` A ) )
4		fveq2 6191	. . . . 5 |- ( x = A -> ( T ` x ) = ( T ` A ) )
5	3, 4	oveq12d 6668	. . . 4 |- ( x = A -> ( ( S ` x ) +h ( T ` x ) ) = ( ( S ` A ) +h ( T ` A ) ) )
6		eqid 2622	. . . 4 |- ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) = ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) )
7		ovex 6678	. . . 4 |- ( ( S ` A ) +h ( T ` A ) ) e. _V
8	5, 6, 7	fvmpt 6282	. . 3 |- ( A e. ~H -> ( ( x e. ~H |-> ( ( S ` x ) +h ( T ` x ) ) ) ` A ) = ( ( S ` A ) +h ( T ` A ) ) )
9	2, 8	sylan9eq 2676	. 2 |- ( ( ( S : ~H --> ~H /\ T : ~H --> ~H ) /\ A e. ~H ) -> ( ( S +op T ) ` A ) = ( ( S ` A ) +h ( T ` A ) ) )
10	9	3impa 1259	1 |- ( ( S : ~H --> ~H /\ T : ~H --> ~H /\ A e. ~H ) -> ( ( S +op T ) ` A ) = ( ( S ` A ) +h ( T ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ~Hchil 27776   +h cva 27777   +op chos 27795

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-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-reu 2919  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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-hosum 28589

This theorem is referenced by:  hoscl  28604  hoaddcomi  28631  hodsi  28634  hoaddassi  28635  hocadddiri  28638  hoaddid1i  28645  honegsubi  28655  hoadddi  28662  hoadddir  28663  lnophsi  28860  hmops  28879  adjadd  28952  nmoptrii  28953  leopadd  28991  pjsdii  29014  pjscji  29029  pjtoi  29038