Theorem hoaddcomi 28631

Description: Commutativity of sum of Hilbert space operators. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hoeq.1	|- S : ~H --> ~H
hoeq.2	|- T : ~H --> ~H

Assertion

Ref	Expression
hoaddcomi	|- ( S +op T ) = ( T +op S )

Proof of Theorem hoaddcomi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hoeq.1	. . . . . 6 |- S : ~H --> ~H
2	1	ffvelrni 6358	. . . . 5 |- ( x e. ~H -> ( S ` x ) e. ~H )
3		hoeq.2	. . . . . 6 |- T : ~H --> ~H
4	3	ffvelrni 6358	. . . . 5 |- ( x e. ~H -> ( T ` x ) e. ~H )
5		ax-hvcom 27858	. . . . 5 |- ( ( ( S ` x ) e. ~H /\ ( T ` x ) e. ~H ) -> ( ( S ` x ) +h ( T ` x ) ) = ( ( T ` x ) +h ( S ` x ) ) )
6	2, 4, 5	syl2anc 693	. . . 4 |- ( x e. ~H -> ( ( S ` x ) +h ( T ` x ) ) = ( ( T ` x ) +h ( S ` x ) ) )
7		hosval 28599	. . . . 5 |- ( ( S : ~H --> ~H /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( S +op T ) ` x ) = ( ( S ` x ) +h ( T ` x ) ) )
8	1, 3, 7	mp3an12 1414	. . . 4 |- ( x e. ~H -> ( ( S +op T ) ` x ) = ( ( S ` x ) +h ( T ` x ) ) )
9		hosval 28599	. . . . 5 |- ( ( T : ~H --> ~H /\ S : ~H --> ~H /\ x e. ~H ) -> ( ( T +op S ) ` x ) = ( ( T ` x ) +h ( S ` x ) ) )
10	3, 1, 9	mp3an12 1414	. . . 4 |- ( x e. ~H -> ( ( T +op S ) ` x ) = ( ( T ` x ) +h ( S ` x ) ) )
11	6, 8, 10	3eqtr4d 2666	. . 3 |- ( x e. ~H -> ( ( S +op T ) ` x ) = ( ( T +op S ) ` x ) )
12	11	rgen 2922	. 2 |- A. x e. ~H ( ( S +op T ) ` x ) = ( ( T +op S ) ` x )
13	1, 3	hoaddcli 28627	. . 3 |- ( S +op T ) : ~H --> ~H
14	3, 1	hoaddcli 28627	. . 3 |- ( T +op S ) : ~H --> ~H
15	13, 14	hoeqi 28620	. 2 |- ( A. x e. ~H ( ( S +op T ) ` x ) = ( ( T +op S ) ` x ) <-> ( S +op T ) = ( T +op S ) )
16	12, 15	mpbi 220	1 |- ( S +op T ) = ( T +op S )

Syntax hints:   = wceq 1483   e. wcel 1990   A.wral 2912   -->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  ax-hfvadd 27857  ax-hvcom 27858

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:  hoaddcom  28633  hoadd12i  28636  hoadd32i  28637  hoaddsubi  28680  hosd1i  28681  hosubeq0i  28685