Theorem hocsubdiri 28639

Description: Distributive law for Hilbert space operator difference. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hods.1	|- R : ~H --> ~H
hods.2	|- S : ~H --> ~H
hods.3	|- T : ~H --> ~H

Assertion

Ref	Expression
hocsubdiri	|- ( ( R -op S ) o. T ) = ( ( R o. T ) -op ( S o. T ) )

Proof of Theorem hocsubdiri

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hods.1	. . . . . 6 |- R : ~H --> ~H
2		hods.2	. . . . . 6 |- S : ~H --> ~H
3	1, 2	hosubcli 28628	. . . . 5 |- ( R -op S ) : ~H --> ~H
4		hods.3	. . . . 5 |- T : ~H --> ~H
5	3, 4	hocoi 28623	. . . 4 |- ( x e. ~H -> ( ( ( R -op S ) o. T ) ` x ) = ( ( R -op S ) ` ( T ` x ) ) )
6	1, 4	hocofi 28625	. . . . . 6 |- ( R o. T ) : ~H --> ~H
7	2, 4	hocofi 28625	. . . . . 6 |- ( S o. T ) : ~H --> ~H
8		hodval 28601	. . . . . 6 |- ( ( ( R o. T ) : ~H --> ~H /\ ( S o. T ) : ~H --> ~H /\ x e. ~H ) -> ( ( ( R o. T ) -op ( S o. T ) ) ` x ) = ( ( ( R o. T ) ` x ) -h ( ( S o. T ) ` x ) ) )
9	6, 7, 8	mp3an12 1414	. . . . 5 |- ( x e. ~H -> ( ( ( R o. T ) -op ( S o. T ) ) ` x ) = ( ( ( R o. T ) ` x ) -h ( ( S o. T ) ` x ) ) )
10	4	ffvelrni 6358	. . . . . . 7 |- ( x e. ~H -> ( T ` x ) e. ~H )
11		hodval 28601	. . . . . . . 8 |- ( ( R : ~H --> ~H /\ S : ~H --> ~H /\ ( T ` x ) e. ~H ) -> ( ( R -op S ) ` ( T ` x ) ) = ( ( R ` ( T ` x ) ) -h ( S ` ( T ` x ) ) ) )
12	1, 2, 11	mp3an12 1414	. . . . . . 7 |- ( ( T ` x ) e. ~H -> ( ( R -op S ) ` ( T ` x ) ) = ( ( R ` ( T ` x ) ) -h ( S ` ( T ` x ) ) ) )
13	10, 12	syl 17	. . . . . 6 |- ( x e. ~H -> ( ( R -op S ) ` ( T ` x ) ) = ( ( R ` ( T ` x ) ) -h ( S ` ( T ` x ) ) ) )
14	1, 4	hocoi 28623	. . . . . . 7 |- ( x e. ~H -> ( ( R o. T ) ` x ) = ( R ` ( T ` x ) ) )
15	2, 4	hocoi 28623	. . . . . . 7 |- ( x e. ~H -> ( ( S o. T ) ` x ) = ( S ` ( T ` x ) ) )
16	14, 15	oveq12d 6668	. . . . . 6 |- ( x e. ~H -> ( ( ( R o. T ) ` x ) -h ( ( S o. T ) ` x ) ) = ( ( R ` ( T ` x ) ) -h ( S ` ( T ` x ) ) ) )
17	13, 16	eqtr4d 2659	. . . . 5 |- ( x e. ~H -> ( ( R -op S ) ` ( T ` x ) ) = ( ( ( R o. T ) ` x ) -h ( ( S o. T ) ` x ) ) )
18	9, 17	eqtr4d 2659	. . . 4 |- ( x e. ~H -> ( ( ( R o. T ) -op ( S o. T ) ) ` x ) = ( ( R -op S ) ` ( T ` x ) ) )
19	5, 18	eqtr4d 2659	. . 3 |- ( x e. ~H -> ( ( ( R -op S ) o. T ) ` x ) = ( ( ( R o. T ) -op ( S o. T ) ) ` x ) )
20	19	rgen 2922	. 2 |- A. x e. ~H ( ( ( R -op S ) o. T ) ` x ) = ( ( ( R o. T ) -op ( S o. T ) ) ` x )
21	3, 4	hocofi 28625	. . 3 |- ( ( R -op S ) o. T ) : ~H --> ~H
22	6, 7	hosubcli 28628	. . 3 |- ( ( R o. T ) -op ( S o. T ) ) : ~H --> ~H
23	21, 22	hoeqi 28620	. 2 |- ( A. x e. ~H ( ( ( R -op S ) o. T ) ` x ) = ( ( ( R o. T ) -op ( S o. T ) ) ` x ) <-> ( ( R -op S ) o. T ) = ( ( R o. T ) -op ( S o. T ) ) )
24	20, 23	mpbi 220	1 |- ( ( R -op S ) o. T ) = ( ( R o. T ) -op ( S o. T ) )

Syntax hints:   = wceq 1483   e. wcel 1990   A.wral 2912   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   ~Hchil 27776   -h cmv 27782   -op chod 27797

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-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-hilex 27856  ax-hfvadd 27857  ax-hfvmul 27862

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-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-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269  df-hvsub 27828  df-hodif 28591

This theorem is referenced by:  hocsubdir  28644  unierri  28963  pjclem3  29056