Theorem honegsubi 28655

Description: Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hodseq.2	|- S : ~H --> ~H
hodseq.3	|- T : ~H --> ~H

Assertion

Ref	Expression
honegsubi	|- ( S +op ( -u 1 .op T ) ) = ( S -op T )

Proof of Theorem honegsubi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hodseq.2	. . . . . 6 |- S : ~H --> ~H
2		neg1cn 11124	. . . . . . 7 |- -u 1 e. CC
3		hodseq.3	. . . . . . 7 |- T : ~H --> ~H
4		homulcl 28618	. . . . . . 7 |- ( ( -u 1 e. CC /\ T : ~H --> ~H ) -> ( -u 1 .op T ) : ~H --> ~H )
5	2, 3, 4	mp2an 708	. . . . . 6 |- ( -u 1 .op T ) : ~H --> ~H
6		hosval 28599	. . . . . 6 |- ( ( S : ~H --> ~H /\ ( -u 1 .op T ) : ~H --> ~H /\ x e. ~H ) -> ( ( S +op ( -u 1 .op T ) ) ` x ) = ( ( S ` x ) +h ( ( -u 1 .op T ) ` x ) ) )
7	1, 5, 6	mp3an12 1414	. . . . 5 |- ( x e. ~H -> ( ( S +op ( -u 1 .op T ) ) ` x ) = ( ( S ` x ) +h ( ( -u 1 .op T ) ` x ) ) )
8	1	ffvelrni 6358	. . . . . . 7 |- ( x e. ~H -> ( S ` x ) e. ~H )
9	3	ffvelrni 6358	. . . . . . 7 |- ( x e. ~H -> ( T ` x ) e. ~H )
10		hvsubval 27873	. . . . . . 7 |- ( ( ( S ` x ) e. ~H /\ ( T ` x ) e. ~H ) -> ( ( S ` x ) -h ( T ` x ) ) = ( ( S ` x ) +h ( -u 1 .h ( T ` x ) ) ) )
11	8, 9, 10	syl2anc 693	. . . . . 6 |- ( x e. ~H -> ( ( S ` x ) -h ( T ` x ) ) = ( ( S ` x ) +h ( -u 1 .h ( T ` x ) ) ) )
12		homval 28600	. . . . . . . 8 |- ( ( -u 1 e. CC /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( -u 1 .op T ) ` x ) = ( -u 1 .h ( T ` x ) ) )
13	2, 3, 12	mp3an12 1414	. . . . . . 7 |- ( x e. ~H -> ( ( -u 1 .op T ) ` x ) = ( -u 1 .h ( T ` x ) ) )
14	13	oveq2d 6666	. . . . . 6 |- ( x e. ~H -> ( ( S ` x ) +h ( ( -u 1 .op T ) ` x ) ) = ( ( S ` x ) +h ( -u 1 .h ( T ` x ) ) ) )
15	11, 14	eqtr4d 2659	. . . . 5 |- ( x e. ~H -> ( ( S ` x ) -h ( T ` x ) ) = ( ( S ` x ) +h ( ( -u 1 .op T ) ` x ) ) )
16	7, 15	eqtr4d 2659	. . . 4 |- ( x e. ~H -> ( ( S +op ( -u 1 .op T ) ) ` x ) = ( ( S ` x ) -h ( T ` x ) ) )
17		hodval 28601	. . . . 5 |- ( ( S : ~H --> ~H /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( S -op T ) ` x ) = ( ( S ` x ) -h ( T ` x ) ) )
18	1, 3, 17	mp3an12 1414	. . . 4 |- ( x e. ~H -> ( ( S -op T ) ` x ) = ( ( S ` x ) -h ( T ` x ) ) )
19	16, 18	eqtr4d 2659	. . 3 |- ( x e. ~H -> ( ( S +op ( -u 1 .op T ) ) ` x ) = ( ( S -op T ) ` x ) )
20	19	rgen 2922	. 2 |- A. x e. ~H ( ( S +op ( -u 1 .op T ) ) ` x ) = ( ( S -op T ) ` x )
21	1, 5	hoaddcli 28627	. . 3 |- ( S +op ( -u 1 .op T ) ) : ~H --> ~H
22	1, 3	hosubcli 28628	. . 3 |- ( S -op T ) : ~H --> ~H
23	21, 22	hoeqi 28620	. 2 |- ( A. x e. ~H ( ( S +op ( -u 1 .op T ) ) ` x ) = ( ( S -op T ) ` x ) <-> ( S +op ( -u 1 .op T ) ) = ( S -op T ) )
24	20, 23	mpbi 220	1 |- ( S +op ( -u 1 .op T ) ) = ( S -op T )

Syntax hints:   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   -ucneg 10267   ~Hchil 27776   +h cva 27777   .h csm 27778   -h cmv 27782   +op chos 27795   .op chot 27796   -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-hosum 28589  df-homul 28590  df-hodif 28591

This theorem is referenced by:  honegsub  28658  hosubeq0i  28685  lnophdi  28861  bdophdi  28956  nmoptri2i  28958