Theorem prsrlem1 6919

Description: Decomposing signed reals into positive reals. Lemma for addsrpr 6922 and mulsrpr 6923. (Contributed by Jim Kingdon, 30-Dec-2019.)

Assertion

Ref	Expression
prsrlem1	|- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( ( ( ( w e. P. /\ v e. P. ) /\ ( s e. P. /\ f e. P. ) ) /\ ( ( u e. P. /\ t e. P. ) /\ ( g e. P. /\ h e. P. ) ) ) /\ ( ( w +P. f ) = ( v +P. s ) /\ ( u +P. h ) = ( t +P. g ) ) ) )

Distinct variable group:   f, g, h, s, t, u, v, w

Allowed substitution hints:   A( w, v, u, t, f, g, h, s)   B( w, v, u, t, f, g, h, s)

Proof of Theorem prsrlem1

Dummy variables a b c d x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enrer 6912	. . . . . 6 |- ~R Er ( P. X. P. )
2		erdm 6139	. . . . . 6 |- ( ~R Er ( P. X. P. ) -> dom ~R = ( P. X. P. ) )
3	1, 2	ax-mp 7	. . . . 5 |- dom ~R = ( P. X. P. )
4		simprll 503	. . . . . 6 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> A = [ <. w , v >. ] ~R )
5		simpll 495	. . . . . 6 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> A e. ( ( P. X. P. ) /. ~R ) )
6	4, 5	eqeltrrd 2156	. . . . 5 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> [ <. w , v >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
7		ecelqsdm 6199	. . . . 5 |- ( ( dom ~R = ( P. X. P. ) /\ [ <. w , v >. ] ~R e. ( ( P. X. P. ) /. ~R ) ) -> <. w , v >. e. ( P. X. P. ) )
8	3, 6, 7	sylancr 405	. . . 4 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> <. w , v >. e. ( P. X. P. ) )
9		opelxp 4392	. . . 4 |- ( <. w , v >. e. ( P. X. P. ) <-> ( w e. P. /\ v e. P. ) )
10	8, 9	sylib 120	. . 3 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( w e. P. /\ v e. P. ) )
11		simprrl 505	. . . . . 6 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> A = [ <. s , f >. ] ~R )
12	11, 5	eqeltrrd 2156	. . . . 5 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> [ <. s , f >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
13		ecelqsdm 6199	. . . . 5 |- ( ( dom ~R = ( P. X. P. ) /\ [ <. s , f >. ] ~R e. ( ( P. X. P. ) /. ~R ) ) -> <. s , f >. e. ( P. X. P. ) )
14	3, 12, 13	sylancr 405	. . . 4 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> <. s , f >. e. ( P. X. P. ) )
15		opelxp 4392	. . . 4 |- ( <. s , f >. e. ( P. X. P. ) <-> ( s e. P. /\ f e. P. ) )
16	14, 15	sylib 120	. . 3 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( s e. P. /\ f e. P. ) )
17	10, 16	jca 300	. 2 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( ( w e. P. /\ v e. P. ) /\ ( s e. P. /\ f e. P. ) ) )
18		simprlr 504	. . . . . 6 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> B = [ <. u , t >. ] ~R )
19		simplr 496	. . . . . 6 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> B e. ( ( P. X. P. ) /. ~R ) )
20	18, 19	eqeltrrd 2156	. . . . 5 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> [ <. u , t >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
21		ecelqsdm 6199	. . . . 5 |- ( ( dom ~R = ( P. X. P. ) /\ [ <. u , t >. ] ~R e. ( ( P. X. P. ) /. ~R ) ) -> <. u , t >. e. ( P. X. P. ) )
22	3, 20, 21	sylancr 405	. . . 4 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> <. u , t >. e. ( P. X. P. ) )
23		opelxp 4392	. . . 4 |- ( <. u , t >. e. ( P. X. P. ) <-> ( u e. P. /\ t e. P. ) )
24	22, 23	sylib 120	. . 3 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( u e. P. /\ t e. P. ) )
25		simprrr 506	. . . . . 6 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> B = [ <. g , h >. ] ~R )
26	25, 19	eqeltrrd 2156	. . . . 5 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> [ <. g , h >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
27		ecelqsdm 6199	. . . . 5 |- ( ( dom ~R = ( P. X. P. ) /\ [ <. g , h >. ] ~R e. ( ( P. X. P. ) /. ~R ) ) -> <. g , h >. e. ( P. X. P. ) )
28	3, 26, 27	sylancr 405	. . . 4 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> <. g , h >. e. ( P. X. P. ) )
29		opelxp 4392	. . . 4 |- ( <. g , h >. e. ( P. X. P. ) <-> ( g e. P. /\ h e. P. ) )
30	28, 29	sylib 120	. . 3 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( g e. P. /\ h e. P. ) )
31	24, 30	jca 300	. 2 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( ( u e. P. /\ t e. P. ) /\ ( g e. P. /\ h e. P. ) ) )
32	4, 11	eqtr3d 2115	. . . . 5 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> [ <. w , v >. ] ~R = [ <. s , f >. ] ~R )
33	1	a1i 9	. . . . . 6 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ~R Er ( P. X. P. ) )
34	33, 8	erth 6173	. . . . 5 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( <. w , v >. ~R <. s , f >. <-> [ <. w , v >. ] ~R = [ <. s , f >. ] ~R ) )
35	32, 34	mpbird 165	. . . 4 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> <. w , v >. ~R <. s , f >. )
36		df-enr 6903	. . . . . 6 |- ~R = { <. x , y >. | ( ( x e. ( P. X. P. ) /\ y e. ( P. X. P. ) ) /\ E. a E. b E. c E. d ( ( x = <. a , b >. /\ y = <. c , d >. ) /\ ( a +P. d ) = ( b +P. c ) ) ) }
37	36	ecopoveq 6224	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ ( s e. P. /\ f e. P. ) ) -> ( <. w , v >. ~R <. s , f >. <-> ( w +P. f ) = ( v +P. s ) ) )
38	10, 16, 37	syl2anc 403	. . . 4 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( <. w , v >. ~R <. s , f >. <-> ( w +P. f ) = ( v +P. s ) ) )
39	35, 38	mpbid 145	. . 3 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( w +P. f ) = ( v +P. s ) )
40	18, 25	eqtr3d 2115	. . . . 5 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> [ <. u , t >. ] ~R = [ <. g , h >. ] ~R )
41	33, 22	erth 6173	. . . . 5 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( <. u , t >. ~R <. g , h >. <-> [ <. u , t >. ] ~R = [ <. g , h >. ] ~R ) )
42	40, 41	mpbird 165	. . . 4 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> <. u , t >. ~R <. g , h >. )
43	36	ecopoveq 6224	. . . . 5 |- ( ( ( u e. P. /\ t e. P. ) /\ ( g e. P. /\ h e. P. ) ) -> ( <. u , t >. ~R <. g , h >. <-> ( u +P. h ) = ( t +P. g ) ) )
44	24, 30, 43	syl2anc 403	. . . 4 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( <. u , t >. ~R <. g , h >. <-> ( u +P. h ) = ( t +P. g ) ) )
45	42, 44	mpbid 145	. . 3 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( u +P. h ) = ( t +P. g ) )
46	39, 45	jca 300	. 2 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( ( w +P. f ) = ( v +P. s ) /\ ( u +P. h ) = ( t +P. g ) ) )
47	17, 31, 46	jca31 302	1 |- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( ( ( ( w e. P. /\ v e. P. ) /\ ( s e. P. /\ f e. P. ) ) /\ ( ( u e. P. /\ t e. P. ) /\ ( g e. P. /\ h e. P. ) ) ) /\ ( ( w +P. f ) = ( v +P. s ) /\ ( u +P. h ) = ( t +P. g ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   <.cop 3401   class class class wbr 3785   X. cxp 4361   dom cdm 4363  (class class class)co 5532   Er wer 6126   [cec 6127   /.cqs 6128   P.cnp 6481   +P. cpp 6483   ~R cer 6486

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-iplp 6658  df-enr 6903

This theorem is referenced by:  addsrmo  6920  mulsrmo  6921