Theorem prsrlem1 8059

Description: Decomposing signed reals into positive reals. Lemma for addsrpr 8062 and mulsrpr 8063. (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 8052	. . . . . 6 |- ~R Er ( P. X. P. )
2		erdm 6779	. . . . . 6 |- ( ~R Er ( P. X. P. ) -> dom ~R = ( P. X. P. ) )
3	1, 2	ax-mp 5	. . . . 5 |- dom ~R = ( P. X. P. )
4		simprll 539	. . . . . 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 527	. . . . . 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 2312	. . . . 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 6841	. . . . 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 414	. . . 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 4781	. . . 4 |- ( <. w , v >. e. ( P. X. P. ) <-> ( w e. P. /\ v e. P. ) )
10	8, 9	sylib 122	. . 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 541	. . . . . 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 2312	. . . . 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 6841	. . . . 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 414	. . . 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 4781	. . . 4 |- ( <. s , f >. e. ( P. X. P. ) <-> ( s e. P. /\ f e. P. ) )
16	14, 15	sylib 122	. . 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 306	. 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 540	. . . . . 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 529	. . . . . 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 2312	. . . . 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 6841	. . . . 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 414	. . . 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 4781	. . . 4 |- ( <. u , t >. e. ( P. X. P. ) <-> ( u e. P. /\ t e. P. ) )
24	22, 23	sylib 122	. . 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 542	. . . . . 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 2312	. . . . 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 6841	. . . . 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 414	. . . 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 4781	. . . 4 |- ( <. g , h >. e. ( P. X. P. ) <-> ( g e. P. /\ h e. P. ) )
30	28, 29	sylib 122	. . 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 306	. 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 2269	. . . . 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 6815	. . . . 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 167	. . . 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 8043	. . . . . 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 6866	. . . . 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 411	. . . 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 147	. . 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 2269	. . . . 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 6815	. . . . 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 167	. . . 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 6866	. . . . 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 411	. . . 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 147	. . 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 306	. 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 309	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 104   <-> wb 105   = wceq 1398   e. wcel 2205   <.cop 3694   class class class wbr 4111   X. cxp 4749   dom cdm 4751  (class class class)co 6052   Er wer 6766   [cec 6767   /.cqs 6768   P.cnp 7608   +P. cpp 7610   ~R cer 7613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-1o 6649  df-2o 6650  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7621  df-pli 7622  df-mi 7623  df-lti 7624  df-plpq 7661  df-mpq 7662  df-enq 7664  df-nqqs 7665  df-plqqs 7666  df-mqqs 7667  df-1nqqs 7668  df-rq 7669  df-ltnqqs 7670  df-enq0 7741  df-nq0 7742  df-0nq0 7743  df-plq0 7744  df-mq0 7745  df-inp 7783  df-iplp 7785  df-enr 8043

This theorem is referenced by:  addsrmo  8060  mulsrmo  8061