Theorem prsrlem1 9893

Description: Decomposing signed reals into positive reals. Lemma for addsrpr 9896 and mulsrpr 9897. (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 9886	. . . . . 6 |- ~R Er ( P. X. P. )
2		erdm 7752	. . . . . 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 802	. . . . . 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 790	. . . . . 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 2702	. . . . 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 7817	. . . . 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 695	. . . 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 5146	. . . 4 |- ( <. w , v >. e. ( P. X. P. ) <-> ( w e. P. /\ v e. P. ) )
10	8, 9	sylib 208	. . 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 804	. . . . . 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 2702	. . . . 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 7817	. . . . 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 695	. . . 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 5146	. . . 4 |- ( <. s , f >. e. ( P. X. P. ) <-> ( s e. P. /\ f e. P. ) )
16	14, 15	sylib 208	. . 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 554	. 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 803	. . . . . 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 792	. . . . . 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 2702	. . . . 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 7817	. . . . 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 695	. . . 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 5146	. . . 4 |- ( <. u , t >. e. ( P. X. P. ) <-> ( u e. P. /\ t e. P. ) )
24	22, 23	sylib 208	. . 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 805	. . . . . 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 2702	. . . . 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 7817	. . . . 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 695	. . . 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 5146	. . . 4 |- ( <. g , h >. e. ( P. X. P. ) <-> ( g e. P. /\ h e. P. ) )
30	28, 29	sylib 208	. . 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 554	. 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 2658	. . . . 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 11	. . . . . 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 7791	. . . . 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 247	. . . 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 9877	. . . . . 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 7848	. . . . 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 693	. . . 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 222	. . 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 2658	. . . . 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 7791	. . . . 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 247	. . . 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 7848	. . . . 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 693	. . . 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 222	. . 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 554	. 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 557	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   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   X. cxp 5112   dom cdm 5114  (class class class)co 6650   Er wer 7739   [cec 7740   /.cqs 7741   P.cnp 9681   +P. cpp 9683   ~R cer 9686

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

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-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805  df-ltp 9807  df-enr 9877

This theorem is referenced by:  addsrmo  9894  mulsrmo  9895