Theorem map2psrpr 9931

Description: Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
map2psrpr.2	|- C e. R.

Assertion

Ref	Expression
map2psrpr	|- ( ( C +R -1R ) <R A <-> E. x e. P. ( C +R [ <. x , 1P >. ] ~R ) = A )

Distinct variable groups:   x, A   x, C

Proof of Theorem map2psrpr

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelsr 9889	. . . . 5 |- <R C_ ( R. X. R. )
2	1	brel 5168	. . . 4 |- ( ( C +R -1R ) <R A -> ( ( C +R -1R ) e. R. /\ A e. R. ) )
3	2	simprd 479	. . 3 |- ( ( C +R -1R ) <R A -> A e. R. )
4		map2psrpr.2	. . . . . 6 |- C e. R.
5		ltasr 9921	. . . . . 6 |- ( C e. R. -> ( -1R <R ( ( C .R -1R ) +R A ) <-> ( C +R -1R ) <R ( C +R ( ( C .R -1R ) +R A ) ) ) )
6	4, 5	ax-mp 5	. . . . 5 |- ( -1R <R ( ( C .R -1R ) +R A ) <-> ( C +R -1R ) <R ( C +R ( ( C .R -1R ) +R A ) ) )
7		pn0sr 9922	. . . . . . . . . 10 |- ( C e. R. -> ( C +R ( C .R -1R ) ) = 0R )
8	4, 7	ax-mp 5	. . . . . . . . 9 |- ( C +R ( C .R -1R ) ) = 0R
9	8	oveq1i 6660	. . . . . . . 8 |- ( ( C +R ( C .R -1R ) ) +R A ) = ( 0R +R A )
10		addasssr 9909	. . . . . . . 8 |- ( ( C +R ( C .R -1R ) ) +R A ) = ( C +R ( ( C .R -1R ) +R A ) )
11		addcomsr 9908	. . . . . . . 8 |- ( 0R +R A ) = ( A +R 0R )
12	9, 10, 11	3eqtr3i 2652	. . . . . . 7 |- ( C +R ( ( C .R -1R ) +R A ) ) = ( A +R 0R )
13		0idsr 9918	. . . . . . 7 |- ( A e. R. -> ( A +R 0R ) = A )
14	12, 13	syl5eq 2668	. . . . . 6 |- ( A e. R. -> ( C +R ( ( C .R -1R ) +R A ) ) = A )
15	14	breq2d 4665	. . . . 5 |- ( A e. R. -> ( ( C +R -1R ) <R ( C +R ( ( C .R -1R ) +R A ) ) <-> ( C +R -1R ) <R A ) )
16	6, 15	syl5bb 272	. . . 4 |- ( A e. R. -> ( -1R <R ( ( C .R -1R ) +R A ) <-> ( C +R -1R ) <R A ) )
17		m1r 9903	. . . . . . . 8 |- -1R e. R.
18		mulclsr 9905	. . . . . . . 8 |- ( ( C e. R. /\ -1R e. R. ) -> ( C .R -1R ) e. R. )
19	4, 17, 18	mp2an 708	. . . . . . 7 |- ( C .R -1R ) e. R.
20		addclsr 9904	. . . . . . 7 |- ( ( ( C .R -1R ) e. R. /\ A e. R. ) -> ( ( C .R -1R ) +R A ) e. R. )
21	19, 20	mpan 706	. . . . . 6 |- ( A e. R. -> ( ( C .R -1R ) +R A ) e. R. )
22		df-nr 9878	. . . . . . 7 |- R. = ( ( P. X. P. ) /. ~R )
23		breq2 4657	. . . . . . . 8 |- ( [ <. y , z >. ] ~R = ( ( C .R -1R ) +R A ) -> ( -1R <R [ <. y , z >. ] ~R <-> -1R <R ( ( C .R -1R ) +R A ) ) )
24		eqeq2 2633	. . . . . . . . 9 |- ( [ <. y , z >. ] ~R = ( ( C .R -1R ) +R A ) -> ( [ <. x , 1P >. ] ~R = [ <. y , z >. ] ~R <-> [ <. x , 1P >. ] ~R = ( ( C .R -1R ) +R A ) ) )
25	24	rexbidv 3052	. . . . . . . 8 |- ( [ <. y , z >. ] ~R = ( ( C .R -1R ) +R A ) -> ( E. x e. P. [ <. x , 1P >. ] ~R = [ <. y , z >. ] ~R <-> E. x e. P. [ <. x , 1P >. ] ~R = ( ( C .R -1R ) +R A ) ) )
26	23, 25	imbi12d 334	. . . . . . 7 |- ( [ <. y , z >. ] ~R = ( ( C .R -1R ) +R A ) -> ( ( -1R <R [ <. y , z >. ] ~R -> E. x e. P. [ <. x , 1P >. ] ~R = [ <. y , z >. ] ~R ) <-> ( -1R <R ( ( C .R -1R ) +R A ) -> E. x e. P. [ <. x , 1P >. ] ~R = ( ( C .R -1R ) +R A ) ) ) )
27		df-m1r 9884	. . . . . . . . . . 11 |- -1R = [ <. 1P , ( 1P +P. 1P ) >. ] ~R
28	27	breq1i 4660	. . . . . . . . . 10 |- ( -1R <R [ <. y , z >. ] ~R <-> [ <. 1P , ( 1P +P. 1P ) >. ] ~R <R [ <. y , z >. ] ~R )
29		addasspr 9844	. . . . . . . . . . . 12 |- ( ( 1P +P. 1P ) +P. y ) = ( 1P +P. ( 1P +P. y ) )
30	29	breq2i 4661	. . . . . . . . . . 11 |- ( ( 1P +P. z ) <P ( ( 1P +P. 1P ) +P. y ) <-> ( 1P +P. z ) <P ( 1P +P. ( 1P +P. y ) ) )
31		ltsrpr 9898	. . . . . . . . . . 11 |- ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R <R [ <. y , z >. ] ~R <-> ( 1P +P. z ) <P ( ( 1P +P. 1P ) +P. y ) )
32		1pr 9837	. . . . . . . . . . . 12 |- 1P e. P.
33		ltapr 9867	. . . . . . . . . . . 12 |- ( 1P e. P. -> ( z <P ( 1P +P. y ) <-> ( 1P +P. z ) <P ( 1P +P. ( 1P +P. y ) ) ) )
34	32, 33	ax-mp 5	. . . . . . . . . . 11 |- ( z <P ( 1P +P. y ) <-> ( 1P +P. z ) <P ( 1P +P. ( 1P +P. y ) ) )
35	30, 31, 34	3bitr4i 292	. . . . . . . . . 10 |- ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R <R [ <. y , z >. ] ~R <-> z <P ( 1P +P. y ) )
36	28, 35	bitri 264	. . . . . . . . 9 |- ( -1R <R [ <. y , z >. ] ~R <-> z <P ( 1P +P. y ) )
37		ltexpri 9865	. . . . . . . . 9 |- ( z <P ( 1P +P. y ) -> E. x e. P. ( z +P. x ) = ( 1P +P. y ) )
38	36, 37	sylbi 207	. . . . . . . 8 |- ( -1R <R [ <. y , z >. ] ~R -> E. x e. P. ( z +P. x ) = ( 1P +P. y ) )
39		enreceq 9887	. . . . . . . . . . . 12 |- ( ( ( x e. P. /\ 1P e. P. ) /\ ( y e. P. /\ z e. P. ) ) -> ( [ <. x , 1P >. ] ~R = [ <. y , z >. ] ~R <-> ( x +P. z ) = ( 1P +P. y ) ) )
40	32, 39	mpanl2 717	. . . . . . . . . . 11 |- ( ( x e. P. /\ ( y e. P. /\ z e. P. ) ) -> ( [ <. x , 1P >. ] ~R = [ <. y , z >. ] ~R <-> ( x +P. z ) = ( 1P +P. y ) ) )
41		addcompr 9843	. . . . . . . . . . . 12 |- ( z +P. x ) = ( x +P. z )
42	41	eqeq1i 2627	. . . . . . . . . . 11 |- ( ( z +P. x ) = ( 1P +P. y ) <-> ( x +P. z ) = ( 1P +P. y ) )
43	40, 42	syl6bbr 278	. . . . . . . . . 10 |- ( ( x e. P. /\ ( y e. P. /\ z e. P. ) ) -> ( [ <. x , 1P >. ] ~R = [ <. y , z >. ] ~R <-> ( z +P. x ) = ( 1P +P. y ) ) )
44	43	ancoms 469	. . . . . . . . 9 |- ( ( ( y e. P. /\ z e. P. ) /\ x e. P. ) -> ( [ <. x , 1P >. ] ~R = [ <. y , z >. ] ~R <-> ( z +P. x ) = ( 1P +P. y ) ) )
45	44	rexbidva 3049	. . . . . . . 8 |- ( ( y e. P. /\ z e. P. ) -> ( E. x e. P. [ <. x , 1P >. ] ~R = [ <. y , z >. ] ~R <-> E. x e. P. ( z +P. x ) = ( 1P +P. y ) ) )
46	38, 45	syl5ibr 236	. . . . . . 7 |- ( ( y e. P. /\ z e. P. ) -> ( -1R <R [ <. y , z >. ] ~R -> E. x e. P. [ <. x , 1P >. ] ~R = [ <. y , z >. ] ~R ) )
47	22, 26, 46	ecoptocl 7837	. . . . . 6 |- ( ( ( C .R -1R ) +R A ) e. R. -> ( -1R <R ( ( C .R -1R ) +R A ) -> E. x e. P. [ <. x , 1P >. ] ~R = ( ( C .R -1R ) +R A ) ) )
48	21, 47	syl 17	. . . . 5 |- ( A e. R. -> ( -1R <R ( ( C .R -1R ) +R A ) -> E. x e. P. [ <. x , 1P >. ] ~R = ( ( C .R -1R ) +R A ) ) )
49		oveq2 6658	. . . . . . . 8 |- ( [ <. x , 1P >. ] ~R = ( ( C .R -1R ) +R A ) -> ( C +R [ <. x , 1P >. ] ~R ) = ( C +R ( ( C .R -1R ) +R A ) ) )
50	49, 14	sylan9eqr 2678	. . . . . . 7 |- ( ( A e. R. /\ [ <. x , 1P >. ] ~R = ( ( C .R -1R ) +R A ) ) -> ( C +R [ <. x , 1P >. ] ~R ) = A )
51	50	ex 450	. . . . . 6 |- ( A e. R. -> ( [ <. x , 1P >. ] ~R = ( ( C .R -1R ) +R A ) -> ( C +R [ <. x , 1P >. ] ~R ) = A ) )
52	51	reximdv 3016	. . . . 5 |- ( A e. R. -> ( E. x e. P. [ <. x , 1P >. ] ~R = ( ( C .R -1R ) +R A ) -> E. x e. P. ( C +R [ <. x , 1P >. ] ~R ) = A ) )
53	48, 52	syld 47	. . . 4 |- ( A e. R. -> ( -1R <R ( ( C .R -1R ) +R A ) -> E. x e. P. ( C +R [ <. x , 1P >. ] ~R ) = A ) )
54	16, 53	sylbird 250	. . 3 |- ( A e. R. -> ( ( C +R -1R ) <R A -> E. x e. P. ( C +R [ <. x , 1P >. ] ~R ) = A ) )
55	3, 54	mpcom 38	. 2 |- ( ( C +R -1R ) <R A -> E. x e. P. ( C +R [ <. x , 1P >. ] ~R ) = A )
56	4	mappsrpr 9929	. . . . 5 |- ( ( C +R -1R ) <R ( C +R [ <. x , 1P >. ] ~R ) <-> x e. P. )
57		breq2 4657	. . . . 5 |- ( ( C +R [ <. x , 1P >. ] ~R ) = A -> ( ( C +R -1R ) <R ( C +R [ <. x , 1P >. ] ~R ) <-> ( C +R -1R ) <R A ) )
58	56, 57	syl5bbr 274	. . . 4 |- ( ( C +R [ <. x , 1P >. ] ~R ) = A -> ( x e. P. <-> ( C +R -1R ) <R A ) )
59	58	biimpac 503	. . 3 |- ( ( x e. P. /\ ( C +R [ <. x , 1P >. ] ~R ) = A ) -> ( C +R -1R ) <R A )
60	59	rexlimiva 3028	. 2 |- ( E. x e. P. ( C +R [ <. x , 1P >. ] ~R ) = A -> ( C +R -1R ) <R A )
61	55, 60	impbii 199	1 |- ( ( C +R -1R ) <R A <-> E. x e. P. ( C +R [ <. x , 1P >. ] ~R ) = A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   <.cop 4183   class class class wbr 4653  (class class class)co 6650   [cec 7740   P.cnp 9681   1Pc1p 9682   +P. cpp 9683   <P cltp 9685   ~R cer 9686   R.cnr 9687   0Rc0r 9688   -1Rcm1r 9690   +R cplr 9691   .R cmr 9692   <R cltr 9693

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-1p 9804  df-plp 9805  df-mp 9806  df-ltp 9807  df-enr 9877  df-nr 9878  df-plr 9879  df-mr 9880  df-ltr 9881  df-0r 9882  df-1r 9883  df-m1r 9884

This theorem is referenced by:  supsrlem  9932