Theorem mappsrpr 9929

Description: Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)

Hypothesis

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

Assertion

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

Proof of Theorem mappsrpr

Step	Hyp	Ref	Expression
1		df-m1r 9884	. . . 4 |- -1R = [ <. 1P , ( 1P +P. 1P ) >. ] ~R
2	1	breq1i 4660	. . 3 |- ( -1R <R [ <. A , 1P >. ] ~R <-> [ <. 1P , ( 1P +P. 1P ) >. ] ~R <R [ <. A , 1P >. ] ~R )
3		ltsrpr 9898	. . 3 |- ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R <R [ <. A , 1P >. ] ~R <-> ( 1P +P. 1P ) <P ( ( 1P +P. 1P ) +P. A ) )
4	2, 3	bitri 264	. 2 |- ( -1R <R [ <. A , 1P >. ] ~R <-> ( 1P +P. 1P ) <P ( ( 1P +P. 1P ) +P. A ) )
5		mappsrpr.2	. . 3 |- C e. R.
6		ltasr 9921	. . 3 |- ( C e. R. -> ( -1R <R [ <. A , 1P >. ] ~R <-> ( C +R -1R ) <R ( C +R [ <. A , 1P >. ] ~R ) ) )
7	5, 6	ax-mp 5	. 2 |- ( -1R <R [ <. A , 1P >. ] ~R <-> ( C +R -1R ) <R ( C +R [ <. A , 1P >. ] ~R ) )
8		ltrelpr 9820	. . . . . 6 |- <P C_ ( P. X. P. )
9	8	brel 5168	. . . . 5 |- ( ( 1P +P. 1P ) <P ( ( 1P +P. 1P ) +P. A ) -> ( ( 1P +P. 1P ) e. P. /\ ( ( 1P +P. 1P ) +P. A ) e. P. ) )
10	9	simprd 479	. . . 4 |- ( ( 1P +P. 1P ) <P ( ( 1P +P. 1P ) +P. A ) -> ( ( 1P +P. 1P ) +P. A ) e. P. )
11		dmplp 9834	. . . . . 6 |- dom +P. = ( P. X. P. )
12		0npr 9814	. . . . . 6 |- -. (/) e. P.
13	11, 12	ndmovrcl 6820	. . . . 5 |- ( ( ( 1P +P. 1P ) +P. A ) e. P. -> ( ( 1P +P. 1P ) e. P. /\ A e. P. ) )
14	13	simprd 479	. . . 4 |- ( ( ( 1P +P. 1P ) +P. A ) e. P. -> A e. P. )
15	10, 14	syl 17	. . 3 |- ( ( 1P +P. 1P ) <P ( ( 1P +P. 1P ) +P. A ) -> A e. P. )
16		1pr 9837	. . . . 5 |- 1P e. P.
17		addclpr 9840	. . . . 5 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
18	16, 16, 17	mp2an 708	. . . 4 |- ( 1P +P. 1P ) e. P.
19		ltaddpr 9856	. . . 4 |- ( ( ( 1P +P. 1P ) e. P. /\ A e. P. ) -> ( 1P +P. 1P ) <P ( ( 1P +P. 1P ) +P. A ) )
20	18, 19	mpan 706	. . 3 |- ( A e. P. -> ( 1P +P. 1P ) <P ( ( 1P +P. 1P ) +P. A ) )
21	15, 20	impbii 199	. 2 |- ( ( 1P +P. 1P ) <P ( ( 1P +P. 1P ) +P. A ) <-> A e. P. )
22	4, 7, 21	3bitr3i 290	1 |- ( ( C +R -1R ) <R ( C +R [ <. A , 1P >. ] ~R ) <-> A e. P. )

Syntax hints:   <-> wb 196   e. wcel 1990   <.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   -1Rcm1r 9690   +R cplr 9691   <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-ltp 9807  df-enr 9877  df-nr 9878  df-plr 9879  df-ltr 9881  df-m1r 9884

This theorem is referenced by:  map2psrpr  9931  supsrlem  9932