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Theorem mappsrpr 8909
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
StepHypRef Expression
1 df-m1r 8867 . . . 4  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
21breq1i 4153 . . 3  |-  ( -1R 
<R  [ <. A ,  1P >. ]  ~R  <->  [ <. 1P , 
( 1P  +P.  1P ) >. ]  ~R  <R  [
<. A ,  1P >. ]  ~R  )
3 ltsrpr 8878 . . 3  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  <R  [ <. A ,  1P >. ]  ~R  <->  ( 1P  +P.  1P )  <P  (
( 1P  +P.  1P )  +P.  A ) )
42, 3bitri 241 . 2  |-  ( -1R 
<R  [ <. A ,  1P >. ]  ~R  <->  ( 1P  +P.  1P )  <P  (
( 1P  +P.  1P )  +P.  A ) )
5 mappsrpr.2 . . 3  |-  C  e. 
R.
6 ltasr 8901 . . 3  |-  ( C  e.  R.  ->  ( -1R  <R  [ <. A ,  1P >. ]  ~R  <->  ( C  +R  -1R )  <R  ( C  +R  [ <. A ,  1P >. ]  ~R  )
) )
75, 6ax-mp 8 . 2  |-  ( -1R 
<R  [ <. A ,  1P >. ]  ~R  <->  ( C  +R  -1R )  <R  ( C  +R  [ <. A ,  1P >. ]  ~R  )
)
8 ltrelpr 8801 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
98brel 4859 . . . . 5  |-  ( ( 1P  +P.  1P ) 
<P  ( ( 1P  +P.  1P )  +P.  A )  ->  ( ( 1P 
+P.  1P )  e.  P.  /\  ( ( 1P  +P.  1P )  +P.  A )  e.  P. ) )
109simprd 450 . . . 4  |-  ( ( 1P  +P.  1P ) 
<P  ( ( 1P  +P.  1P )  +P.  A )  ->  ( ( 1P 
+P.  1P )  +P.  A
)  e.  P. )
11 dmplp 8815 . . . . . 6  |-  dom  +P.  =  ( P.  X.  P. )
12 0npr 8795 . . . . . 6  |-  -.  (/)  e.  P.
1311, 12ndmovrcl 6165 . . . . 5  |-  ( ( ( 1P  +P.  1P )  +P.  A )  e. 
P.  ->  ( ( 1P 
+P.  1P )  e.  P.  /\  A  e.  P. )
)
1413simprd 450 . . . 4  |-  ( ( ( 1P  +P.  1P )  +P.  A )  e. 
P.  ->  A  e.  P. )
1510, 14syl 16 . . 3  |-  ( ( 1P  +P.  1P ) 
<P  ( ( 1P  +P.  1P )  +P.  A )  ->  A  e.  P. )
16 1pr 8818 . . . . 5  |-  1P  e.  P.
17 addclpr 8821 . . . . 5  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
1816, 16, 17mp2an 654 . . . 4  |-  ( 1P 
+P.  1P )  e.  P.
19 ltaddpr 8837 . . . 4  |-  ( ( ( 1P  +P.  1P )  e.  P.  /\  A  e.  P. )  ->  ( 1P  +P.  1P )  <P 
( ( 1P  +P.  1P )  +P.  A ) )
2018, 19mpan 652 . . 3  |-  ( A  e.  P.  ->  ( 1P  +P.  1P )  <P 
( ( 1P  +P.  1P )  +P.  A ) )
2115, 20impbii 181 . 2  |-  ( ( 1P  +P.  1P ) 
<P  ( ( 1P  +P.  1P )  +P.  A )  <-> 
A  e.  P. )
224, 7, 213bitr3i 267 1  |-  ( ( C  +R  -1R )  <R  ( C  +R  [ <. A ,  1P >. ]  ~R  )  <->  A  e.  P. )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1717   <.cop 3753   class class class wbr 4146  (class class class)co 6013   [cec 6832   P.cnp 8660   1Pc1p 8661    +P. cpp 8662    <P cltp 8664    ~R cer 8667   R.cnr 8668   -1Rcm1r 8671    +R cplr 8672    <R cltr 8674
This theorem is referenced by:  map2psrpr  8911  supsrlem  8912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-omul 6658  df-er 6834  df-ec 6836  df-qs 6840  df-ni 8675  df-pli 8676  df-mi 8677  df-lti 8678  df-plpq 8711  df-mpq 8712  df-ltpq 8713  df-enq 8714  df-nq 8715  df-erq 8716  df-plq 8717  df-mq 8718  df-1nq 8719  df-rq 8720  df-ltnq 8721  df-np 8784  df-1p 8785  df-plp 8786  df-ltp 8788  df-plpr 8858  df-enr 8860  df-nr 8861  df-plr 8862  df-ltr 8864  df-m1r 8867
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