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Theorem mappsrpr 8746
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 8704 . . . 4  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
21breq1i 4046 . . 3  |-  ( -1R 
<R  [ <. A ,  1P >. ]  ~R  <->  [ <. 1P , 
( 1P  +P.  1P ) >. ]  ~R  <R  [
<. A ,  1P >. ]  ~R  )
3 ltsrpr 8715 . . 3  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  <R  [ <. A ,  1P >. ]  ~R  <->  ( 1P  +P.  1P )  <P  (
( 1P  +P.  1P )  +P.  A ) )
42, 3bitri 240 . 2  |-  ( -1R 
<R  [ <. A ,  1P >. ]  ~R  <->  ( 1P  +P.  1P )  <P  (
( 1P  +P.  1P )  +P.  A ) )
5 mappsrpr.2 . . 3  |-  C  e. 
R.
6 ltasr 8738 . . 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 8638 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
98brel 4753 . . . . 5  |-  ( ( 1P  +P.  1P ) 
<P  ( ( 1P  +P.  1P )  +P.  A )  ->  ( ( 1P 
+P.  1P )  e.  P.  /\  ( ( 1P  +P.  1P )  +P.  A )  e.  P. ) )
109simprd 449 . . . 4  |-  ( ( 1P  +P.  1P ) 
<P  ( ( 1P  +P.  1P )  +P.  A )  ->  ( ( 1P 
+P.  1P )  +P.  A
)  e.  P. )
11 dmplp 8652 . . . . . 6  |-  dom  +P.  =  ( P.  X.  P. )
12 0npr 8632 . . . . . 6  |-  -.  (/)  e.  P.
1311, 12ndmovrcl 6022 . . . . 5  |-  ( ( ( 1P  +P.  1P )  +P.  A )  e. 
P.  ->  ( ( 1P 
+P.  1P )  e.  P.  /\  A  e.  P. )
)
1413simprd 449 . . . 4  |-  ( ( ( 1P  +P.  1P )  +P.  A )  e. 
P.  ->  A  e.  P. )
1510, 14syl 15 . . 3  |-  ( ( 1P  +P.  1P ) 
<P  ( ( 1P  +P.  1P )  +P.  A )  ->  A  e.  P. )
16 1pr 8655 . . . . 5  |-  1P  e.  P.
17 addclpr 8658 . . . . 5  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
1816, 16, 17mp2an 653 . . . 4  |-  ( 1P 
+P.  1P )  e.  P.
19 ltaddpr 8674 . . . 4  |-  ( ( ( 1P  +P.  1P )  e.  P.  /\  A  e.  P. )  ->  ( 1P  +P.  1P )  <P 
( ( 1P  +P.  1P )  +P.  A ) )
2018, 19mpan 651 . . 3  |-  ( A  e.  P.  ->  ( 1P  +P.  1P )  <P 
( ( 1P  +P.  1P )  +P.  A ) )
2115, 20impbii 180 . 2  |-  ( ( 1P  +P.  1P ) 
<P  ( ( 1P  +P.  1P )  +P.  A )  <-> 
A  e.  P. )
224, 7, 213bitr3i 266 1  |-  ( ( C  +R  -1R )  <R  ( C  +R  [ <. A ,  1P >. ]  ~R  )  <->  A  e.  P. )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1696   <.cop 3656   class class class wbr 4039  (class class class)co 5874   [cec 6674   P.cnp 8497   1Pc1p 8498    +P. cpp 8499    <P cltp 8501    ~R cer 8504   R.cnr 8505   -1Rcm1r 8508    +R cplr 8509    <R cltr 8511
This theorem is referenced by:  map2psrpr  8748  supsrlem  8749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-1p 8622  df-plp 8623  df-ltp 8625  df-plpr 8695  df-enr 8697  df-nr 8698  df-plr 8699  df-ltr 8701  df-m1r 8704
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