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Theorem map2psrpr 8748
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.
StepHypRef Expression
1 ltrelsr 8709 . . . . 5  |-  <R  C_  ( R.  X.  R. )
21brel 4753 . . . 4  |-  ( ( C  +R  -1R )  <R  A  ->  ( ( C  +R  -1R )  e. 
R.  /\  A  e.  R. ) )
32simprd 449 . . 3  |-  ( ( C  +R  -1R )  <R  A  ->  A  e.  R. )
4 map2psrpr.2 . . . . . 6  |-  C  e. 
R.
5 ltasr 8738 . . . . . 6  |-  ( C  e.  R.  ->  ( -1R  <R  ( ( C  .R  -1R )  +R  A )  <->  ( C  +R  -1R )  <R  ( C  +R  ( ( C  .R  -1R )  +R  A ) ) ) )
64, 5ax-mp 8 . . . . 5  |-  ( -1R 
<R  ( ( C  .R  -1R )  +R  A
)  <->  ( C  +R  -1R )  <R  ( C  +R  ( ( C  .R  -1R )  +R  A ) ) )
7 pn0sr 8739 . . . . . . . . . 10  |-  ( C  e.  R.  ->  ( C  +R  ( C  .R  -1R ) )  =  0R )
84, 7ax-mp 8 . . . . . . . . 9  |-  ( C  +R  ( C  .R  -1R ) )  =  0R
98oveq1i 5884 . . . . . . . 8  |-  ( ( C  +R  ( C  .R  -1R ) )  +R  A )  =  ( 0R  +R  A
)
10 addasssr 8726 . . . . . . . 8  |-  ( ( C  +R  ( C  .R  -1R ) )  +R  A )  =  ( C  +R  (
( C  .R  -1R )  +R  A ) )
11 addcomsr 8725 . . . . . . . 8  |-  ( 0R 
+R  A )  =  ( A  +R  0R )
129, 10, 113eqtr3i 2324 . . . . . . 7  |-  ( C  +R  ( ( C  .R  -1R )  +R  A ) )  =  ( A  +R  0R )
13 0idsr 8735 . . . . . . 7  |-  ( A  e.  R.  ->  ( A  +R  0R )  =  A )
1412, 13syl5eq 2340 . . . . . 6  |-  ( A  e.  R.  ->  ( C  +R  ( ( C  .R  -1R )  +R  A ) )  =  A )
1514breq2d 4051 . . . . 5  |-  ( A  e.  R.  ->  (
( C  +R  -1R )  <R  ( C  +R  ( ( C  .R  -1R )  +R  A
) )  <->  ( C  +R  -1R )  <R  A ) )
166, 15syl5bb 248 . . . 4  |-  ( A  e.  R.  ->  ( -1R  <R  ( ( C  .R  -1R )  +R  A )  <->  ( C  +R  -1R )  <R  A ) )
17 m1r 8720 . . . . . . . 8  |-  -1R  e.  R.
18 mulclsr 8722 . . . . . . . 8  |-  ( ( C  e.  R.  /\  -1R  e.  R. )  -> 
( C  .R  -1R )  e.  R. )
194, 17, 18mp2an 653 . . . . . . 7  |-  ( C  .R  -1R )  e. 
R.
20 addclsr 8721 . . . . . . 7  |-  ( ( ( C  .R  -1R )  e.  R.  /\  A  e.  R. )  ->  (
( C  .R  -1R )  +R  A )  e. 
R. )
2119, 20mpan 651 . . . . . 6  |-  ( A  e.  R.  ->  (
( C  .R  -1R )  +R  A )  e. 
R. )
22 df-nr 8698 . . . . . . 7  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
23 breq2 4043 . . . . . . . 8  |-  ( [
<. y ,  z >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( -1R  <R  [ <. y ,  z >. ]  ~R  <->  -1R 
<R  ( ( C  .R  -1R )  +R  A
) ) )
24 eqeq2 2305 . . . . . . . . 9  |-  ( [
<. y ,  z >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  <->  [ <. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A ) ) )
2524rexbidv 2577 . . . . . . . 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 ) ) )
2623, 25imbi12d 311 . . . . . . 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 8704 . . . . . . . . . . 11  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
2827breq1i 4046 . . . . . . . . . 10  |-  ( -1R 
<R  [ <. y ,  z
>. ]  ~R  <->  [ <. 1P , 
( 1P  +P.  1P ) >. ]  ~R  <R  [
<. y ,  z >. ]  ~R  )
29 addasspr 8662 . . . . . . . . . . . 12  |-  ( ( 1P  +P.  1P )  +P.  y )  =  ( 1P  +P.  ( 1P  +P.  y ) )
3029breq2i 4047 . . . . . . . . . . 11  |-  ( ( 1P  +P.  z ) 
<P  ( ( 1P  +P.  1P )  +P.  y )  <-> 
( 1P  +P.  z
)  <P  ( 1P  +P.  ( 1P  +P.  y ) ) )
31 ltsrpr 8715 . . . . . . . . . . 11  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  <R  [ <. y ,  z
>. ]  ~R  <->  ( 1P  +P.  z )  <P  (
( 1P  +P.  1P )  +P.  y ) )
32 1pr 8655 . . . . . . . . . . . 12  |-  1P  e.  P.
33 ltapr 8685 . . . . . . . . . . . 12  |-  ( 1P  e.  P.  ->  (
z  <P  ( 1P  +P.  y )  <->  ( 1P  +P.  z )  <P  ( 1P  +P.  ( 1P  +P.  y ) ) ) )
3432, 33ax-mp 8 . . . . . . . . . . 11  |-  ( z 
<P  ( 1P  +P.  y
)  <->  ( 1P  +P.  z )  <P  ( 1P  +P.  ( 1P  +P.  y ) ) )
3530, 31, 343bitr4i 268 . . . . . . . . . 10  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  <R  [ <. y ,  z
>. ]  ~R  <->  z  <P  ( 1P  +P.  y ) )
3628, 35bitri 240 . . . . . . . . 9  |-  ( -1R 
<R  [ <. y ,  z
>. ]  ~R  <->  z  <P  ( 1P  +P.  y ) )
37 ltexpri 8683 . . . . . . . . 9  |-  ( z 
<P  ( 1P  +P.  y
)  ->  E. x  e.  P.  ( z  +P.  x )  =  ( 1P  +P.  y ) )
3836, 37sylbi 187 . . . . . . . 8  |-  ( -1R 
<R  [ <. y ,  z
>. ]  ~R  ->  E. x  e.  P.  ( z  +P.  x )  =  ( 1P  +P.  y ) )
39 enreceq 8707 . . . . . . . . . . . 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 ) ) )
4032, 39mpanl2 662 . . . . . . . . . . 11  |-  ( ( x  e.  P.  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  <->  ( x  +P.  z )  =  ( 1P  +P.  y ) ) )
41 addcompr 8661 . . . . . . . . . . . 12  |-  ( z  +P.  x )  =  ( x  +P.  z
)
4241eqeq1i 2303 . . . . . . . . . . 11  |-  ( ( z  +P.  x )  =  ( 1P  +P.  y )  <->  ( x  +P.  z )  =  ( 1P  +P.  y ) )
4340, 42syl6bbr 254 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  <->  ( z  +P.  x )  =  ( 1P  +P.  y ) ) )
4443ancoms 439 . . . . . . . . 9  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  x  e.  P. )  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z
>. ]  ~R  <->  ( z  +P.  x )  =  ( 1P  +P.  y ) ) )
4544rexbidva 2573 . . . . . . . 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 ) ) )
4638, 45syl5ibr 212 . . . . . . 7  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( -1R  <R  [ <. y ,  z >. ]  ~R  ->  E. x  e.  P.  [
<. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  ) )
4722, 26, 46ecoptocl 6764 . . . . . 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 ) ) )
4821, 47syl 15 . . . . 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 5882 . . . . . . . 8  |-  ( [
<. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  ( C  +R  ( ( C  .R  -1R )  +R  A
) ) )
5049, 14sylan9eqr 2350 . . . . . . 7  |-  ( ( A  e.  R.  /\  [
<. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A ) )  -> 
( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A )
5150ex 423 . . . . . 6  |-  ( A  e.  R.  ->  ( [ <. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  -> 
( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A ) )
5251reximdv 2667 . . . . 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 ) )
5348, 52syld 40 . . . 4  |-  ( A  e.  R.  ->  ( -1R  <R  ( ( C  .R  -1R )  +R  A )  ->  E. x  e.  P.  ( C  +R  [
<. x ,  1P >. ]  ~R  )  =  A ) )
5416, 53sylbird 226 . . 3  |-  ( A  e.  R.  ->  (
( C  +R  -1R )  <R  A  ->  E. x  e.  P.  ( C  +R  [
<. x ,  1P >. ]  ~R  )  =  A ) )
553, 54mpcom 32 . 2  |-  ( ( C  +R  -1R )  <R  A  ->  E. x  e.  P.  ( C  +R  [
<. x ,  1P >. ]  ~R  )  =  A )
564mappsrpr 8746 . . . . 5  |-  ( ( C  +R  -1R )  <R  ( C  +R  [ <. x ,  1P >. ]  ~R  )  <->  x  e.  P. )
57 breq2 4043 . . . . 5  |-  ( ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A  ->  ( ( C  +R  -1R )  <R  ( C  +R  [
<. x ,  1P >. ]  ~R  )  <->  ( C  +R  -1R )  <R  A ) )
5856, 57syl5bbr 250 . . . 4  |-  ( ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A  ->  ( x  e.  P.  <->  ( C  +R  -1R )  <R  A ) )
5958biimpac 472 . . 3  |-  ( ( x  e.  P.  /\  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A )  ->  ( C  +R  -1R )  <R  A )
6059rexlimiva 2675 . 2  |-  ( E. x  e.  P.  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A  ->  ( C  +R  -1R )  <R  A )
6155, 60impbii 180 1  |-  ( ( C  +R  -1R )  <R  A  <->  E. x  e.  P.  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   <.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   0Rc0r 8506   -1Rcm1r 8508    +R cplr 8509    .R cmr 8510    <R cltr 8511
This theorem is referenced by:  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-mp 8624  df-ltp 8625  df-plpr 8695  df-mpr 8696  df-enr 8697  df-nr 8698  df-plr 8699  df-mr 8700  df-ltr 8701  df-0r 8702  df-1r 8703  df-m1r 8704
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