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Theorem map2psrpr 8941
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 8902 . . . . 5  |-  <R  C_  ( R.  X.  R. )
21brel 4885 . . . 4  |-  ( ( C  +R  -1R )  <R  A  ->  ( ( C  +R  -1R )  e. 
R.  /\  A  e.  R. ) )
32simprd 450 . . 3  |-  ( ( C  +R  -1R )  <R  A  ->  A  e.  R. )
4 map2psrpr.2 . . . . . 6  |-  C  e. 
R.
5 ltasr 8931 . . . . . 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 8932 . . . . . . . . . 10  |-  ( C  e.  R.  ->  ( C  +R  ( C  .R  -1R ) )  =  0R )
84, 7ax-mp 8 . . . . . . . . 9  |-  ( C  +R  ( C  .R  -1R ) )  =  0R
98oveq1i 6050 . . . . . . . 8  |-  ( ( C  +R  ( C  .R  -1R ) )  +R  A )  =  ( 0R  +R  A
)
10 addasssr 8919 . . . . . . . 8  |-  ( ( C  +R  ( C  .R  -1R ) )  +R  A )  =  ( C  +R  (
( C  .R  -1R )  +R  A ) )
11 addcomsr 8918 . . . . . . . 8  |-  ( 0R 
+R  A )  =  ( A  +R  0R )
129, 10, 113eqtr3i 2432 . . . . . . 7  |-  ( C  +R  ( ( C  .R  -1R )  +R  A ) )  =  ( A  +R  0R )
13 0idsr 8928 . . . . . . 7  |-  ( A  e.  R.  ->  ( A  +R  0R )  =  A )
1412, 13syl5eq 2448 . . . . . 6  |-  ( A  e.  R.  ->  ( C  +R  ( ( C  .R  -1R )  +R  A ) )  =  A )
1514breq2d 4184 . . . . 5  |-  ( A  e.  R.  ->  (
( C  +R  -1R )  <R  ( C  +R  ( ( C  .R  -1R )  +R  A
) )  <->  ( C  +R  -1R )  <R  A ) )
166, 15syl5bb 249 . . . 4  |-  ( A  e.  R.  ->  ( -1R  <R  ( ( C  .R  -1R )  +R  A )  <->  ( C  +R  -1R )  <R  A ) )
17 m1r 8913 . . . . . . . 8  |-  -1R  e.  R.
18 mulclsr 8915 . . . . . . . 8  |-  ( ( C  e.  R.  /\  -1R  e.  R. )  -> 
( C  .R  -1R )  e.  R. )
194, 17, 18mp2an 654 . . . . . . 7  |-  ( C  .R  -1R )  e. 
R.
20 addclsr 8914 . . . . . . 7  |-  ( ( ( C  .R  -1R )  e.  R.  /\  A  e.  R. )  ->  (
( C  .R  -1R )  +R  A )  e. 
R. )
2119, 20mpan 652 . . . . . 6  |-  ( A  e.  R.  ->  (
( C  .R  -1R )  +R  A )  e. 
R. )
22 df-nr 8891 . . . . . . 7  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
23 breq2 4176 . . . . . . . 8  |-  ( [
<. y ,  z >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( -1R  <R  [ <. y ,  z >. ]  ~R  <->  -1R 
<R  ( ( C  .R  -1R )  +R  A
) ) )
24 eqeq2 2413 . . . . . . . . 9  |-  ( [
<. y ,  z >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  <->  [ <. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A ) ) )
2524rexbidv 2687 . . . . . . . 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 312 . . . . . . 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 8897 . . . . . . . . . . 11  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
2827breq1i 4179 . . . . . . . . . 10  |-  ( -1R 
<R  [ <. y ,  z
>. ]  ~R  <->  [ <. 1P , 
( 1P  +P.  1P ) >. ]  ~R  <R  [
<. y ,  z >. ]  ~R  )
29 addasspr 8855 . . . . . . . . . . . 12  |-  ( ( 1P  +P.  1P )  +P.  y )  =  ( 1P  +P.  ( 1P  +P.  y ) )
3029breq2i 4180 . . . . . . . . . . 11  |-  ( ( 1P  +P.  z ) 
<P  ( ( 1P  +P.  1P )  +P.  y )  <-> 
( 1P  +P.  z
)  <P  ( 1P  +P.  ( 1P  +P.  y ) ) )
31 ltsrpr 8908 . . . . . . . . . . 11  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  <R  [ <. y ,  z
>. ]  ~R  <->  ( 1P  +P.  z )  <P  (
( 1P  +P.  1P )  +P.  y ) )
32 1pr 8848 . . . . . . . . . . . 12  |-  1P  e.  P.
33 ltapr 8878 . . . . . . . . . . . 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 269 . . . . . . . . . 10  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  <R  [ <. y ,  z
>. ]  ~R  <->  z  <P  ( 1P  +P.  y ) )
3628, 35bitri 241 . . . . . . . . 9  |-  ( -1R 
<R  [ <. y ,  z
>. ]  ~R  <->  z  <P  ( 1P  +P.  y ) )
37 ltexpri 8876 . . . . . . . . 9  |-  ( z 
<P  ( 1P  +P.  y
)  ->  E. x  e.  P.  ( z  +P.  x )  =  ( 1P  +P.  y ) )
3836, 37sylbi 188 . . . . . . . 8  |-  ( -1R 
<R  [ <. y ,  z
>. ]  ~R  ->  E. x  e.  P.  ( z  +P.  x )  =  ( 1P  +P.  y ) )
39 enreceq 8900 . . . . . . . . . . . 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 663 . . . . . . . . . . 11  |-  ( ( x  e.  P.  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  <->  ( x  +P.  z )  =  ( 1P  +P.  y ) ) )
41 addcompr 8854 . . . . . . . . . . . 12  |-  ( z  +P.  x )  =  ( x  +P.  z
)
4241eqeq1i 2411 . . . . . . . . . . 11  |-  ( ( z  +P.  x )  =  ( 1P  +P.  y )  <->  ( x  +P.  z )  =  ( 1P  +P.  y ) )
4340, 42syl6bbr 255 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  <->  ( z  +P.  x )  =  ( 1P  +P.  y ) ) )
4443ancoms 440 . . . . . . . . 9  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  x  e.  P. )  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z
>. ]  ~R  <->  ( z  +P.  x )  =  ( 1P  +P.  y ) ) )
4544rexbidva 2683 . . . . . . . 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 213 . . . . . . 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 6953 . . . . . 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 16 . . . . 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 6048 . . . . . . . 8  |-  ( [
<. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  ( C  +R  ( ( C  .R  -1R )  +R  A
) ) )
5049, 14sylan9eqr 2458 . . . . . . 7  |-  ( ( A  e.  R.  /\  [
<. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A ) )  -> 
( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A )
5150ex 424 . . . . . 6  |-  ( A  e.  R.  ->  ( [ <. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  -> 
( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A ) )
5251reximdv 2777 . . . . 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 42 . . . 4  |-  ( A  e.  R.  ->  ( -1R  <R  ( ( C  .R  -1R )  +R  A )  ->  E. x  e.  P.  ( C  +R  [
<. x ,  1P >. ]  ~R  )  =  A ) )
5416, 53sylbird 227 . . 3  |-  ( A  e.  R.  ->  (
( C  +R  -1R )  <R  A  ->  E. x  e.  P.  ( C  +R  [
<. x ,  1P >. ]  ~R  )  =  A ) )
553, 54mpcom 34 . 2  |-  ( ( C  +R  -1R )  <R  A  ->  E. x  e.  P.  ( C  +R  [
<. x ,  1P >. ]  ~R  )  =  A )
564mappsrpr 8939 . . . . 5  |-  ( ( C  +R  -1R )  <R  ( C  +R  [ <. x ,  1P >. ]  ~R  )  <->  x  e.  P. )
57 breq2 4176 . . . . 5  |-  ( ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A  ->  ( ( C  +R  -1R )  <R  ( C  +R  [
<. x ,  1P >. ]  ~R  )  <->  ( C  +R  -1R )  <R  A ) )
5856, 57syl5bbr 251 . . . 4  |-  ( ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A  ->  ( x  e.  P.  <->  ( C  +R  -1R )  <R  A ) )
5958biimpac 473 . . 3  |-  ( ( x  e.  P.  /\  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A )  ->  ( C  +R  -1R )  <R  A )
6059rexlimiva 2785 . 2  |-  ( E. x  e.  P.  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A  ->  ( C  +R  -1R )  <R  A )
6155, 60impbii 181 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   <.cop 3777   class class class wbr 4172  (class class class)co 6040   [cec 6862   P.cnp 8690   1Pc1p 8691    +P. cpp 8692    <P cltp 8694    ~R cer 8697   R.cnr 8698   0Rc0r 8699   -1Rcm1r 8701    +R cplr 8702    .R cmr 8703    <R cltr 8704
This theorem is referenced by:  supsrlem  8942
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-omul 6688  df-er 6864  df-ec 6866  df-qs 6870  df-ni 8705  df-pli 8706  df-mi 8707  df-lti 8708  df-plpq 8741  df-mpq 8742  df-ltpq 8743  df-enq 8744  df-nq 8745  df-erq 8746  df-plq 8747  df-mq 8748  df-1nq 8749  df-rq 8750  df-ltnq 8751  df-np 8814  df-1p 8815  df-plp 8816  df-mp 8817  df-ltp 8818  df-plpr 8888  df-mpr 8889  df-enr 8890  df-nr 8891  df-plr 8892  df-mr 8893  df-ltr 8894  df-0r 8895  df-1r 8896  df-m1r 8897
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