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Theorem map2psrpr 9273
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 9234 . . . . 5  |-  <R  C_  ( R.  X.  R. )
21brel 4883 . . . 4  |-  ( ( C  +R  -1R )  <R  A  ->  ( ( C  +R  -1R )  e. 
R.  /\  A  e.  R. ) )
32simprd 460 . . 3  |-  ( ( C  +R  -1R )  <R  A  ->  A  e.  R. )
4 map2psrpr.2 . . . . . 6  |-  C  e. 
R.
5 ltasr 9263 . . . . . 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 5 . . . . 5  |-  ( -1R 
<R  ( ( C  .R  -1R )  +R  A
)  <->  ( C  +R  -1R )  <R  ( C  +R  ( ( C  .R  -1R )  +R  A ) ) )
7 pn0sr 9264 . . . . . . . . . 10  |-  ( C  e.  R.  ->  ( C  +R  ( C  .R  -1R ) )  =  0R )
84, 7ax-mp 5 . . . . . . . . 9  |-  ( C  +R  ( C  .R  -1R ) )  =  0R
98oveq1i 6100 . . . . . . . 8  |-  ( ( C  +R  ( C  .R  -1R ) )  +R  A )  =  ( 0R  +R  A
)
10 addasssr 9251 . . . . . . . 8  |-  ( ( C  +R  ( C  .R  -1R ) )  +R  A )  =  ( C  +R  (
( C  .R  -1R )  +R  A ) )
11 addcomsr 9250 . . . . . . . 8  |-  ( 0R 
+R  A )  =  ( A  +R  0R )
129, 10, 113eqtr3i 2469 . . . . . . 7  |-  ( C  +R  ( ( C  .R  -1R )  +R  A ) )  =  ( A  +R  0R )
13 0idsr 9260 . . . . . . 7  |-  ( A  e.  R.  ->  ( A  +R  0R )  =  A )
1412, 13syl5eq 2485 . . . . . 6  |-  ( A  e.  R.  ->  ( C  +R  ( ( C  .R  -1R )  +R  A ) )  =  A )
1514breq2d 4301 . . . . 5  |-  ( A  e.  R.  ->  (
( C  +R  -1R )  <R  ( C  +R  ( ( C  .R  -1R )  +R  A
) )  <->  ( C  +R  -1R )  <R  A ) )
166, 15syl5bb 257 . . . 4  |-  ( A  e.  R.  ->  ( -1R  <R  ( ( C  .R  -1R )  +R  A )  <->  ( C  +R  -1R )  <R  A ) )
17 m1r 9245 . . . . . . . 8  |-  -1R  e.  R.
18 mulclsr 9247 . . . . . . . 8  |-  ( ( C  e.  R.  /\  -1R  e.  R. )  -> 
( C  .R  -1R )  e.  R. )
194, 17, 18mp2an 667 . . . . . . 7  |-  ( C  .R  -1R )  e. 
R.
20 addclsr 9246 . . . . . . 7  |-  ( ( ( C  .R  -1R )  e.  R.  /\  A  e.  R. )  ->  (
( C  .R  -1R )  +R  A )  e. 
R. )
2119, 20mpan 665 . . . . . 6  |-  ( A  e.  R.  ->  (
( C  .R  -1R )  +R  A )  e. 
R. )
22 df-nr 9223 . . . . . . 7  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
23 breq2 4293 . . . . . . . 8  |-  ( [
<. y ,  z >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( -1R  <R  [ <. y ,  z >. ]  ~R  <->  -1R 
<R  ( ( C  .R  -1R )  +R  A
) ) )
24 eqeq2 2450 . . . . . . . . 9  |-  ( [
<. y ,  z >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  <->  [ <. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A ) ) )
2524rexbidv 2734 . . . . . . . 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 320 . . . . . . 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 9229 . . . . . . . . . . 11  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
2827breq1i 4296 . . . . . . . . . 10  |-  ( -1R 
<R  [ <. y ,  z
>. ]  ~R  <->  [ <. 1P , 
( 1P  +P.  1P ) >. ]  ~R  <R  [
<. y ,  z >. ]  ~R  )
29 addasspr 9187 . . . . . . . . . . . 12  |-  ( ( 1P  +P.  1P )  +P.  y )  =  ( 1P  +P.  ( 1P  +P.  y ) )
3029breq2i 4297 . . . . . . . . . . 11  |-  ( ( 1P  +P.  z ) 
<P  ( ( 1P  +P.  1P )  +P.  y )  <-> 
( 1P  +P.  z
)  <P  ( 1P  +P.  ( 1P  +P.  y ) ) )
31 ltsrpr 9240 . . . . . . . . . . 11  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  <R  [ <. y ,  z
>. ]  ~R  <->  ( 1P  +P.  z )  <P  (
( 1P  +P.  1P )  +P.  y ) )
32 1pr 9180 . . . . . . . . . . . 12  |-  1P  e.  P.
33 ltapr 9210 . . . . . . . . . . . 12  |-  ( 1P  e.  P.  ->  (
z  <P  ( 1P  +P.  y )  <->  ( 1P  +P.  z )  <P  ( 1P  +P.  ( 1P  +P.  y ) ) ) )
3432, 33ax-mp 5 . . . . . . . . . . 11  |-  ( z 
<P  ( 1P  +P.  y
)  <->  ( 1P  +P.  z )  <P  ( 1P  +P.  ( 1P  +P.  y ) ) )
3530, 31, 343bitr4i 277 . . . . . . . . . 10  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  <R  [ <. y ,  z
>. ]  ~R  <->  z  <P  ( 1P  +P.  y ) )
3628, 35bitri 249 . . . . . . . . 9  |-  ( -1R 
<R  [ <. y ,  z
>. ]  ~R  <->  z  <P  ( 1P  +P.  y ) )
37 ltexpri 9208 . . . . . . . . 9  |-  ( z 
<P  ( 1P  +P.  y
)  ->  E. x  e.  P.  ( z  +P.  x )  =  ( 1P  +P.  y ) )
3836, 37sylbi 195 . . . . . . . 8  |-  ( -1R 
<R  [ <. y ,  z
>. ]  ~R  ->  E. x  e.  P.  ( z  +P.  x )  =  ( 1P  +P.  y ) )
39 enreceq 9232 . . . . . . . . . . . 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 676 . . . . . . . . . . 11  |-  ( ( x  e.  P.  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  <->  ( x  +P.  z )  =  ( 1P  +P.  y ) ) )
41 addcompr 9186 . . . . . . . . . . . 12  |-  ( z  +P.  x )  =  ( x  +P.  z
)
4241eqeq1i 2448 . . . . . . . . . . 11  |-  ( ( z  +P.  x )  =  ( 1P  +P.  y )  <->  ( x  +P.  z )  =  ( 1P  +P.  y ) )
4340, 42syl6bbr 263 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  <->  ( z  +P.  x )  =  ( 1P  +P.  y ) ) )
4443ancoms 450 . . . . . . . . 9  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  x  e.  P. )  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z
>. ]  ~R  <->  ( z  +P.  x )  =  ( 1P  +P.  y ) ) )
4544rexbidva 2730 . . . . . . . 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 221 . . . . . . 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 7186 . . . . . 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 6098 . . . . . . . 8  |-  ( [
<. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  ( C  +R  ( ( C  .R  -1R )  +R  A
) ) )
5049, 14sylan9eqr 2495 . . . . . . 7  |-  ( ( A  e.  R.  /\  [
<. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A ) )  -> 
( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A )
5150ex 434 . . . . . 6  |-  ( A  e.  R.  ->  ( [ <. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  -> 
( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A ) )
5251reximdv 2825 . . . . 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 44 . . . 4  |-  ( A  e.  R.  ->  ( -1R  <R  ( ( C  .R  -1R )  +R  A )  ->  E. x  e.  P.  ( C  +R  [
<. x ,  1P >. ]  ~R  )  =  A ) )
5416, 53sylbird 235 . . 3  |-  ( A  e.  R.  ->  (
( C  +R  -1R )  <R  A  ->  E. x  e.  P.  ( C  +R  [
<. x ,  1P >. ]  ~R  )  =  A ) )
553, 54mpcom 36 . 2  |-  ( ( C  +R  -1R )  <R  A  ->  E. x  e.  P.  ( C  +R  [
<. x ,  1P >. ]  ~R  )  =  A )
564mappsrpr 9271 . . . . 5  |-  ( ( C  +R  -1R )  <R  ( C  +R  [ <. x ,  1P >. ]  ~R  )  <->  x  e.  P. )
57 breq2 4293 . . . . 5  |-  ( ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A  ->  ( ( C  +R  -1R )  <R  ( C  +R  [
<. x ,  1P >. ]  ~R  )  <->  ( C  +R  -1R )  <R  A ) )
5856, 57syl5bbr 259 . . . 4  |-  ( ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A  ->  ( x  e.  P.  <->  ( C  +R  -1R )  <R  A ) )
5958biimpac 483 . . 3  |-  ( ( x  e.  P.  /\  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A )  ->  ( C  +R  -1R )  <R  A )
6059rexlimiva 2834 . 2  |-  ( E. x  e.  P.  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A  ->  ( C  +R  -1R )  <R  A )
6155, 60impbii 188 1  |-  ( ( C  +R  -1R )  <R  A  <->  E. x  e.  P.  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   E.wrex 2714   <.cop 3880   class class class wbr 4289  (class class class)co 6090   [cec 7095   P.cnp 9022   1Pc1p 9023    +P. cpp 9024    <P cltp 9026    ~R cer 9029   R.cnr 9030   0Rc0r 9031   -1Rcm1r 9033    +R cplr 9034    .R cmr 9035    <R cltr 9036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-omul 6921  df-er 7097  df-ec 7099  df-qs 7103  df-ni 9037  df-pli 9038  df-mi 9039  df-lti 9040  df-plpq 9073  df-mpq 9074  df-ltpq 9075  df-enq 9076  df-nq 9077  df-erq 9078  df-plq 9079  df-mq 9080  df-1nq 9081  df-rq 9082  df-ltnq 9083  df-np 9146  df-1p 9147  df-plp 9148  df-mp 9149  df-ltp 9150  df-plpr 9220  df-mpr 9221  df-enr 9222  df-nr 9223  df-plr 9224  df-mr 9225  df-ltr 9226  df-0r 9227  df-1r 9228  df-m1r 9229
This theorem is referenced by:  supsrlem  9274
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