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Theorem map2psrpr 9487
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 9445 . . . . 5  |-  <R  C_  ( R.  X.  R. )
21brel 5048 . . . 4  |-  ( ( C  +R  -1R )  <R  A  ->  ( ( C  +R  -1R )  e. 
R.  /\  A  e.  R. ) )
32simprd 463 . . 3  |-  ( ( C  +R  -1R )  <R  A  ->  A  e.  R. )
4 map2psrpr.2 . . . . . 6  |-  C  e. 
R.
5 ltasr 9477 . . . . . 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 9478 . . . . . . . . . 10  |-  ( C  e.  R.  ->  ( C  +R  ( C  .R  -1R ) )  =  0R )
84, 7ax-mp 5 . . . . . . . . 9  |-  ( C  +R  ( C  .R  -1R ) )  =  0R
98oveq1i 6294 . . . . . . . 8  |-  ( ( C  +R  ( C  .R  -1R ) )  +R  A )  =  ( 0R  +R  A
)
10 addasssr 9465 . . . . . . . 8  |-  ( ( C  +R  ( C  .R  -1R ) )  +R  A )  =  ( C  +R  (
( C  .R  -1R )  +R  A ) )
11 addcomsr 9464 . . . . . . . 8  |-  ( 0R 
+R  A )  =  ( A  +R  0R )
129, 10, 113eqtr3i 2504 . . . . . . 7  |-  ( C  +R  ( ( C  .R  -1R )  +R  A ) )  =  ( A  +R  0R )
13 0idsr 9474 . . . . . . 7  |-  ( A  e.  R.  ->  ( A  +R  0R )  =  A )
1412, 13syl5eq 2520 . . . . . 6  |-  ( A  e.  R.  ->  ( C  +R  ( ( C  .R  -1R )  +R  A ) )  =  A )
1514breq2d 4459 . . . . 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 9459 . . . . . . . 8  |-  -1R  e.  R.
18 mulclsr 9461 . . . . . . . 8  |-  ( ( C  e.  R.  /\  -1R  e.  R. )  -> 
( C  .R  -1R )  e.  R. )
194, 17, 18mp2an 672 . . . . . . 7  |-  ( C  .R  -1R )  e. 
R.
20 addclsr 9460 . . . . . . 7  |-  ( ( ( C  .R  -1R )  e.  R.  /\  A  e.  R. )  ->  (
( C  .R  -1R )  +R  A )  e. 
R. )
2119, 20mpan 670 . . . . . 6  |-  ( A  e.  R.  ->  (
( C  .R  -1R )  +R  A )  e. 
R. )
22 df-nr 9434 . . . . . . 7  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
23 breq2 4451 . . . . . . . 8  |-  ( [
<. y ,  z >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( -1R  <R  [ <. y ,  z >. ]  ~R  <->  -1R 
<R  ( ( C  .R  -1R )  +R  A
) ) )
24 eqeq2 2482 . . . . . . . . 9  |-  ( [
<. y ,  z >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  <->  [ <. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A ) ) )
2524rexbidv 2973 . . . . . . . 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 9440 . . . . . . . . . . 11  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
2827breq1i 4454 . . . . . . . . . 10  |-  ( -1R 
<R  [ <. y ,  z
>. ]  ~R  <->  [ <. 1P , 
( 1P  +P.  1P ) >. ]  ~R  <R  [
<. y ,  z >. ]  ~R  )
29 addasspr 9400 . . . . . . . . . . . 12  |-  ( ( 1P  +P.  1P )  +P.  y )  =  ( 1P  +P.  ( 1P  +P.  y ) )
3029breq2i 4455 . . . . . . . . . . 11  |-  ( ( 1P  +P.  z ) 
<P  ( ( 1P  +P.  1P )  +P.  y )  <-> 
( 1P  +P.  z
)  <P  ( 1P  +P.  ( 1P  +P.  y ) ) )
31 ltsrpr 9454 . . . . . . . . . . 11  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  <R  [ <. y ,  z
>. ]  ~R  <->  ( 1P  +P.  z )  <P  (
( 1P  +P.  1P )  +P.  y ) )
32 1pr 9393 . . . . . . . . . . . 12  |-  1P  e.  P.
33 ltapr 9423 . . . . . . . . . . . 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 9421 . . . . . . . . 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 9443 . . . . . . . . . . . 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 681 . . . . . . . . . . 11  |-  ( ( x  e.  P.  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  <->  ( x  +P.  z )  =  ( 1P  +P.  y ) ) )
41 addcompr 9399 . . . . . . . . . . . 12  |-  ( z  +P.  x )  =  ( x  +P.  z
)
4241eqeq1i 2474 . . . . . . . . . . 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 453 . . . . . . . . 9  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  x  e.  P. )  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z
>. ]  ~R  <->  ( z  +P.  x )  =  ( 1P  +P.  y ) ) )
4544rexbidva 2970 . . . . . . . 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 7401 . . . . . 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 6292 . . . . . . . 8  |-  ( [
<. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  ( C  +R  ( ( C  .R  -1R )  +R  A
) ) )
5049, 14sylan9eqr 2530 . . . . . . 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 2937 . . . . 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 9485 . . . . 5  |-  ( ( C  +R  -1R )  <R  ( C  +R  [ <. x ,  1P >. ]  ~R  )  <->  x  e.  P. )
57 breq2 4451 . . . . 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 486 . . 3  |-  ( ( x  e.  P.  /\  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A )  ->  ( C  +R  -1R )  <R  A )
6059rexlimiva 2951 . 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 1379    e. wcel 1767   E.wrex 2815   <.cop 4033   class class class wbr 4447  (class class class)co 6284   [cec 7309   P.cnp 9237   1Pc1p 9238    +P. cpp 9239    <P cltp 9241    ~R cer 9242   R.cnr 9243   0Rc0r 9244   -1Rcm1r 9246    +R cplr 9247    .R cmr 9248    <R cltr 9249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-omul 7135  df-er 7311  df-ec 7313  df-qs 7317  df-ni 9250  df-pli 9251  df-mi 9252  df-lti 9253  df-plpq 9286  df-mpq 9287  df-ltpq 9288  df-enq 9289  df-nq 9290  df-erq 9291  df-plq 9292  df-mq 9293  df-1nq 9294  df-rq 9295  df-ltnq 9296  df-np 9359  df-1p 9360  df-plp 9361  df-mp 9362  df-ltp 9363  df-enr 9433  df-nr 9434  df-plr 9435  df-mr 9436  df-ltr 9437  df-0r 9438  df-1r 9439  df-m1r 9440
This theorem is referenced by:  supsrlem  9488
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