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Theorem map2psrpr 9533
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 9491 . . . . 5  |-  <R  C_  ( R.  X.  R. )
21brel 4903 . . . 4  |-  ( ( C  +R  -1R )  <R  A  ->  ( ( C  +R  -1R )  e. 
R.  /\  A  e.  R. ) )
32simprd 464 . . 3  |-  ( ( C  +R  -1R )  <R  A  ->  A  e.  R. )
4 map2psrpr.2 . . . . . 6  |-  C  e. 
R.
5 ltasr 9523 . . . . . 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 9524 . . . . . . . . . 10  |-  ( C  e.  R.  ->  ( C  +R  ( C  .R  -1R ) )  =  0R )
84, 7ax-mp 5 . . . . . . . . 9  |-  ( C  +R  ( C  .R  -1R ) )  =  0R
98oveq1i 6315 . . . . . . . 8  |-  ( ( C  +R  ( C  .R  -1R ) )  +R  A )  =  ( 0R  +R  A
)
10 addasssr 9511 . . . . . . . 8  |-  ( ( C  +R  ( C  .R  -1R ) )  +R  A )  =  ( C  +R  (
( C  .R  -1R )  +R  A ) )
11 addcomsr 9510 . . . . . . . 8  |-  ( 0R 
+R  A )  =  ( A  +R  0R )
129, 10, 113eqtr3i 2466 . . . . . . 7  |-  ( C  +R  ( ( C  .R  -1R )  +R  A ) )  =  ( A  +R  0R )
13 0idsr 9520 . . . . . . 7  |-  ( A  e.  R.  ->  ( A  +R  0R )  =  A )
1412, 13syl5eq 2482 . . . . . 6  |-  ( A  e.  R.  ->  ( C  +R  ( ( C  .R  -1R )  +R  A ) )  =  A )
1514breq2d 4438 . . . . 5  |-  ( A  e.  R.  ->  (
( C  +R  -1R )  <R  ( C  +R  ( ( C  .R  -1R )  +R  A
) )  <->  ( C  +R  -1R )  <R  A ) )
166, 15syl5bb 260 . . . 4  |-  ( A  e.  R.  ->  ( -1R  <R  ( ( C  .R  -1R )  +R  A )  <->  ( C  +R  -1R )  <R  A ) )
17 m1r 9505 . . . . . . . 8  |-  -1R  e.  R.
18 mulclsr 9507 . . . . . . . 8  |-  ( ( C  e.  R.  /\  -1R  e.  R. )  -> 
( C  .R  -1R )  e.  R. )
194, 17, 18mp2an 676 . . . . . . 7  |-  ( C  .R  -1R )  e. 
R.
20 addclsr 9506 . . . . . . 7  |-  ( ( ( C  .R  -1R )  e.  R.  /\  A  e.  R. )  ->  (
( C  .R  -1R )  +R  A )  e. 
R. )
2119, 20mpan 674 . . . . . 6  |-  ( A  e.  R.  ->  (
( C  .R  -1R )  +R  A )  e. 
R. )
22 df-nr 9480 . . . . . . 7  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
23 breq2 4430 . . . . . . . 8  |-  ( [
<. y ,  z >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( -1R  <R  [ <. y ,  z >. ]  ~R  <->  -1R 
<R  ( ( C  .R  -1R )  +R  A
) ) )
24 eqeq2 2444 . . . . . . . . 9  |-  ( [
<. y ,  z >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  <->  [ <. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A ) ) )
2524rexbidv 2946 . . . . . . . 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 321 . . . . . . 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 9486 . . . . . . . . . . 11  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
2827breq1i 4433 . . . . . . . . . 10  |-  ( -1R 
<R  [ <. y ,  z
>. ]  ~R  <->  [ <. 1P , 
( 1P  +P.  1P ) >. ]  ~R  <R  [
<. y ,  z >. ]  ~R  )
29 addasspr 9446 . . . . . . . . . . . 12  |-  ( ( 1P  +P.  1P )  +P.  y )  =  ( 1P  +P.  ( 1P  +P.  y ) )
3029breq2i 4434 . . . . . . . . . . 11  |-  ( ( 1P  +P.  z ) 
<P  ( ( 1P  +P.  1P )  +P.  y )  <-> 
( 1P  +P.  z
)  <P  ( 1P  +P.  ( 1P  +P.  y ) ) )
31 ltsrpr 9500 . . . . . . . . . . 11  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  <R  [ <. y ,  z
>. ]  ~R  <->  ( 1P  +P.  z )  <P  (
( 1P  +P.  1P )  +P.  y ) )
32 1pr 9439 . . . . . . . . . . . 12  |-  1P  e.  P.
33 ltapr 9469 . . . . . . . . . . . 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 280 . . . . . . . . . 10  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  <R  [ <. y ,  z
>. ]  ~R  <->  z  <P  ( 1P  +P.  y ) )
3628, 35bitri 252 . . . . . . . . 9  |-  ( -1R 
<R  [ <. y ,  z
>. ]  ~R  <->  z  <P  ( 1P  +P.  y ) )
37 ltexpri 9467 . . . . . . . . 9  |-  ( z 
<P  ( 1P  +P.  y
)  ->  E. x  e.  P.  ( z  +P.  x )  =  ( 1P  +P.  y ) )
3836, 37sylbi 198 . . . . . . . 8  |-  ( -1R 
<R  [ <. y ,  z
>. ]  ~R  ->  E. x  e.  P.  ( z  +P.  x )  =  ( 1P  +P.  y ) )
39 enreceq 9489 . . . . . . . . . . . 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 685 . . . . . . . . . . 11  |-  ( ( x  e.  P.  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  <->  ( x  +P.  z )  =  ( 1P  +P.  y ) ) )
41 addcompr 9445 . . . . . . . . . . . 12  |-  ( z  +P.  x )  =  ( x  +P.  z
)
4241eqeq1i 2436 . . . . . . . . . . 11  |-  ( ( z  +P.  x )  =  ( 1P  +P.  y )  <->  ( x  +P.  z )  =  ( 1P  +P.  y ) )
4340, 42syl6bbr 266 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z >. ]  ~R  <->  ( z  +P.  x )  =  ( 1P  +P.  y ) ) )
4443ancoms 454 . . . . . . . . 9  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  x  e.  P. )  ->  ( [ <. x ,  1P >. ]  ~R  =  [ <. y ,  z
>. ]  ~R  <->  ( z  +P.  x )  =  ( 1P  +P.  y ) ) )
4544rexbidva 2943 . . . . . . . 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 224 . . . . . . 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 7461 . . . . . 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 17 . . . . 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 6313 . . . . . . . 8  |-  ( [
<. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  ->  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  ( C  +R  ( ( C  .R  -1R )  +R  A
) ) )
5049, 14sylan9eqr 2492 . . . . . . 7  |-  ( ( A  e.  R.  /\  [
<. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A ) )  -> 
( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A )
5150ex 435 . . . . . 6  |-  ( A  e.  R.  ->  ( [ <. x ,  1P >. ]  ~R  =  ( ( C  .R  -1R )  +R  A )  -> 
( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A ) )
5251reximdv 2906 . . . . 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 45 . . . 4  |-  ( A  e.  R.  ->  ( -1R  <R  ( ( C  .R  -1R )  +R  A )  ->  E. x  e.  P.  ( C  +R  [
<. x ,  1P >. ]  ~R  )  =  A ) )
5416, 53sylbird 238 . . 3  |-  ( A  e.  R.  ->  (
( C  +R  -1R )  <R  A  ->  E. x  e.  P.  ( C  +R  [
<. x ,  1P >. ]  ~R  )  =  A ) )
553, 54mpcom 37 . 2  |-  ( ( C  +R  -1R )  <R  A  ->  E. x  e.  P.  ( C  +R  [
<. x ,  1P >. ]  ~R  )  =  A )
564mappsrpr 9531 . . . . 5  |-  ( ( C  +R  -1R )  <R  ( C  +R  [ <. x ,  1P >. ]  ~R  )  <->  x  e.  P. )
57 breq2 4430 . . . . 5  |-  ( ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A  ->  ( ( C  +R  -1R )  <R  ( C  +R  [
<. x ,  1P >. ]  ~R  )  <->  ( C  +R  -1R )  <R  A ) )
5856, 57syl5bbr 262 . . . 4  |-  ( ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A  ->  ( x  e.  P.  <->  ( C  +R  -1R )  <R  A ) )
5958biimpac 488 . . 3  |-  ( ( x  e.  P.  /\  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A )  ->  ( C  +R  -1R )  <R  A )
6059rexlimiva 2920 . 2  |-  ( E. x  e.  P.  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A  ->  ( C  +R  -1R )  <R  A )
6155, 60impbii 190 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 187    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783   <.cop 4008   class class class wbr 4426  (class class class)co 6305   [cec 7369   P.cnp 9283   1Pc1p 9284    +P. cpp 9285    <P cltp 9287    ~R cer 9288   R.cnr 9289   0Rc0r 9290   -1Rcm1r 9292    +R cplr 9293    .R cmr 9294    <R cltr 9295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-omul 7195  df-er 7371  df-ec 7373  df-qs 7377  df-ni 9296  df-pli 9297  df-mi 9298  df-lti 9299  df-plpq 9332  df-mpq 9333  df-ltpq 9334  df-enq 9335  df-nq 9336  df-erq 9337  df-plq 9338  df-mq 9339  df-1nq 9340  df-rq 9341  df-ltnq 9342  df-np 9405  df-1p 9406  df-plp 9407  df-mp 9408  df-ltp 9409  df-enr 9479  df-nr 9480  df-plr 9481  df-mr 9482  df-ltr 9483  df-0r 9484  df-1r 9485  df-m1r 9486
This theorem is referenced by:  supsrlem  9534
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