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Theorem srpospr 7231
Description: Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.)
Assertion
Ref Expression
srpospr  |-  ( ( A  e.  R.  /\  0R  <R  A )  ->  E! x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )
Distinct variable group:    x, A

Proof of Theorem srpospr
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 7176 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq2 3815 . . . 4  |-  ( [
<. a ,  b >. ]  ~R  =  A  -> 
( 0R  <R  [ <. a ,  b >. ]  ~R  <->  0R 
<R  A ) )
3 eqeq2 2092 . . . . 5  |-  ( [
<. a ,  b >. ]  ~R  =  A  -> 
( [ <. (
x  +P.  1P ) ,  1P >. ]  ~R  =  [ <. a ,  b
>. ]  ~R  <->  [ <. (
x  +P.  1P ) ,  1P >. ]  ~R  =  A ) )
43reubidv 2543 . . . 4  |-  ( [
<. a ,  b >. ]  ~R  =  A  -> 
( E! x  e. 
P.  [ <. (
x  +P.  1P ) ,  1P >. ]  ~R  =  [ <. a ,  b
>. ]  ~R  <->  E! x  e.  P.  [ <. (
x  +P.  1P ) ,  1P >. ]  ~R  =  A ) )
52, 4imbi12d 232 . . 3  |-  ( [
<. a ,  b >. ]  ~R  =  A  -> 
( ( 0R  <R  [
<. a ,  b >. ]  ~R  ->  E! x  e.  P.  [ <. (
x  +P.  1P ) ,  1P >. ]  ~R  =  [ <. a ,  b
>. ]  ~R  )  <->  ( 0R  <R  A  ->  E! x  e.  P.  [ <. (
x  +P.  1P ) ,  1P >. ]  ~R  =  A ) ) )
6 gt0srpr 7197 . . . . . . . 8  |-  ( 0R 
<R  [ <. a ,  b
>. ]  ~R  <->  b  <P  a )
76biimpi 118 . . . . . . 7  |-  ( 0R 
<R  [ <. a ,  b
>. ]  ~R  ->  b  <P  a )
87adantl 271 . . . . . 6  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  0R  <R  [ <. a ,  b >. ]  ~R  )  ->  b  <P  a
)
9 lteupri 7079 . . . . . 6  |-  ( b 
<P  a  ->  E! x  e.  P.  ( b  +P.  x )  =  a )
108, 9syl 14 . . . . 5  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  0R  <R  [ <. a ,  b >. ]  ~R  )  ->  E! x  e. 
P.  ( b  +P.  x )  =  a )
11 simpr 108 . . . . . . . . 9  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  x  e.  P. )
12 1pr 7016 . . . . . . . . . 10  |-  1P  e.  P.
1312a1i 9 . . . . . . . . 9  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  1P  e.  P. )
14 addclpr 6999 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  +P.  1P )  e.  P. )
1511, 13, 14syl2anc 403 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
x  +P.  1P )  e.  P. )
16 simplll 500 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  a  e.  P. )
17 simpllr 501 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  b  e.  P. )
18 enreceq 7185 . . . . . . . 8  |-  ( ( ( ( x  +P.  1P )  e.  P.  /\  1P  e.  P. )  /\  ( a  e.  P.  /\  b  e.  P. )
)  ->  ( [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  [ <. a ,  b
>. ]  ~R  <->  ( (
x  +P.  1P )  +P.  b )  =  ( 1P  +P.  a ) ) )
1915, 13, 16, 17, 18syl22anc 1171 . . . . . . 7  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  ( [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  [ <. a ,  b >. ]  ~R  <->  ( ( x  +P.  1P )  +P.  b )  =  ( 1P  +P.  a
) ) )
20 addcomprg 7040 . . . . . . . . . . . 12  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  +P.  1P )  =  ( 1P  +P.  x ) )
2111, 13, 20syl2anc 403 . . . . . . . . . . 11  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
x  +P.  1P )  =  ( 1P  +P.  x ) )
2221oveq1d 5606 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
( x  +P.  1P )  +P.  b )  =  ( ( 1P  +P.  x )  +P.  b
) )
23 addassprg 7041 . . . . . . . . . . 11  |-  ( ( 1P  e.  P.  /\  x  e.  P.  /\  b  e.  P. )  ->  (
( 1P  +P.  x
)  +P.  b )  =  ( 1P  +P.  ( x  +P.  b ) ) )
2413, 11, 17, 23syl3anc 1170 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
( 1P  +P.  x
)  +P.  b )  =  ( 1P  +P.  ( x  +P.  b ) ) )
2522, 24eqtrd 2115 . . . . . . . . 9  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
( x  +P.  1P )  +P.  b )  =  ( 1P  +P.  (
x  +P.  b )
) )
2625eqeq1d 2091 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
( ( x  +P.  1P )  +P.  b )  =  ( 1P  +P.  a )  <->  ( 1P  +P.  ( x  +P.  b
) )  =  ( 1P  +P.  a ) ) )
27 addclpr 6999 . . . . . . . . . . 11  |-  ( ( x  e.  P.  /\  b  e.  P. )  ->  ( x  +P.  b
)  e.  P. )
2811, 17, 27syl2anc 403 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
x  +P.  b )  e.  P. )
29 addcanprg 7078 . . . . . . . . . 10  |-  ( ( 1P  e.  P.  /\  ( x  +P.  b )  e.  P.  /\  a  e.  P. )  ->  (
( 1P  +P.  (
x  +P.  b )
)  =  ( 1P 
+P.  a )  -> 
( x  +P.  b
)  =  a ) )
3013, 28, 16, 29syl3anc 1170 . . . . . . . . 9  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
( 1P  +P.  (
x  +P.  b )
)  =  ( 1P 
+P.  a )  -> 
( x  +P.  b
)  =  a ) )
31 oveq2 5599 . . . . . . . . 9  |-  ( ( x  +P.  b )  =  a  ->  ( 1P  +P.  ( x  +P.  b ) )  =  ( 1P  +P.  a
) )
3230, 31impbid1 140 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
( 1P  +P.  (
x  +P.  b )
)  =  ( 1P 
+P.  a )  <->  ( x  +P.  b )  =  a ) )
3326, 32bitrd 186 . . . . . . 7  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
( ( x  +P.  1P )  +P.  b )  =  ( 1P  +P.  a )  <->  ( x  +P.  b )  =  a ) )
34 addcomprg 7040 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  b  e.  P. )  ->  ( x  +P.  b
)  =  ( b  +P.  x ) )
3511, 17, 34syl2anc 403 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
x  +P.  b )  =  ( b  +P.  x ) )
3635eqeq1d 2091 . . . . . . 7  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
( x  +P.  b
)  =  a  <->  ( b  +P.  x )  =  a ) )
3719, 33, 363bitrrd 213 . . . . . 6  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
( b  +P.  x
)  =  a  <->  [ <. (
x  +P.  1P ) ,  1P >. ]  ~R  =  [ <. a ,  b
>. ]  ~R  ) )
3837reubidva 2542 . . . . 5  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  0R  <R  [ <. a ,  b >. ]  ~R  )  ->  ( E! x  e.  P.  ( b  +P.  x )  =  a  <-> 
E! x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  [ <. a ,  b
>. ]  ~R  ) )
3910, 38mpbid 145 . . . 4  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  0R  <R  [ <. a ,  b >. ]  ~R  )  ->  E! x  e. 
P.  [ <. (
x  +P.  1P ) ,  1P >. ]  ~R  =  [ <. a ,  b
>. ]  ~R  )
4039ex 113 . . 3  |-  ( ( a  e.  P.  /\  b  e.  P. )  ->  ( 0R  <R  [ <. a ,  b >. ]  ~R  ->  E! x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  [ <. a ,  b
>. ]  ~R  ) )
411, 5, 40ecoptocl 6309 . 2  |-  ( A  e.  R.  ->  ( 0R  <R  A  ->  E! x  e.  P.  [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A ) )
4241imp 122 1  |-  ( ( A  e.  R.  /\  0R  <R  A )  ->  E! x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   E!wreu 2355   <.cop 3425   class class class wbr 3811  (class class class)co 5591   [cec 6220   P.cnp 6753   1Pc1p 6754    +P. cpp 6755    <P cltp 6757    ~R cer 6758   R.cnr 6759   0Rc0r 6760    <R cltr 6765
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-2o 6114  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-enq0 6886  df-nq0 6887  df-0nq0 6888  df-plq0 6889  df-mq0 6890  df-inp 6928  df-i1p 6929  df-iplp 6930  df-iltp 6932  df-enr 7175  df-nr 7176  df-ltr 7179  df-0r 7180
This theorem is referenced by:  prsrriota  7236  caucvgsrlemcl  7237  caucvgsrlemgt1  7243
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