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Theorem srpospr 7555
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 7499 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq2 3901 . . . 4  |-  ( [
<. a ,  b >. ]  ~R  =  A  -> 
( 0R  <R  [ <. a ,  b >. ]  ~R  <->  0R 
<R  A ) )
3 eqeq2 2125 . . . . 5  |-  ( [
<. a ,  b >. ]  ~R  =  A  -> 
( [ <. (
x  +P.  1P ) ,  1P >. ]  ~R  =  [ <. a ,  b
>. ]  ~R  <->  [ <. (
x  +P.  1P ) ,  1P >. ]  ~R  =  A ) )
43reubidv 2589 . . . 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 233 . . 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 7520 . . . . . . . 8  |-  ( 0R 
<R  [ <. a ,  b
>. ]  ~R  <->  b  <P  a )
76biimpi 119 . . . . . . 7  |-  ( 0R 
<R  [ <. a ,  b
>. ]  ~R  ->  b  <P  a )
87adantl 273 . . . . . 6  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  0R  <R  [ <. a ,  b >. ]  ~R  )  ->  b  <P  a
)
9 lteupri 7389 . . . . . 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 109 . . . . . . . . 9  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  x  e.  P. )
12 1pr 7326 . . . . . . . . . 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 7309 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  +P.  1P )  e.  P. )
1511, 13, 14syl2anc 406 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
x  +P.  1P )  e.  P. )
16 simplll 505 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  a  e.  P. )
17 simpllr 506 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  b  e.  P. )
18 enreceq 7508 . . . . . . . 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 1200 . . . . . . 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 7350 . . . . . . . . . . . 12  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  +P.  1P )  =  ( 1P  +P.  x ) )
2111, 13, 20syl2anc 406 . . . . . . . . . . 11  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
x  +P.  1P )  =  ( 1P  +P.  x ) )
2221oveq1d 5755 . . . . . . . . . 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 7351 . . . . . . . . . . 11  |-  ( ( 1P  e.  P.  /\  x  e.  P.  /\  b  e.  P. )  ->  (
( 1P  +P.  x
)  +P.  b )  =  ( 1P  +P.  ( x  +P.  b ) ) )
2413, 11, 17, 23syl3anc 1199 . . . . . . . . . 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 2148 . . . . . . . . 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 2124 . . . . . . . 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 7309 . . . . . . . . . . 11  |-  ( ( x  e.  P.  /\  b  e.  P. )  ->  ( x  +P.  b
)  e.  P. )
2811, 17, 27syl2anc 406 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
x  +P.  b )  e.  P. )
29 addcanprg 7388 . . . . . . . . . 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 1199 . . . . . . . . 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 5748 . . . . . . . . 9  |-  ( ( x  +P.  b )  =  a  ->  ( 1P  +P.  ( x  +P.  b ) )  =  ( 1P  +P.  a
) )
3230, 31impbid1 141 . . . . . . . 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 187 . . . . . . 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 7350 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  b  e.  P. )  ->  ( x  +P.  b
)  =  ( b  +P.  x ) )
3511, 17, 34syl2anc 406 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
x  +P.  b )  =  ( b  +P.  x ) )
3635eqeq1d 2124 . . . . . . 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 214 . . . . . 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 2588 . . . . 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 146 . . . 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 114 . . 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 6482 . 2  |-  ( A  e.  R.  ->  ( 0R  <R  A  ->  E! x  e.  P.  [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A ) )
4241imp 123 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 103    <-> wb 104    = wceq 1314    e. wcel 1463   E!wreu 2393   <.cop 3498   class class class wbr 3897  (class class class)co 5740   [cec 6393   P.cnp 7063   1Pc1p 7064    +P. cpp 7065    <P cltp 7067    ~R cer 7068   R.cnr 7069   0Rc0r 7070    <R cltr 7075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-i1p 7239  df-iplp 7240  df-iltp 7242  df-enr 7498  df-nr 7499  df-ltr 7502  df-0r 7503
This theorem is referenced by:  prsrriota  7560  caucvgsrlemcl  7561  caucvgsrlemgt1  7567
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