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Theorem srpospr 6924
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 6869 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq2 3795 . . . 4  |-  ( [
<. a ,  b >. ]  ~R  =  A  -> 
( 0R  <R  [ <. a ,  b >. ]  ~R  <->  0R 
<R  A ) )
3 eqeq2 2065 . . . . 5  |-  ( [
<. a ,  b >. ]  ~R  =  A  -> 
( [ <. (
x  +P.  1P ) ,  1P >. ]  ~R  =  [ <. a ,  b
>. ]  ~R  <->  [ <. (
x  +P.  1P ) ,  1P >. ]  ~R  =  A ) )
43reubidv 2510 . . . 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 227 . . 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 6890 . . . . . . . 8  |-  ( 0R 
<R  [ <. a ,  b
>. ]  ~R  <->  b  <P  a )
76biimpi 117 . . . . . . 7  |-  ( 0R 
<R  [ <. a ,  b
>. ]  ~R  ->  b  <P  a )
87adantl 266 . . . . . 6  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  0R  <R  [ <. a ,  b >. ]  ~R  )  ->  b  <P  a
)
9 lteupri 6772 . . . . . 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 107 . . . . . . . . 9  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  x  e.  P. )
12 1pr 6709 . . . . . . . . . 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 6692 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  +P.  1P )  e.  P. )
1511, 13, 14syl2anc 397 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
x  +P.  1P )  e.  P. )
16 simplll 493 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  a  e.  P. )
17 simpllr 494 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  b  e.  P. )
18 enreceq 6878 . . . . . . . 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 1147 . . . . . . 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 6733 . . . . . . . . . . . 12  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  +P.  1P )  =  ( 1P  +P.  x ) )
2111, 13, 20syl2anc 397 . . . . . . . . . . 11  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
x  +P.  1P )  =  ( 1P  +P.  x ) )
2221oveq1d 5554 . . . . . . . . . 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 6734 . . . . . . . . . . 11  |-  ( ( 1P  e.  P.  /\  x  e.  P.  /\  b  e.  P. )  ->  (
( 1P  +P.  x
)  +P.  b )  =  ( 1P  +P.  ( x  +P.  b ) ) )
2413, 11, 17, 23syl3anc 1146 . . . . . . . . . 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 2088 . . . . . . . . 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 2064 . . . . . . . 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 6692 . . . . . . . . . . 11  |-  ( ( x  e.  P.  /\  b  e.  P. )  ->  ( x  +P.  b
)  e.  P. )
2811, 17, 27syl2anc 397 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
x  +P.  b )  e.  P. )
29 addcanprg 6771 . . . . . . . . . 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 1146 . . . . . . . . 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 5547 . . . . . . . . 9  |-  ( ( x  +P.  b )  =  a  ->  ( 1P  +P.  ( x  +P.  b ) )  =  ( 1P  +P.  a
) )
3230, 31impbid1 134 . . . . . . . 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 181 . . . . . . 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 6733 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  b  e.  P. )  ->  ( x  +P.  b
)  =  ( b  +P.  x ) )
3511, 17, 34syl2anc 397 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  0R  <R  [
<. a ,  b >. ]  ~R  )  /\  x  e.  P. )  ->  (
x  +P.  b )  =  ( b  +P.  x ) )
3635eqeq1d 2064 . . . . . . 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 208 . . . . . 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 2509 . . . . 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 139 . . . 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 112 . . 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 6223 . 2  |-  ( A  e.  R.  ->  ( 0R  <R  A  ->  E! x  e.  P.  [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A ) )
4241imp 119 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 101    <-> wb 102    = wceq 1259    e. wcel 1409   E!wreu 2325   <.cop 3405   class class class wbr 3791  (class class class)co 5539   [cec 6134   P.cnp 6446   1Pc1p 6447    +P. cpp 6448    <P cltp 6450    ~R cer 6451   R.cnr 6452   0Rc0r 6453    <R cltr 6458
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-i1p 6622  df-iplp 6623  df-iltp 6625  df-enr 6868  df-nr 6869  df-ltr 6872  df-0r 6873
This theorem is referenced by:  prsrriota  6929  caucvgsrlemcl  6930  caucvgsrlemgt1  6936
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