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Theorem prsrriota 7394
Description: Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
Assertion
Ref Expression
prsrriota  |-  ( ( A  e.  R.  /\  0R  <R  A )  ->  [ <. ( ( iota_ x  e.  P.  [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  +P.  1P ) ,  1P >. ]  ~R  =  A )
Distinct variable group:    x, A

Proof of Theorem prsrriota
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 srpospr 7389 . . 3  |-  ( ( A  e.  R.  /\  0R  <R  A )  ->  E! y  e.  P.  [
<. ( y  +P.  1P ) ,  1P >. ]  ~R  =  A )
2 reurex 2581 . . 3  |-  ( E! y  e.  P.  [ <. ( y  +P.  1P ) ,  1P >. ]  ~R  =  A  ->  E. y  e.  P.  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  =  A )
31, 2syl 14 . 2  |-  ( ( A  e.  R.  /\  0R  <R  A )  ->  E. y  e.  P.  [
<. ( y  +P.  1P ) ,  1P >. ]  ~R  =  A )
4 simprr 500 . . . . 5  |-  ( ( ( A  e.  R.  /\  0R  <R  A )  /\  ( y  e.  P.  /\ 
[ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  =  A ) )  ->  [ <. ( y  +P.  1P ) ,  1P >. ]  ~R  =  A )
5 simprl 499 . . . . . 6  |-  ( ( ( A  e.  R.  /\  0R  <R  A )  /\  ( y  e.  P.  /\ 
[ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  =  A ) )  ->  y  e.  P. )
6 srpospr 7389 . . . . . . 7  |-  ( ( A  e.  R.  /\  0R  <R  A )  ->  E! x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )
76adantr 271 . . . . . 6  |-  ( ( ( A  e.  R.  /\  0R  <R  A )  /\  ( y  e.  P.  /\ 
[ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  =  A ) )  ->  E! x  e.  P.  [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )
8 oveq1 5673 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  +P.  1P )  =  ( y  +P. 
1P ) )
98opeq1d 3634 . . . . . . . . 9  |-  ( x  =  y  ->  <. (
x  +P.  1P ) ,  1P >.  =  <. ( y  +P.  1P ) ,  1P >. )
109eceq1d 6342 . . . . . . . 8  |-  ( x  =  y  ->  [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  )
1110eqeq1d 2097 . . . . . . 7  |-  ( x  =  y  ->  ( [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A  <->  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  =  A ) )
1211riota2 5644 . . . . . 6  |-  ( ( y  e.  P.  /\  E! x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  ->  ( [ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  =  A  <-> 
( iota_ x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  =  y ) )
135, 7, 12syl2anc 404 . . . . 5  |-  ( ( ( A  e.  R.  /\  0R  <R  A )  /\  ( y  e.  P.  /\ 
[ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  =  A ) )  ->  ( [ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  =  A  <-> 
( iota_ x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  =  y ) )
144, 13mpbid 146 . . . 4  |-  ( ( ( A  e.  R.  /\  0R  <R  A )  /\  ( y  e.  P.  /\ 
[ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  =  A ) )  ->  ( iota_ x  e.  P.  [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  =  y )
15 oveq1 5673 . . . . . 6  |-  ( (
iota_ x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  =  y  ->  ( ( iota_ x  e.  P.  [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  +P.  1P )  =  ( y  +P.  1P ) )
1615opeq1d 3634 . . . . 5  |-  ( (
iota_ x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  =  y  ->  <. ( ( iota_ x  e.  P.  [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  +P.  1P ) ,  1P >.  =  <. ( y  +P.  1P ) ,  1P >. )
1716eceq1d 6342 . . . 4  |-  ( (
iota_ x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  =  y  ->  [ <. (
( iota_ x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  )
1814, 17syl 14 . . 3  |-  ( ( ( A  e.  R.  /\  0R  <R  A )  /\  ( y  e.  P.  /\ 
[ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  =  A ) )  ->  [ <. ( ( iota_ x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  )
1918, 4eqtrd 2121 . 2  |-  ( ( ( A  e.  R.  /\  0R  <R  A )  /\  ( y  e.  P.  /\ 
[ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  =  A ) )  ->  [ <. ( ( iota_ x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  +P.  1P ) ,  1P >. ]  ~R  =  A )
203, 19rexlimddv 2494 1  |-  ( ( A  e.  R.  /\  0R  <R  A )  ->  [ <. ( ( iota_ x  e.  P.  [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  +P.  1P ) ,  1P >. ]  ~R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   E.wrex 2361   E!wreu 2362   <.cop 3453   class class class wbr 3851   iota_crio 5621  (class class class)co 5666   [cec 6304   P.cnp 6911   1Pc1p 6912    +P. cpp 6913    ~R cer 6916   R.cnr 6917   0Rc0r 6918    <R cltr 6923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-2o 6196  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-pli 6925  df-mi 6926  df-lti 6927  df-plpq 6964  df-mpq 6965  df-enq 6967  df-nqqs 6968  df-plqqs 6969  df-mqqs 6970  df-1nqqs 6971  df-rq 6972  df-ltnqqs 6973  df-enq0 7044  df-nq0 7045  df-0nq0 7046  df-plq0 7047  df-mq0 7048  df-inp 7086  df-i1p 7087  df-iplp 7088  df-iltp 7090  df-enr 7333  df-nr 7334  df-ltr 7337  df-0r 7338
This theorem is referenced by:  caucvgsrlemfv  7397  caucvgsrlemgt1  7401
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