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Theorem prsrriota 7589
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 7584 . . 3  |-  ( ( A  e.  R.  /\  0R  <R  A )  ->  E! y  e.  P.  [
<. ( y  +P.  1P ) ,  1P >. ]  ~R  =  A )
2 reurex 2642 . . 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 521 . . . . 5  |-  ( ( ( A  e.  R.  /\  0R  <R  A )  /\  ( y  e.  P.  /\ 
[ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  =  A ) )  ->  [ <. ( y  +P.  1P ) ,  1P >. ]  ~R  =  A )
5 simprl 520 . . . . . 6  |-  ( ( ( A  e.  R.  /\  0R  <R  A )  /\  ( y  e.  P.  /\ 
[ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  =  A ) )  ->  y  e.  P. )
6 srpospr 7584 . . . . . . 7  |-  ( ( A  e.  R.  /\  0R  <R  A )  ->  E! x  e.  P.  [
<. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )
76adantr 274 . . . . . 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 5774 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  +P.  1P )  =  ( y  +P. 
1P ) )
98opeq1d 3706 . . . . . . . . 9  |-  ( x  =  y  ->  <. (
x  +P.  1P ) ,  1P >.  =  <. ( y  +P.  1P ) ,  1P >. )
109eceq1d 6458 . . . . . . . 8  |-  ( x  =  y  ->  [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  )
1110eqeq1d 2146 . . . . . . 7  |-  ( x  =  y  ->  ( [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A  <->  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  =  A ) )
1211riota2 5745 . . . . . 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 408 . . . . 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 5774 . . . . . 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 3706 . . . . 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 6458 . . . 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 2170 . 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 2552 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 1331    e. wcel 1480   E.wrex 2415   E!wreu 2416   <.cop 3525   class class class wbr 3924   iota_crio 5722  (class class class)co 5767   [cec 6420   P.cnp 7092   1Pc1p 7093    +P. cpp 7094    ~R cer 7097   R.cnr 7098   0Rc0r 7099    <R cltr 7104
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-i1p 7268  df-iplp 7269  df-iltp 7271  df-enr 7527  df-nr 7528  df-ltr 7531  df-0r 7532
This theorem is referenced by:  caucvgsrlemfv  7592  caucvgsrlemgt1  7596
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