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Theorem prsrriota 7750
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 7745 . . 3  |-  ( ( A  e.  R.  /\  0R  <R  A )  ->  E! y  e.  P.  [
<. ( y  +P.  1P ) ,  1P >. ]  ~R  =  A )
2 reurex 2683 . . 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 527 . . . . 5  |-  ( ( ( A  e.  R.  /\  0R  <R  A )  /\  ( y  e.  P.  /\ 
[ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  =  A ) )  ->  [ <. ( y  +P.  1P ) ,  1P >. ]  ~R  =  A )
5 simprl 526 . . . . . 6  |-  ( ( ( A  e.  R.  /\  0R  <R  A )  /\  ( y  e.  P.  /\ 
[ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  =  A ) )  ->  y  e.  P. )
6 srpospr 7745 . . . . . . 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 5860 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  +P.  1P )  =  ( y  +P. 
1P ) )
98opeq1d 3771 . . . . . . . . 9  |-  ( x  =  y  ->  <. (
x  +P.  1P ) ,  1P >.  =  <. ( y  +P.  1P ) ,  1P >. )
109eceq1d 6549 . . . . . . . 8  |-  ( x  =  y  ->  [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  )
1110eqeq1d 2179 . . . . . . 7  |-  ( x  =  y  ->  ( [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A  <->  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  =  A ) )
1211riota2 5831 . . . . . 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 409 . . . . 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 5860 . . . . . 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 3771 . . . . 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 6549 . . . 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 2203 . 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 2592 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 1348    e. wcel 2141   E.wrex 2449   E!wreu 2450   <.cop 3586   class class class wbr 3989   iota_crio 5808  (class class class)co 5853   [cec 6511   P.cnp 7253   1Pc1p 7254    +P. cpp 7255    ~R cer 7258   R.cnr 7259   0Rc0r 7260    <R cltr 7265
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-i1p 7429  df-iplp 7430  df-iltp 7432  df-enr 7688  df-nr 7689  df-ltr 7692  df-0r 7693
This theorem is referenced by:  caucvgsrlemfv  7753  caucvgsrlemgt1  7757
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