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Theorem leordtvallem1 17198
Description: Lemma for leordtval 17201. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
leordtval.1  |-  A  =  ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )
Assertion
Ref Expression
leordtvallem1  |-  A  =  ran  ( x  e. 
RR*  |->  { y  e. 
RR*  |  -.  y  <_  x } )
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)

Proof of Theorem leordtvallem1
StepHypRef Expression
1 leordtval.1 . 2  |-  A  =  ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )
2 iocssxr 10928 . . . . . 6  |-  ( x (,]  +oo )  C_  RR*
3 dfss1 3490 . . . . . 6  |-  ( ( x (,]  +oo )  C_ 
RR* 
<->  ( RR*  i^i  (
x (,]  +oo ) )  =  ( x (,] 
+oo ) )
42, 3mpbi 200 . . . . 5  |-  ( RR*  i^i  ( x (,]  +oo ) )  =  ( x (,]  +oo )
5 simpl 444 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  x  e.  RR* )
6 pnfxr 10647 . . . . . . . 8  |-  +oo  e.  RR*
7 elioc1 10892 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  +oo  e.  RR* )  ->  (
y  e.  ( x (,]  +oo )  <->  ( y  e.  RR*  /\  x  < 
y  /\  y  <_  +oo ) ) )
85, 6, 7sylancl 644 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  ( x (,]  +oo )  <->  ( y  e.  RR*  /\  x  < 
y  /\  y  <_  +oo ) ) )
9 simpr 448 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  y  e.  RR* )
10 pnfge 10661 . . . . . . . . . . 11  |-  ( y  e.  RR*  ->  y  <_  +oo )
1110adantl 453 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  y  <_  +oo )
129, 11jca 519 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  RR*  /\  y  <_  +oo ) )
1312biantrurd 495 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  <->  ( (
y  e.  RR*  /\  y  <_  +oo )  /\  x  <  y ) ) )
14 3anan32 948 . . . . . . . 8  |-  ( ( y  e.  RR*  /\  x  <  y  /\  y  <_  +oo )  <->  ( ( y  e.  RR*  /\  y  <_  +oo )  /\  x  <  y ) )
1513, 14syl6bbr 255 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  <->  ( y  e.  RR*  /\  x  < 
y  /\  y  <_  +oo ) ) )
16 xrltnle 9079 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  <->  -.  y  <_  x ) )
178, 15, 163bitr2d 273 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  ( x (,]  +oo )  <->  -.  y  <_  x ) )
1817rabbi2dva 3494 . . . . 5  |-  ( x  e.  RR*  ->  ( RR*  i^i  ( x (,]  +oo ) )  =  {
y  e.  RR*  |  -.  y  <_  x } )
194, 18syl5eqr 2435 . . . 4  |-  ( x  e.  RR*  ->  ( x (,]  +oo )  =  {
y  e.  RR*  |  -.  y  <_  x } )
2019mpteq2ia 4234 . . 3  |-  ( x  e.  RR*  |->  ( x (,]  +oo ) )  =  ( x  e.  RR*  |->  { y  e.  RR*  |  -.  y  <_  x } )
2120rneqi 5038 . 2  |-  ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  =  ran  ( x  e. 
RR*  |->  { y  e. 
RR*  |  -.  y  <_  x } )
221, 21eqtri 2409 1  |-  A  =  ran  ( x  e. 
RR*  |->  { y  e. 
RR*  |  -.  y  <_  x } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {crab 2655    i^i cin 3264    C_ wss 3265   class class class wbr 4155    e. cmpt 4209   ran crn 4821  (class class class)co 6022    +oocpnf 9052   RR*cxr 9054    < clt 9055    <_ cle 9056   (,]cioc 10851
This theorem is referenced by:  leordtval2  17200  leordtval  17201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-ioc 10855
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