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Theorem leordtvallem1 17266
Description: Lemma for leordtval 17269. (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 10986 . . . . . 6  |-  ( x (,]  +oo )  C_  RR*
3 dfss1 3537 . . . . . 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 10705 . . . . . . . 8  |-  +oo  e.  RR*
7 elioc1 10950 . . . . . . . 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 10719 . . . . . . . . . . 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 9136 . . . . . . 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 3541 . . . . 5  |-  ( x  e.  RR*  ->  ( RR*  i^i  ( x (,]  +oo ) )  =  {
y  e.  RR*  |  -.  y  <_  x } )
194, 18syl5eqr 2481 . . . 4  |-  ( x  e.  RR*  ->  ( x (,]  +oo )  =  {
y  e.  RR*  |  -.  y  <_  x } )
2019mpteq2ia 4283 . . 3  |-  ( x  e.  RR*  |->  ( x (,]  +oo ) )  =  ( x  e.  RR*  |->  { y  e.  RR*  |  -.  y  <_  x } )
2120rneqi 5088 . 2  |-  ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  =  ran  ( x  e. 
RR*  |->  { y  e. 
RR*  |  -.  y  <_  x } )
221, 21eqtri 2455 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 1652    e. wcel 1725   {crab 2701    i^i cin 3311    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   ran crn 4871  (class class class)co 6073    +oocpnf 9109   RR*cxr 9111    < clt 9112    <_ cle 9113   (,]cioc 10909
This theorem is referenced by:  leordtval2  17268  leordtval  17269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-ioc 10913
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