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Theorem leordtvallem1 16940
Description: Lemma for leordtval 16943. (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 10733 . . . . . 6  |-  ( x (,]  +oo )  C_  RR*
3 dfss1 3373 . . . . . 6  |-  ( ( x (,]  +oo )  C_ 
RR* 
<->  ( RR*  i^i  (
x (,]  +oo ) )  =  ( x (,] 
+oo ) )
42, 3mpbi 199 . . . . 5  |-  ( RR*  i^i  ( x (,]  +oo ) )  =  ( x (,]  +oo )
5 simpl 443 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  x  e.  RR* )
6 pnfxr 10455 . . . . . . . 8  |-  +oo  e.  RR*
7 elioc1 10698 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  +oo  e.  RR* )  ->  (
y  e.  ( x (,]  +oo )  <->  ( y  e.  RR*  /\  x  < 
y  /\  y  <_  +oo ) ) )
85, 6, 7sylancl 643 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  ( x (,]  +oo )  <->  ( y  e.  RR*  /\  x  < 
y  /\  y  <_  +oo ) ) )
9 simpr 447 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  y  e.  RR* )
10 pnfge 10469 . . . . . . . . . . 11  |-  ( y  e.  RR*  ->  y  <_  +oo )
1110adantl 452 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  y  <_  +oo )
129, 11jca 518 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  RR*  /\  y  <_  +oo ) )
1312biantrurd 494 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  <->  ( (
y  e.  RR*  /\  y  <_  +oo )  /\  x  <  y ) ) )
14 3anan32 946 . . . . . . . 8  |-  ( ( y  e.  RR*  /\  x  <  y  /\  y  <_  +oo )  <->  ( ( y  e.  RR*  /\  y  <_  +oo )  /\  x  <  y ) )
1513, 14syl6bbr 254 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  <->  ( y  e.  RR*  /\  x  < 
y  /\  y  <_  +oo ) ) )
16 xrltnle 8891 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  <->  -.  y  <_  x ) )
178, 15, 163bitr2d 272 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  ( x (,]  +oo )  <->  -.  y  <_  x ) )
1817rabbi2dva 3377 . . . . 5  |-  ( x  e.  RR*  ->  ( RR*  i^i  ( x (,]  +oo ) )  =  {
y  e.  RR*  |  -.  y  <_  x } )
194, 18syl5eqr 2329 . . . 4  |-  ( x  e.  RR*  ->  ( x (,]  +oo )  =  {
y  e.  RR*  |  -.  y  <_  x } )
2019mpteq2ia 4102 . . 3  |-  ( x  e.  RR*  |->  ( x (,]  +oo ) )  =  ( x  e.  RR*  |->  { y  e.  RR*  |  -.  y  <_  x } )
2120rneqi 4905 . 2  |-  ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  =  ran  ( x  e. 
RR*  |->  { y  e. 
RR*  |  -.  y  <_  x } )
221, 21eqtri 2303 1  |-  A  =  ran  ( x  e. 
RR*  |->  { y  e. 
RR*  |  -.  y  <_  x } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    i^i cin 3151    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   ran crn 4690  (class class class)co 5858    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868   (,]cioc 10657
This theorem is referenced by:  leordtval2  16942  leordtval  16943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-ioc 10661
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