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Theorem leordtvallem1 19470
Description: Lemma for leordtval 19473. (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 11597 . . . . . 6  |-  ( x (,] +oo )  C_  RR*
3 dfss1 3696 . . . . . 6  |-  ( ( x (,] +oo )  C_ 
RR* 
<->  ( RR*  i^i  (
x (,] +oo )
)  =  ( x (,] +oo ) )
42, 3mpbi 208 . . . . 5  |-  ( RR*  i^i  ( x (,] +oo ) )  =  ( x (,] +oo )
5 simpl 457 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  x  e.  RR* )
6 pnfxr 11310 . . . . . . . 8  |- +oo  e.  RR*
7 elioc1 11560 . . . . . . . 8  |-  ( ( x  e.  RR*  /\ +oo  e.  RR* )  ->  (
y  e.  ( x (,] +oo )  <->  ( y  e.  RR*  /\  x  < 
y  /\  y  <_ +oo ) ) )
85, 6, 7sylancl 662 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  ( x (,] +oo )  <->  ( y  e.  RR*  /\  x  < 
y  /\  y  <_ +oo ) ) )
9 simpr 461 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  y  e.  RR* )
10 pnfge 11328 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  y  <_ +oo )
119, 10jccir 539 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  RR*  /\  y  <_ +oo ) )
1211biantrurd 508 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  <->  ( (
y  e.  RR*  /\  y  <_ +oo )  /\  x  <  y ) ) )
13 3anan32 980 . . . . . . . 8  |-  ( ( y  e.  RR*  /\  x  <  y  /\  y  <_ +oo )  <->  ( ( y  e.  RR*  /\  y  <_ +oo )  /\  x  <  y ) )
1412, 13syl6bbr 263 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  <->  ( y  e.  RR*  /\  x  < 
y  /\  y  <_ +oo ) ) )
15 xrltnle 9642 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  <->  -.  y  <_  x ) )
168, 14, 153bitr2d 281 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  ( x (,] +oo )  <->  -.  y  <_  x ) )
1716rabbi2dva 3699 . . . . 5  |-  ( x  e.  RR*  ->  ( RR*  i^i  ( x (,] +oo ) )  =  {
y  e.  RR*  |  -.  y  <_  x } )
184, 17syl5eqr 2515 . . . 4  |-  ( x  e.  RR*  ->  ( x (,] +oo )  =  { y  e.  RR*  |  -.  y  <_  x } )
1918mpteq2ia 4522 . . 3  |-  ( x  e.  RR*  |->  ( x (,] +oo ) )  =  ( x  e. 
RR*  |->  { y  e. 
RR*  |  -.  y  <_  x } )
2019rneqi 5220 . 2  |-  ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  =  ran  ( x  e.  RR*  |->  { y  e.  RR*  |  -.  y  <_  x } )
211, 20eqtri 2489 1  |-  A  =  ran  ( x  e. 
RR*  |->  { y  e. 
RR*  |  -.  y  <_  x } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   {crab 2811    i^i cin 3468    C_ wss 3469   class class class wbr 4440    |-> cmpt 4498   ran crn 4993  (class class class)co 6275   +oocpnf 9614   RR*cxr 9616    < clt 9617    <_ cle 9618   (,]cioc 11519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-ioc 11523
This theorem is referenced by:  leordtval2  19472  leordtval  19473
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