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Theorem leordtvallem1 19896
Description: Lemma for leordtval 19899. (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 11579 . . . . . 6  |-  ( x (,] +oo )  C_  RR*
3 dfss1 3643 . . . . . 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 455 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  x  e.  RR* )
6 pnfxr 11292 . . . . . . . 8  |- +oo  e.  RR*
7 elioc1 11542 . . . . . . . 8  |-  ( ( x  e.  RR*  /\ +oo  e.  RR* )  ->  (
y  e.  ( x (,] +oo )  <->  ( y  e.  RR*  /\  x  < 
y  /\  y  <_ +oo ) ) )
85, 6, 7sylancl 660 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  ( x (,] +oo )  <->  ( y  e.  RR*  /\  x  < 
y  /\  y  <_ +oo ) ) )
9 simpr 459 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  y  e.  RR* )
10 pnfge 11310 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  y  <_ +oo )
119, 10jccir 537 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  RR*  /\  y  <_ +oo ) )
1211biantrurd 506 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  <->  ( (
y  e.  RR*  /\  y  <_ +oo )  /\  x  <  y ) ) )
13 3anan32 986 . . . . . . . 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 9603 . . . . . . 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 3646 . . . . 5  |-  ( x  e.  RR*  ->  ( RR*  i^i  ( x (,] +oo ) )  =  {
y  e.  RR*  |  -.  y  <_  x } )
184, 17syl5eqr 2457 . . . 4  |-  ( x  e.  RR*  ->  ( x (,] +oo )  =  { y  e.  RR*  |  -.  y  <_  x } )
1918mpteq2ia 4476 . . 3  |-  ( x  e.  RR*  |->  ( x (,] +oo ) )  =  ( x  e. 
RR*  |->  { y  e. 
RR*  |  -.  y  <_  x } )
2019rneqi 5171 . 2  |-  ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  =  ran  ( x  e.  RR*  |->  { y  e.  RR*  |  -.  y  <_  x } )
211, 20eqtri 2431 1  |-  A  =  ran  ( x  e. 
RR*  |->  { y  e. 
RR*  |  -.  y  <_  x } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   {crab 2757    i^i cin 3412    C_ wss 3413   class class class wbr 4394    |-> cmpt 4452   ran crn 4943  (class class class)co 6234   +oocpnf 9575   RR*cxr 9577    < clt 9578    <_ cle 9579   (,]cioc 11501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-ioc 11505
This theorem is referenced by:  leordtval2  19898  leordtval  19899
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