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Theorem leisorel 12471
Description: Version of isorel 6208 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)
Assertion
Ref Expression
leisorel  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  <_  D  <->  ( F `  C )  <_  ( F `  D )
) )

Proof of Theorem leisorel
StepHypRef Expression
1 leiso 12470 . . . 4  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  ( F  Isom  <  ,  <  ( A ,  B )  <-> 
F  Isom  <_  ,  <_  ( A ,  B ) ) )
21biimpcd 224 . . 3  |-  ( F 
Isom  <  ,  <  ( A ,  B )  ->  ( ( A  C_  RR* 
/\  B  C_  RR* )  ->  F  Isom  <_  ,  <_  ( A ,  B ) ) )
3 isorel 6208 . . . 4  |-  ( ( F  Isom  <_  ,  <_  ( A ,  B )  /\  ( C  e.  A  /\  D  e.  A ) )  -> 
( C  <_  D  <->  ( F `  C )  <_  ( F `  D ) ) )
43ex 434 . . 3  |-  ( F 
Isom  <_  ,  <_  ( A ,  B )  ->  ( ( C  e.  A  /\  D  e.  A )  ->  ( C  <_  D  <->  ( F `  C )  <_  ( F `  D )
) ) )
52, 4syl6 33 . 2  |-  ( F 
Isom  <  ,  <  ( A ,  B )  ->  ( ( A  C_  RR* 
/\  B  C_  RR* )  ->  ( ( C  e.  A  /\  D  e.  A )  ->  ( C  <_  D  <->  ( F `  C )  <_  ( F `  D )
) ) ) )
653imp 1190 1  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  <_  D  <->  ( F `  C )  <_  ( F `  D )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767    C_ wss 3476   class class class wbr 4447   ` cfv 5586    Isom wiso 5587   RR*cxr 9623    < clt 9624    <_ cle 9625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-le 9630
This theorem is referenced by:  seqcoll  12474  isercolllem2  13447  isercoll  13449  summolem2a  13496  xrhmeo  21181  prodmolem2a  28643  fourierdlem52  31459
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