ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  leisorel Unicode version

Theorem leisorel 10592
Description: Version of isorel 5709 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 simp1 981 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  F  Isom  <  ,  <  ( A ,  B )
)
2 simp3r 1010 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  D  e.  A )
3 simp3l 1009 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  C  e.  A )
4 isorel 5709 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( D  e.  A  /\  C  e.  A ) )  -> 
( D  <  C  <->  ( F `  D )  <  ( F `  C ) ) )
54notbid 656 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( D  e.  A  /\  C  e.  A ) )  -> 
( -.  D  < 
C  <->  -.  ( F `  D )  <  ( F `  C )
) )
61, 2, 3, 5syl12anc 1214 . 2  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( -.  D  <  C  <->  -.  ( F `  D )  <  ( F `  C
) ) )
7 simp2l 1007 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  A  C_ 
RR* )
87, 3sseldd 3098 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  C  e.  RR* )
97, 2sseldd 3098 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  D  e.  RR* )
10 xrlenlt 7841 . . 3  |-  ( ( C  e.  RR*  /\  D  e.  RR* )  ->  ( C  <_  D  <->  -.  D  <  C ) )
118, 9, 10syl2anc 408 . 2  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  <_  D  <->  -.  D  <  C ) )
12 simp2r 1008 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  B  C_ 
RR* )
13 isof1o 5708 . . . . . 6  |-  ( F 
Isom  <  ,  <  ( A ,  B )  ->  F : A -1-1-onto-> B )
14 f1of 5367 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
151, 13, 143syl 17 . . . . 5  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  F : A --> B )
1615, 3ffvelrnd 5556 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( F `  C )  e.  B )
1712, 16sseldd 3098 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( F `  C )  e.  RR* )
1815, 2ffvelrnd 5556 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( F `  D )  e.  B )
1912, 18sseldd 3098 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( F `  D )  e.  RR* )
20 xrlenlt 7841 . . 3  |-  ( ( ( F `  C
)  e.  RR*  /\  ( F `  D )  e.  RR* )  ->  (
( F `  C
)  <_  ( F `  D )  <->  -.  ( F `  D )  <  ( F `  C
) ) )
2117, 19, 20syl2anc 408 . 2  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  (
( F `  C
)  <_  ( F `  D )  <->  -.  ( F `  D )  <  ( F `  C
) ) )
226, 11, 213bitr4d 219 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    e. wcel 1480    C_ wss 3071   class class class wbr 3929   -->wf 5119   -1-1-onto->wf1o 5122   ` cfv 5123    Isom wiso 5124   RR*cxr 7811    < clt 7812    <_ cle 7813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-f1o 5130  df-fv 5131  df-isom 5132  df-le 7818
This theorem is referenced by:  seq3coll  10597  summodclem2a  11162  prodmodclem2a  11357
  Copyright terms: Public domain W3C validator