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Theorem leisorel 10835
Description: Version of isorel 5825 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 999 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  F  Isom  <  ,  <  ( A ,  B )
)
2 simp3r 1028 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  D  e.  A )
3 simp3l 1027 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  C  e.  A )
4 isorel 5825 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( D  e.  A  /\  C  e.  A ) )  -> 
( D  <  C  <->  ( F `  D )  <  ( F `  C ) ) )
54notbid 668 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( D  e.  A  /\  C  e.  A ) )  -> 
( -.  D  < 
C  <->  -.  ( F `  D )  <  ( F `  C )
) )
61, 2, 3, 5syl12anc 1247 . 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 1025 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  A  C_ 
RR* )
87, 3sseldd 3171 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  C  e.  RR* )
97, 2sseldd 3171 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  D  e.  RR* )
10 xrlenlt 8040 . . 3  |-  ( ( C  e.  RR*  /\  D  e.  RR* )  ->  ( C  <_  D  <->  -.  D  <  C ) )
118, 9, 10syl2anc 411 . 2  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  <_  D  <->  -.  D  <  C ) )
12 simp2r 1026 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  B  C_ 
RR* )
13 isof1o 5824 . . . . . 6  |-  ( F 
Isom  <  ,  <  ( A ,  B )  ->  F : A -1-1-onto-> B )
14 f1of 5476 . . . . . 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, 3ffvelcdmd 5668 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( F `  C )  e.  B )
1712, 16sseldd 3171 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( F `  C )  e.  RR* )
1815, 2ffvelcdmd 5668 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( F `  D )  e.  B )
1912, 18sseldd 3171 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( F `  D )  e.  RR* )
20 xrlenlt 8040 . . 3  |-  ( ( ( F `  C
)  e.  RR*  /\  ( F `  D )  e.  RR* )  ->  (
( F `  C
)  <_  ( F `  D )  <->  -.  ( F `  D )  <  ( F `  C
) ) )
2117, 19, 20syl2anc 411 . 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 220 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 104    <-> wb 105    /\ w3a 980    e. wcel 2160    C_ wss 3144   class class class wbr 4018   -->wf 5227   -1-1-onto->wf1o 5230   ` cfv 5231    Isom wiso 5232   RR*cxr 8009    < clt 8010    <_ cle 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-f1o 5238  df-fv 5239  df-isom 5240  df-le 8016
This theorem is referenced by:  seq3coll  10840  summodclem2a  11407  prodmodclem2a  11602
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