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Theorem leisorel 10357
Description: Version of isorel 5625 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 946 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  F  Isom  <  ,  <  ( A ,  B )
)
2 simp3r 975 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  D  e.  A )
3 simp3l 974 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  C  e.  A )
4 isorel 5625 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( D  e.  A  /\  C  e.  A ) )  -> 
( D  <  C  <->  ( F `  D )  <  ( F `  C ) ) )
54notbid 630 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( D  e.  A  /\  C  e.  A ) )  -> 
( -.  D  < 
C  <->  -.  ( F `  D )  <  ( F `  C )
) )
61, 2, 3, 5syl12anc 1179 . 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 972 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  A  C_ 
RR* )
87, 3sseldd 3040 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  C  e.  RR* )
97, 2sseldd 3040 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  D  e.  RR* )
10 xrlenlt 7648 . . 3  |-  ( ( C  e.  RR*  /\  D  e.  RR* )  ->  ( C  <_  D  <->  -.  D  <  C ) )
118, 9, 10syl2anc 404 . 2  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  <_  D  <->  -.  D  <  C ) )
12 simp2r 973 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  B  C_ 
RR* )
13 isof1o 5624 . . . . . 6  |-  ( F 
Isom  <  ,  <  ( A ,  B )  ->  F : A -1-1-onto-> B )
14 f1of 5288 . . . . . 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 5474 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( F `  C )  e.  B )
1712, 16sseldd 3040 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( F `  C )  e.  RR* )
1815, 2ffvelrnd 5474 . . . 4  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( F `  D )  e.  B )
1912, 18sseldd 3040 . . 3  |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A  C_  RR* 
/\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( F `  D )  e.  RR* )
20 xrlenlt 7648 . . 3  |-  ( ( ( F `  C
)  e.  RR*  /\  ( F `  D )  e.  RR* )  ->  (
( F `  C
)  <_  ( F `  D )  <->  -.  ( F `  D )  <  ( F `  C
) ) )
2117, 19, 20syl2anc 404 . 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 927    e. wcel 1445    C_ wss 3013   class class class wbr 3867   -->wf 5045   -1-1-onto->wf1o 5048   ` cfv 5049    Isom wiso 5050   RR*cxr 7618    < clt 7619    <_ cle 7620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-f1o 5056  df-fv 5057  df-isom 5058  df-le 7625
This theorem is referenced by:  seq3coll  10362  summodclem2a  10924
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