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Theorem leiso 12204
Description: Two ways to write a strictly increasing function on the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
Assertion
Ref Expression
leiso  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  ( F  Isom  <  ,  <  ( A ,  B )  <-> 
F  Isom  <_  ,  <_  ( A ,  B ) ) )

Proof of Theorem leiso
StepHypRef Expression
1 df-le 9416 . . . . . . 7  |-  <_  =  ( ( RR*  X.  RR* )  \  `'  <  )
21ineq1i 3543 . . . . . 6  |-  (  <_  i^i  ( A  X.  A
) )  =  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( A  X.  A
) )
3 indif1 3589 . . . . . 6  |-  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( A  X.  A
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  \  `'  <  )
42, 3eqtri 2458 . . . . 5  |-  (  <_  i^i  ( A  X.  A
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  \  `'  <  )
5 xpss12 4940 . . . . . . . 8  |-  ( ( A  C_  RR*  /\  A  C_ 
RR* )  ->  ( A  X.  A )  C_  ( RR*  X.  RR* )
)
65anidms 645 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( A  X.  A )  C_  ( RR*  X.  RR* ) )
7 dfss1 3550 . . . . . . 7  |-  ( ( A  X.  A ) 
C_  ( RR*  X.  RR* ) 
<->  ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  =  ( A  X.  A
) )
86, 7sylib 196 . . . . . 6  |-  ( A 
C_  RR*  ->  ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  =  ( A  X.  A ) )
98difeq1d 3468 . . . . 5  |-  ( A 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( A  X.  A
) )  \  `'  <  )  =  ( ( A  X.  A ) 
\  `'  <  )
)
104, 9syl5req 2483 . . . 4  |-  ( A 
C_  RR*  ->  ( ( A  X.  A )  \  `'  <  )  =  (  <_  i^i  ( A  X.  A ) ) )
11 isoeq2 6006 . . . 4  |-  ( ( ( A  X.  A
)  \  `'  <  )  =  (  <_  i^i  ( A  X.  A
) )  ->  ( F  Isom  ( ( A  X.  A )  \  `'  <  ) ,  ( ( B  X.  B
)  \  `'  <  ) ( A ,  B
)  <->  F  Isom  (  <_  i^i  ( A  X.  A
) ) ,  ( ( B  X.  B
)  \  `'  <  ) ( A ,  B
) ) )
1210, 11syl 16 . . 3  |-  ( A 
C_  RR*  ->  ( F  Isom  ( ( A  X.  A )  \  `'  <  ) ,  ( ( B  X.  B ) 
\  `'  <  )
( A ,  B
)  <->  F  Isom  (  <_  i^i  ( A  X.  A
) ) ,  ( ( B  X.  B
)  \  `'  <  ) ( A ,  B
) ) )
131ineq1i 3543 . . . . . 6  |-  (  <_  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )
14 indif1 3589 . . . . . 6  |-  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
1513, 14eqtri 2458 . . . . 5  |-  (  <_  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
16 xpss12 4940 . . . . . . . 8  |-  ( ( B  C_  RR*  /\  B  C_ 
RR* )  ->  ( B  X.  B )  C_  ( RR*  X.  RR* )
)
1716anidms 645 . . . . . . 7  |-  ( B 
C_  RR*  ->  ( B  X.  B )  C_  ( RR*  X.  RR* ) )
18 dfss1 3550 . . . . . . 7  |-  ( ( B  X.  B ) 
C_  ( RR*  X.  RR* ) 
<->  ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  =  ( B  X.  B
) )
1917, 18sylib 196 . . . . . 6  |-  ( B 
C_  RR*  ->  ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  =  ( B  X.  B ) )
2019difeq1d 3468 . . . . 5  |-  ( B 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( B  X.  B
) )  \  `'  <  )  =  ( ( B  X.  B ) 
\  `'  <  )
)
2115, 20syl5req 2483 . . . 4  |-  ( B 
C_  RR*  ->  ( ( B  X.  B )  \  `'  <  )  =  (  <_  i^i  ( B  X.  B ) ) )
22 isoeq3 6007 . . . 4  |-  ( ( ( B  X.  B
)  \  `'  <  )  =  (  <_  i^i  ( B  X.  B
) )  ->  ( F  Isom  (  <_  i^i  ( A  X.  A
) ) ,  ( ( B  X.  B
)  \  `'  <  ) ( A ,  B
)  <->  F  Isom  (  <_  i^i  ( A  X.  A
) ) ,  (  <_  i^i  ( B  X.  B ) ) ( A ,  B ) ) )
2321, 22syl 16 . . 3  |-  ( B 
C_  RR*  ->  ( F  Isom  (  <_  i^i  ( A  X.  A ) ) ,  ( ( B  X.  B )  \  `'  <  ) ( A ,  B )  <->  F  Isom  (  <_  i^i  ( A  X.  A ) ) ,  (  <_  i^i  ( B  X.  B ) ) ( A ,  B
) ) )
2412, 23sylan9bb 699 . 2  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  ( F  Isom  ( ( A  X.  A )  \  `'  <  ) ,  ( ( B  X.  B
)  \  `'  <  ) ( A ,  B
)  <->  F  Isom  (  <_  i^i  ( A  X.  A
) ) ,  (  <_  i^i  ( B  X.  B ) ) ( A ,  B ) ) )
25 isocnv2 6017 . . 3  |-  ( F 
Isom  <  ,  <  ( A ,  B )  <->  F 
Isom  `'  <  ,  `'  <  ( A ,  B
) )
26 eqid 2438 . . . 4  |-  ( ( A  X.  A ) 
\  `'  <  )  =  ( ( A  X.  A )  \  `'  <  )
27 eqid 2438 . . . 4  |-  ( ( B  X.  B ) 
\  `'  <  )  =  ( ( B  X.  B )  \  `'  <  )
2826, 27isocnv3 6018 . . 3  |-  ( F 
Isom  `'  <  ,  `'  <  ( A ,  B
)  <->  F  Isom  ( ( A  X.  A ) 
\  `'  <  ) ,  ( ( B  X.  B )  \  `'  <  ) ( A ,  B ) )
2925, 28bitri 249 . 2  |-  ( F 
Isom  <  ,  <  ( A ,  B )  <->  F 
Isom  ( ( A  X.  A )  \  `'  <  ) ,  ( ( B  X.  B
)  \  `'  <  ) ( A ,  B
) )
30 isores1 6020 . . 3  |-  ( F 
Isom  <_  ,  <_  ( A ,  B )  <->  F 
Isom  (  <_  i^i  ( A  X.  A ) ) ,  <_  ( A ,  B ) )
31 isores2 6019 . . 3  |-  ( F 
Isom  (  <_  i^i  ( A  X.  A ) ) ,  <_  ( A ,  B )  <->  F  Isom  (  <_  i^i  ( A  X.  A ) ) ,  (  <_  i^i  ( B  X.  B ) ) ( A ,  B
) )
3230, 31bitri 249 . 2  |-  ( F 
Isom  <_  ,  <_  ( A ,  B )  <->  F 
Isom  (  <_  i^i  ( A  X.  A ) ) ,  (  <_  i^i  ( B  X.  B
) ) ( A ,  B ) )
3324, 29, 323bitr4g 288 1  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  ( F  Isom  <  ,  <  ( A ,  B )  <-> 
F  Isom  <_  ,  <_  ( A ,  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    \ cdif 3320    i^i cin 3322    C_ wss 3323    X. cxp 4833   `'ccnv 4834    Isom wiso 5414   RR*cxr 9409    < clt 9410    <_ cle 9411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-le 9416
This theorem is referenced by:  leisorel  12205  icopnfhmeo  20495  iccpnfhmeo  20497  xrhmeo  20498
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