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Theorem leiso 11663
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 9082 . . . . . . 7  |-  <_  =  ( ( RR*  X.  RR* )  \  `'  <  )
21ineq1i 3498 . . . . . 6  |-  (  <_  i^i  ( A  X.  A
) )  =  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( A  X.  A
) )
3 indif1 3545 . . . . . 6  |-  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( A  X.  A
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  \  `'  <  )
42, 3eqtri 2424 . . . . 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 627 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( A  X.  A )  C_  ( RR*  X.  RR* ) )
7 dfss1 3505 . . . . . . 7  |-  ( ( A  X.  A ) 
C_  ( RR*  X.  RR* ) 
<->  ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  =  ( A  X.  A
) )
86, 7sylib 189 . . . . . 6  |-  ( A 
C_  RR*  ->  ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  =  ( A  X.  A ) )
98difeq1d 3424 . . . . 5  |-  ( A 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( A  X.  A
) )  \  `'  <  )  =  ( ( A  X.  A ) 
\  `'  <  )
)
104, 9syl5req 2449 . . . 4  |-  ( A 
C_  RR*  ->  ( ( A  X.  A )  \  `'  <  )  =  (  <_  i^i  ( A  X.  A ) ) )
11 isoeq2 5999 . . . 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 3498 . . . . . 6  |-  (  <_  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )
14 indif1 3545 . . . . . 6  |-  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
1513, 14eqtri 2424 . . . . 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 627 . . . . . . 7  |-  ( B 
C_  RR*  ->  ( B  X.  B )  C_  ( RR*  X.  RR* ) )
18 dfss1 3505 . . . . . . 7  |-  ( ( B  X.  B ) 
C_  ( RR*  X.  RR* ) 
<->  ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  =  ( B  X.  B
) )
1917, 18sylib 189 . . . . . 6  |-  ( B 
C_  RR*  ->  ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  =  ( B  X.  B ) )
2019difeq1d 3424 . . . . 5  |-  ( B 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( B  X.  B
) )  \  `'  <  )  =  ( ( B  X.  B ) 
\  `'  <  )
)
2115, 20syl5req 2449 . . . 4  |-  ( B 
C_  RR*  ->  ( ( B  X.  B )  \  `'  <  )  =  (  <_  i^i  ( B  X.  B ) ) )
22 isoeq3 6000 . . . 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 681 . 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 6010 . . 3  |-  ( F 
Isom  <  ,  <  ( A ,  B )  <->  F 
Isom  `'  <  ,  `'  <  ( A ,  B
) )
26 eqid 2404 . . . 4  |-  ( ( A  X.  A ) 
\  `'  <  )  =  ( ( A  X.  A )  \  `'  <  )
27 eqid 2404 . . . 4  |-  ( ( B  X.  B ) 
\  `'  <  )  =  ( ( B  X.  B )  \  `'  <  )
2826, 27isocnv3 6011 . . 3  |-  ( F 
Isom  `'  <  ,  `'  <  ( A ,  B
)  <->  F  Isom  ( ( A  X.  A ) 
\  `'  <  ) ,  ( ( B  X.  B )  \  `'  <  ) ( A ,  B ) )
2925, 28bitri 241 . 2  |-  ( F 
Isom  <  ,  <  ( A ,  B )  <->  F 
Isom  ( ( A  X.  A )  \  `'  <  ) ,  ( ( B  X.  B
)  \  `'  <  ) ( A ,  B
) )
30 isores1 6013 . . 3  |-  ( F 
Isom  <_  ,  <_  ( A ,  B )  <->  F 
Isom  (  <_  i^i  ( A  X.  A ) ) ,  <_  ( A ,  B ) )
31 isores2 6012 . . 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 241 . 2  |-  ( F 
Isom  <_  ,  <_  ( A ,  B )  <->  F 
Isom  (  <_  i^i  ( A  X.  A ) ) ,  (  <_  i^i  ( B  X.  B
) ) ( A ,  B ) )
3324, 29, 323bitr4g 280 1  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  ( F  Isom  <  ,  <  ( A ,  B )  <-> 
F  Isom  <_  ,  <_  ( A ,  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    \ cdif 3277    i^i cin 3279    C_ wss 3280    X. cxp 4835   `'ccnv 4836    Isom wiso 5414   RR*cxr 9075    < clt 9076    <_ cle 9077
This theorem is referenced by:  leisorel  11664  icopnfhmeo  18921  iccpnfhmeo  18923  xrhmeo  18924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  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 9082
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