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Theorem leiso 12329
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 9534 . . . . . . 7  |-  <_  =  ( ( RR*  X.  RR* )  \  `'  <  )
21ineq1i 3655 . . . . . 6  |-  (  <_  i^i  ( A  X.  A
) )  =  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( A  X.  A
) )
3 indif1 3701 . . . . . 6  |-  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( A  X.  A
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  \  `'  <  )
42, 3eqtri 2483 . . . . 5  |-  (  <_  i^i  ( A  X.  A
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  \  `'  <  )
5 xpss12 5052 . . . . . . . 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 3662 . . . . . . 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 3580 . . . . 5  |-  ( A 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( A  X.  A
) )  \  `'  <  )  =  ( ( A  X.  A ) 
\  `'  <  )
)
104, 9syl5req 2508 . . . 4  |-  ( A 
C_  RR*  ->  ( ( A  X.  A )  \  `'  <  )  =  (  <_  i^i  ( A  X.  A ) ) )
11 isoeq2 6119 . . . 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 3655 . . . . . 6  |-  (  <_  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )
14 indif1 3701 . . . . . 6  |-  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
1513, 14eqtri 2483 . . . . 5  |-  (  <_  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
16 xpss12 5052 . . . . . . . 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 3662 . . . . . . 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 3580 . . . . 5  |-  ( B 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( B  X.  B
) )  \  `'  <  )  =  ( ( B  X.  B ) 
\  `'  <  )
)
2115, 20syl5req 2508 . . . 4  |-  ( B 
C_  RR*  ->  ( ( B  X.  B )  \  `'  <  )  =  (  <_  i^i  ( B  X.  B ) ) )
22 isoeq3 6120 . . . 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 6130 . . 3  |-  ( F 
Isom  <  ,  <  ( A ,  B )  <->  F 
Isom  `'  <  ,  `'  <  ( A ,  B
) )
26 eqid 2454 . . . 4  |-  ( ( A  X.  A ) 
\  `'  <  )  =  ( ( A  X.  A )  \  `'  <  )
27 eqid 2454 . . . 4  |-  ( ( B  X.  B ) 
\  `'  <  )  =  ( ( B  X.  B )  \  `'  <  )
2826, 27isocnv3 6131 . . 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 6133 . . 3  |-  ( F 
Isom  <_  ,  <_  ( A ,  B )  <->  F 
Isom  (  <_  i^i  ( A  X.  A ) ) ,  <_  ( A ,  B ) )
31 isores2 6132 . . 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 1370    \ cdif 3432    i^i cin 3434    C_ wss 3435    X. cxp 4945   `'ccnv 4946    Isom wiso 5526   RR*cxr 9527    < clt 9528    <_ cle 9529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-le 9534
This theorem is referenced by:  leisorel  12330  icopnfhmeo  20646  iccpnfhmeo  20648  xrhmeo  20649
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