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Theorem leiso 12490
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 9637 . . . . . . 7  |-  <_  =  ( ( RR*  X.  RR* )  \  `'  <  )
21ineq1i 3681 . . . . . 6  |-  (  <_  i^i  ( A  X.  A
) )  =  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( A  X.  A
) )
3 indif1 3727 . . . . . 6  |-  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( A  X.  A
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  \  `'  <  )
42, 3eqtri 2472 . . . . 5  |-  (  <_  i^i  ( A  X.  A
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  \  `'  <  )
5 xpss12 5098 . . . . . . . 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 3688 . . . . . . 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 3606 . . . . 5  |-  ( A 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( A  X.  A
) )  \  `'  <  )  =  ( ( A  X.  A ) 
\  `'  <  )
)
104, 9syl5req 2497 . . . 4  |-  ( A 
C_  RR*  ->  ( ( A  X.  A )  \  `'  <  )  =  (  <_  i^i  ( A  X.  A ) ) )
11 isoeq2 6201 . . . 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 3681 . . . . . 6  |-  (  <_  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )
14 indif1 3727 . . . . . 6  |-  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
1513, 14eqtri 2472 . . . . 5  |-  (  <_  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
16 xpss12 5098 . . . . . . . 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 3688 . . . . . . 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 3606 . . . . 5  |-  ( B 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( B  X.  B
) )  \  `'  <  )  =  ( ( B  X.  B ) 
\  `'  <  )
)
2115, 20syl5req 2497 . . . 4  |-  ( B 
C_  RR*  ->  ( ( B  X.  B )  \  `'  <  )  =  (  <_  i^i  ( B  X.  B ) ) )
22 isoeq3 6202 . . . 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 6212 . . 3  |-  ( F 
Isom  <  ,  <  ( A ,  B )  <->  F 
Isom  `'  <  ,  `'  <  ( A ,  B
) )
26 eqid 2443 . . . 4  |-  ( ( A  X.  A ) 
\  `'  <  )  =  ( ( A  X.  A )  \  `'  <  )
27 eqid 2443 . . . 4  |-  ( ( B  X.  B ) 
\  `'  <  )  =  ( ( B  X.  B )  \  `'  <  )
2826, 27isocnv3 6213 . . 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 6215 . . 3  |-  ( F 
Isom  <_  ,  <_  ( A ,  B )  <->  F 
Isom  (  <_  i^i  ( A  X.  A ) ) ,  <_  ( A ,  B ) )
31 isores2 6214 . . 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 1383    \ cdif 3458    i^i cin 3460    C_ wss 3461    X. cxp 4987   `'ccnv 4988    Isom wiso 5579   RR*cxr 9630    < clt 9631    <_ cle 9632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-le 9637
This theorem is referenced by:  leisorel  12491  icopnfhmeo  21421  iccpnfhmeo  21423  xrhmeo  21424
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