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Theorem leiso 12474
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 9634 . . . . . . 7  |-  <_  =  ( ( RR*  X.  RR* )  \  `'  <  )
21ineq1i 3696 . . . . . 6  |-  (  <_  i^i  ( A  X.  A
) )  =  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( A  X.  A
) )
3 indif1 3742 . . . . . 6  |-  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( A  X.  A
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  \  `'  <  )
42, 3eqtri 2496 . . . . 5  |-  (  <_  i^i  ( A  X.  A
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  \  `'  <  )
5 xpss12 5108 . . . . . . . 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 3703 . . . . . . 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 3621 . . . . 5  |-  ( A 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( A  X.  A
) )  \  `'  <  )  =  ( ( A  X.  A ) 
\  `'  <  )
)
104, 9syl5req 2521 . . . 4  |-  ( A 
C_  RR*  ->  ( ( A  X.  A )  \  `'  <  )  =  (  <_  i^i  ( A  X.  A ) ) )
11 isoeq2 6204 . . . 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 3696 . . . . . 6  |-  (  <_  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )
14 indif1 3742 . . . . . 6  |-  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
1513, 14eqtri 2496 . . . . 5  |-  (  <_  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
16 xpss12 5108 . . . . . . . 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 3703 . . . . . . 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 3621 . . . . 5  |-  ( B 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( B  X.  B
) )  \  `'  <  )  =  ( ( B  X.  B ) 
\  `'  <  )
)
2115, 20syl5req 2521 . . . 4  |-  ( B 
C_  RR*  ->  ( ( B  X.  B )  \  `'  <  )  =  (  <_  i^i  ( B  X.  B ) ) )
22 isoeq3 6205 . . . 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 6215 . . 3  |-  ( F 
Isom  <  ,  <  ( A ,  B )  <->  F 
Isom  `'  <  ,  `'  <  ( A ,  B
) )
26 eqid 2467 . . . 4  |-  ( ( A  X.  A ) 
\  `'  <  )  =  ( ( A  X.  A )  \  `'  <  )
27 eqid 2467 . . . 4  |-  ( ( B  X.  B ) 
\  `'  <  )  =  ( ( B  X.  B )  \  `'  <  )
2826, 27isocnv3 6216 . . 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 6218 . . 3  |-  ( F 
Isom  <_  ,  <_  ( A ,  B )  <->  F 
Isom  (  <_  i^i  ( A  X.  A ) ) ,  <_  ( A ,  B ) )
31 isores2 6217 . . 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 1379    \ cdif 3473    i^i cin 3475    C_ wss 3476    X. cxp 4997   `'ccnv 4998    Isom wiso 5589   RR*cxr 9627    < clt 9628    <_ cle 9629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-le 9634
This theorem is referenced by:  leisorel  12475  icopnfhmeo  21206  iccpnfhmeo  21208  xrhmeo  21209
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