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Theorem gtiso 25947
Description: Two ways to write a strictly decreasing function on the reals. (Contributed by Thierry Arnoux, 6-Apr-2017.)
Assertion
Ref Expression
gtiso  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  ( F  Isom  <  ,  `'  <  ( A ,  B
)  <->  F  Isom  <_  ,  `'  <_  ( A ,  B
) ) )

Proof of Theorem gtiso
StepHypRef Expression
1 eqid 2438 . . . . 5  |-  ( ( A  X.  A ) 
\  <  )  =  ( ( A  X.  A )  \  <  )
2 eqid 2438 . . . . 5  |-  ( ( B  X.  B ) 
\  `'  <  )  =  ( ( B  X.  B )  \  `'  <  )
31, 2isocnv3 6018 . . . 4  |-  ( F 
Isom  <  ,  `'  <  ( A ,  B )  <-> 
F  Isom  ( ( A  X.  A )  \  <  ) ,  ( ( B  X.  B ) 
\  `'  <  )
( A ,  B
) )
43a1i 11 . . 3  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  ( F  Isom  <  ,  `'  <  ( A ,  B
)  <->  F  Isom  ( ( A  X.  A ) 
\  <  ) , 
( ( B  X.  B )  \  `'  <  ) ( A ,  B ) ) )
5 df-le 9416 . . . . . . . . . 10  |-  <_  =  ( ( RR*  X.  RR* )  \  `'  <  )
65cnveqi 5009 . . . . . . . . 9  |-  `'  <_  =  `' ( ( RR*  X. 
RR* )  \  `'  <  )
7 cnvdif 5238 . . . . . . . . 9  |-  `' ( ( RR*  X.  RR* )  \  `'  <  )  =  ( `' ( RR*  X. 
RR* )  \  `' `'  <  )
8 cnvxp 5250 . . . . . . . . . 10  |-  `' (
RR*  X.  RR* )  =  ( RR*  X.  RR* )
9 ltrel 9431 . . . . . . . . . . 11  |-  Rel  <
10 dfrel2 5283 . . . . . . . . . . 11  |-  ( Rel 
< 
<->  `' `'  <  =  < 
)
119, 10mpbi 208 . . . . . . . . . 10  |-  `' `'  <  =  <
128, 11difeq12i 3467 . . . . . . . . 9  |-  ( `' ( RR*  X.  RR* )  \  `' `'  <  )  =  ( ( RR*  X.  RR* )  \  <  )
136, 7, 123eqtri 2462 . . . . . . . 8  |-  `'  <_  =  ( ( RR*  X.  RR* )  \  <  )
1413ineq1i 3543 . . . . . . 7  |-  ( `' 
<_  i^i  ( A  X.  A ) )  =  ( ( ( RR*  X. 
RR* )  \  <  )  i^i  ( A  X.  A ) )
15 indif1 3589 . . . . . . 7  |-  ( ( ( RR*  X.  RR* )  \  <  )  i^i  ( A  X.  A ) )  =  ( ( (
RR*  X.  RR* )  i^i  ( A  X.  A
) )  \  <  )
1614, 15eqtri 2458 . . . . . 6  |-  ( `' 
<_  i^i  ( A  X.  A ) )  =  ( ( ( RR*  X. 
RR* )  i^i  ( A  X.  A ) ) 
\  <  )
17 xpss12 4940 . . . . . . . . 9  |-  ( ( A  C_  RR*  /\  A  C_ 
RR* )  ->  ( A  X.  A )  C_  ( RR*  X.  RR* )
)
1817anidms 645 . . . . . . . 8  |-  ( A 
C_  RR*  ->  ( A  X.  A )  C_  ( RR*  X.  RR* ) )
19 dfss1 3550 . . . . . . . 8  |-  ( ( A  X.  A ) 
C_  ( RR*  X.  RR* ) 
<->  ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  =  ( A  X.  A
) )
2018, 19sylib 196 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  =  ( A  X.  A ) )
2120difeq1d 3468 . . . . . 6  |-  ( A 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( A  X.  A
) )  \  <  )  =  ( ( A  X.  A )  \  <  ) )
2216, 21syl5req 2483 . . . . 5  |-  ( A 
C_  RR*  ->  ( ( A  X.  A )  \  <  )  =  ( `' 
<_  i^i  ( A  X.  A ) ) )
2322adantr 465 . . . 4  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  (
( A  X.  A
)  \  <  )  =  ( `'  <_  i^i  ( A  X.  A
) ) )
24 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 ) ) )
2523, 24syl 16 . . 3  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  ( F  Isom  ( ( A  X.  A )  \  <  ) ,  ( ( B  X.  B ) 
\  `'  <  )
( A ,  B
)  <->  F  Isom  ( `' 
<_  i^i  ( A  X.  A ) ) ,  ( ( B  X.  B )  \  `'  <  ) ( A ,  B ) ) )
265ineq1i 3543 . . . . . . 7  |-  (  <_  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )
27 indif1 3589 . . . . . . 7  |-  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
2826, 27eqtri 2458 . . . . . 6  |-  (  <_  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
29 xpss12 4940 . . . . . . . . 9  |-  ( ( B  C_  RR*  /\  B  C_ 
RR* )  ->  ( B  X.  B )  C_  ( RR*  X.  RR* )
)
3029anidms 645 . . . . . . . 8  |-  ( B 
C_  RR*  ->  ( B  X.  B )  C_  ( RR*  X.  RR* ) )
31 dfss1 3550 . . . . . . . 8  |-  ( ( B  X.  B ) 
C_  ( RR*  X.  RR* ) 
<->  ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  =  ( B  X.  B
) )
3230, 31sylib 196 . . . . . . 7  |-  ( B 
C_  RR*  ->  ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  =  ( B  X.  B ) )
3332difeq1d 3468 . . . . . 6  |-  ( B 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( B  X.  B
) )  \  `'  <  )  =  ( ( B  X.  B ) 
\  `'  <  )
)
3428, 33syl5req 2483 . . . . 5  |-  ( B 
C_  RR*  ->  ( ( B  X.  B )  \  `'  <  )  =  (  <_  i^i  ( B  X.  B ) ) )
3534adantl 466 . . . 4  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  (
( B  X.  B
)  \  `'  <  )  =  (  <_  i^i  ( B  X.  B
) ) )
36 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
) ) )
3735, 36syl 16 . . 3  |-  ( ( A  C_  RR*  /\  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
) ) )
384, 25, 373bitrd 279 . 2  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  ( F  Isom  <  ,  `'  <  ( A ,  B
)  <->  F  Isom  ( `' 
<_  i^i  ( A  X.  A ) ) ,  (  <_  i^i  ( B  X.  B ) ) ( A ,  B
) ) )
39 isocnv2 6017 . . 3  |-  ( F 
Isom  `'  <_  ,  <_  ( A ,  B )  <-> 
F  Isom  `' `'  <_  ,  `'  <_  ( A ,  B )
)
40 isores2 6019 . . . 4  |-  ( F 
Isom  `'  <_  ,  <_  ( A ,  B )  <-> 
F  Isom  `'  <_  ,  (  <_  i^i  ( B  X.  B ) ) ( A ,  B
) )
41 isores1 6020 . . . 4  |-  ( F 
Isom  `'  <_  ,  (  <_  i^i  ( B  X.  B ) ) ( A ,  B )  <-> 
F  Isom  ( `'  <_  i^i  ( A  X.  A ) ) ,  (  <_  i^i  ( B  X.  B ) ) ( A ,  B
) )
4240, 41bitri 249 . . 3  |-  ( F 
Isom  `'  <_  ,  <_  ( A ,  B )  <-> 
F  Isom  ( `'  <_  i^i  ( A  X.  A ) ) ,  (  <_  i^i  ( B  X.  B ) ) ( A ,  B
) )
43 lerel 9433 . . . . 5  |-  Rel  <_
44 dfrel2 5283 . . . . 5  |-  ( Rel 
<_ 
<->  `' `'  <_  =  <_ 
)
4543, 44mpbi 208 . . . 4  |-  `' `'  <_  =  <_
46 isoeq2 6006 . . . 4  |-  ( `' `'  <_  =  <_  ->  ( F  Isom  `' `'  <_  ,  `'  <_  ( A ,  B )  <->  F 
Isom  <_  ,  `'  <_  ( A ,  B ) ) )
4745, 46ax-mp 5 . . 3  |-  ( F 
Isom  `' `'  <_  ,  `'  <_  ( A ,  B
)  <->  F  Isom  <_  ,  `'  <_  ( A ,  B
) )
4839, 42, 473bitr3ri 276 . 2  |-  ( F 
Isom  <_  ,  `'  <_  ( A ,  B )  <-> 
F  Isom  ( `'  <_  i^i  ( A  X.  A ) ) ,  (  <_  i^i  ( B  X.  B ) ) ( A ,  B
) )
4938, 48syl6bbr 263 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   Rel wrel 4840    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-xr 9414  df-ltxr 9415  df-le 9416
This theorem is referenced by:  xrge0iifhmeo  26318
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