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Theorem gtiso 26148
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 2454 . . . . 5  |-  ( ( A  X.  A ) 
\  <  )  =  ( ( A  X.  A )  \  <  )
2 eqid 2454 . . . . 5  |-  ( ( B  X.  B ) 
\  `'  <  )  =  ( ( B  X.  B )  \  `'  <  )
31, 2isocnv3 6133 . . . 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 9536 . . . . . . . . . 10  |-  <_  =  ( ( RR*  X.  RR* )  \  `'  <  )
65cnveqi 5123 . . . . . . . . 9  |-  `'  <_  =  `' ( ( RR*  X. 
RR* )  \  `'  <  )
7 cnvdif 5352 . . . . . . . . 9  |-  `' ( ( RR*  X.  RR* )  \  `'  <  )  =  ( `' ( RR*  X. 
RR* )  \  `' `'  <  )
8 cnvxp 5364 . . . . . . . . . 10  |-  `' (
RR*  X.  RR* )  =  ( RR*  X.  RR* )
9 ltrel 9551 . . . . . . . . . . 11  |-  Rel  <
10 dfrel2 5397 . . . . . . . . . . 11  |-  ( Rel 
< 
<->  `' `'  <  =  < 
)
119, 10mpbi 208 . . . . . . . . . 10  |-  `' `'  <  =  <
128, 11difeq12i 3581 . . . . . . . . 9  |-  ( `' ( RR*  X.  RR* )  \  `' `'  <  )  =  ( ( RR*  X.  RR* )  \  <  )
136, 7, 123eqtri 2487 . . . . . . . 8  |-  `'  <_  =  ( ( RR*  X.  RR* )  \  <  )
1413ineq1i 3657 . . . . . . 7  |-  ( `' 
<_  i^i  ( A  X.  A ) )  =  ( ( ( RR*  X. 
RR* )  \  <  )  i^i  ( A  X.  A ) )
15 indif1 3703 . . . . . . 7  |-  ( ( ( RR*  X.  RR* )  \  <  )  i^i  ( A  X.  A ) )  =  ( ( (
RR*  X.  RR* )  i^i  ( A  X.  A
) )  \  <  )
1614, 15eqtri 2483 . . . . . 6  |-  ( `' 
<_  i^i  ( A  X.  A ) )  =  ( ( ( RR*  X. 
RR* )  i^i  ( A  X.  A ) ) 
\  <  )
17 xpss12 5054 . . . . . . . . 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 3664 . . . . . . . 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 3582 . . . . . 6  |-  ( A 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( A  X.  A
) )  \  <  )  =  ( ( A  X.  A )  \  <  ) )
2216, 21syl5req 2508 . . . . 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 6121 . . . 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 3657 . . . . . . 7  |-  (  <_  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )
27 indif1 3703 . . . . . . 7  |-  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
2826, 27eqtri 2483 . . . . . 6  |-  (  <_  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
29 xpss12 5054 . . . . . . . . 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 3664 . . . . . . . 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 3582 . . . . . 6  |-  ( B 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( B  X.  B
) )  \  `'  <  )  =  ( ( B  X.  B ) 
\  `'  <  )
)
3428, 33syl5req 2508 . . . . 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 6122 . . . 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 6132 . . 3  |-  ( F 
Isom  `'  <_  ,  <_  ( A ,  B )  <-> 
F  Isom  `' `'  <_  ,  `'  <_  ( A ,  B )
)
40 isores2 6134 . . . 4  |-  ( F 
Isom  `'  <_  ,  <_  ( A ,  B )  <-> 
F  Isom  `'  <_  ,  (  <_  i^i  ( B  X.  B ) ) ( A ,  B
) )
41 isores1 6135 . . . 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 9553 . . . . 5  |-  Rel  <_
44 dfrel2 5397 . . . . 5  |-  ( Rel 
<_ 
<->  `' `'  <_  =  <_ 
)
4543, 44mpbi 208 . . . 4  |-  `' `'  <_  =  <_
46 isoeq2 6121 . . . 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 1370    \ cdif 3434    i^i cin 3436    C_ wss 3437    X. cxp 4947   `'ccnv 4948   Rel wrel 4954    Isom wiso 5528   RR*cxr 9529    < clt 9530    <_ cle 9531
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 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
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 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-xr 9534  df-ltxr 9535  df-le 9536
This theorem is referenced by:  xrge0iifhmeo  26512
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