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Theorem gtiso 28162
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 2420 . . . . 5  |-  ( ( A  X.  A ) 
\  <  )  =  ( ( A  X.  A )  \  <  )
2 eqid 2420 . . . . 5  |-  ( ( B  X.  B ) 
\  `'  <  )  =  ( ( B  X.  B )  \  `'  <  )
31, 2isocnv3 6229 . . . 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 9670 . . . . . . . . . 10  |-  <_  =  ( ( RR*  X.  RR* )  \  `'  <  )
65cnveqi 5020 . . . . . . . . 9  |-  `'  <_  =  `' ( ( RR*  X. 
RR* )  \  `'  <  )
7 cnvdif 5253 . . . . . . . . 9  |-  `' ( ( RR*  X.  RR* )  \  `'  <  )  =  ( `' ( RR*  X. 
RR* )  \  `' `'  <  )
8 cnvxp 5265 . . . . . . . . . 10  |-  `' (
RR*  X.  RR* )  =  ( RR*  X.  RR* )
9 ltrel 9685 . . . . . . . . . . 11  |-  Rel  <
10 dfrel2 5297 . . . . . . . . . . 11  |-  ( Rel 
< 
<->  `' `'  <  =  < 
)
119, 10mpbi 211 . . . . . . . . . 10  |-  `' `'  <  =  <
128, 11difeq12i 3578 . . . . . . . . 9  |-  ( `' ( RR*  X.  RR* )  \  `' `'  <  )  =  ( ( RR*  X.  RR* )  \  <  )
136, 7, 123eqtri 2453 . . . . . . . 8  |-  `'  <_  =  ( ( RR*  X.  RR* )  \  <  )
1413ineq1i 3657 . . . . . . 7  |-  ( `' 
<_  i^i  ( A  X.  A ) )  =  ( ( ( RR*  X. 
RR* )  \  <  )  i^i  ( A  X.  A ) )
15 indif1 3714 . . . . . . 7  |-  ( ( ( RR*  X.  RR* )  \  <  )  i^i  ( A  X.  A ) )  =  ( ( (
RR*  X.  RR* )  i^i  ( A  X.  A
) )  \  <  )
1614, 15eqtri 2449 . . . . . 6  |-  ( `' 
<_  i^i  ( A  X.  A ) )  =  ( ( ( RR*  X. 
RR* )  i^i  ( A  X.  A ) ) 
\  <  )
17 xpss12 4951 . . . . . . . . 9  |-  ( ( A  C_  RR*  /\  A  C_ 
RR* )  ->  ( A  X.  A )  C_  ( RR*  X.  RR* )
)
1817anidms 649 . . . . . . . 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 199 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( ( RR*  X.  RR* )  i^i  ( A  X.  A ) )  =  ( A  X.  A ) )
2120difeq1d 3579 . . . . . 6  |-  ( A 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( A  X.  A
) )  \  <  )  =  ( ( A  X.  A )  \  <  ) )
2216, 21syl5req 2474 . . . . 5  |-  ( A 
C_  RR*  ->  ( ( A  X.  A )  \  <  )  =  ( `' 
<_  i^i  ( A  X.  A ) ) )
2322adantr 466 . . . 4  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  (
( A  X.  A
)  \  <  )  =  ( `'  <_  i^i  ( A  X.  A
) ) )
24 isoeq2 6217 . . . 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 17 . . 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 3714 . . . . . . 7  |-  ( ( ( RR*  X.  RR* )  \  `'  <  )  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
2826, 27eqtri 2449 . . . . . 6  |-  (  <_  i^i  ( B  X.  B
) )  =  ( ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  \  `'  <  )
29 xpss12 4951 . . . . . . . . 9  |-  ( ( B  C_  RR*  /\  B  C_ 
RR* )  ->  ( B  X.  B )  C_  ( RR*  X.  RR* )
)
3029anidms 649 . . . . . . . 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 199 . . . . . . 7  |-  ( B 
C_  RR*  ->  ( ( RR*  X.  RR* )  i^i  ( B  X.  B ) )  =  ( B  X.  B ) )
3332difeq1d 3579 . . . . . 6  |-  ( B 
C_  RR*  ->  ( (
( RR*  X.  RR* )  i^i  ( B  X.  B
) )  \  `'  <  )  =  ( ( B  X.  B ) 
\  `'  <  )
)
3428, 33syl5req 2474 . . . . 5  |-  ( B 
C_  RR*  ->  ( ( B  X.  B )  \  `'  <  )  =  (  <_  i^i  ( B  X.  B ) ) )
3534adantl 467 . . . 4  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  (
( B  X.  B
)  \  `'  <  )  =  (  <_  i^i  ( B  X.  B
) ) )
36 isoeq3 6218 . . . 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 17 . . 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 282 . 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 6228 . . 3  |-  ( F 
Isom  `'  <_  ,  <_  ( A ,  B )  <-> 
F  Isom  `' `'  <_  ,  `'  <_  ( A ,  B )
)
40 isores2 6230 . . . 4  |-  ( F 
Isom  `'  <_  ,  <_  ( A ,  B )  <-> 
F  Isom  `'  <_  ,  (  <_  i^i  ( B  X.  B ) ) ( A ,  B
) )
41 isores1 6231 . . . 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 252 . . 3  |-  ( F 
Isom  `'  <_  ,  <_  ( A ,  B )  <-> 
F  Isom  ( `'  <_  i^i  ( A  X.  A ) ) ,  (  <_  i^i  ( B  X.  B ) ) ( A ,  B
) )
43 lerel 9687 . . . . 5  |-  Rel  <_
44 dfrel2 5297 . . . . 5  |-  ( Rel 
<_ 
<->  `' `'  <_  =  <_ 
)
4543, 44mpbi 211 . . . 4  |-  `' `'  <_  =  <_
46 isoeq2 6217 . . . 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 279 . 2  |-  ( F 
Isom  <_  ,  `'  <_  ( A ,  B )  <-> 
F  Isom  ( `'  <_  i^i  ( A  X.  A ) ) ,  (  <_  i^i  ( B  X.  B ) ) ( A ,  B
) )
4938, 48syl6bbr 266 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 187    /\ wa 370    = wceq 1437    \ cdif 3430    i^i cin 3432    C_ wss 3433    X. cxp 4843   `'ccnv 4844   Rel wrel 4850    Isom wiso 5593   RR*cxr 9663    < clt 9664    <_ cle 9665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-xr 9668  df-ltxr 9669  df-le 9670
This theorem is referenced by:  xrge0iifhmeo  28622
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