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Theorem leiso 13243
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 10080 . . . . . . 7 |- <_ = ( ( RR* X. RR* ) \ `' < )
21ineq1i 3810 . . . . . 6 |- ( <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) \ `' < ) i^i ( A X. A ) )
3 indif1 3871 . . . . . 6 |- ( ( ( RR* X. RR* ) \ `' < ) i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ `' < )
42, 3eqtri 2644 . . . . 5 |- ( <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ `' < )
5 xpss12 5225 . . . . . . . 8 |- ( ( A C_ RR* /\ A C_ RR* ) -> ( A X. A ) C_ ( RR* X. RR* ) )
65anidms 677 . . . . . . 7 |- ( A C_ RR* -> ( A X. A ) C_ ( RR* X. RR* ) )
7 sseqin2 3817 . . . . . . 7 |- ( ( A X. A ) C_ ( RR* X. RR* ) <-> ( ( RR* X. RR* ) i^i ( A X. A ) ) = ( A X. A ) )
86, 7sylib 208 . . . . . 6 |- ( A C_ RR* -> ( ( RR* X. RR* ) i^i ( A X. A ) ) = ( A X. A ) )
98difeq1d 3727 . . . . 5 |- ( A C_ RR* -> ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ `' < ) = ( ( A X. A ) \ `' < ) )
104, 9syl5req 2669 . . . 4 |- ( A C_ RR* -> ( ( A X. A ) \ `' < ) = ( <_ i^i ( A X. A ) ) )
11 isoeq2 6568 . . . 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 17 . . 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 3810 . . . . . 6 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) )
14 indif1 3871 . . . . . 6 |- ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
1513, 14eqtri 2644 . . . . 5 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
16 xpss12 5225 . . . . . . . 8 |- ( ( B C_ RR* /\ B C_ RR* ) -> ( B X. B ) C_ ( RR* X. RR* ) )
1716anidms 677 . . . . . . 7 |- ( B C_ RR* -> ( B X. B ) C_ ( RR* X. RR* ) )
18 sseqin2 3817 . . . . . . 7 |- ( ( B X. B ) C_ ( RR* X. RR* ) <-> ( ( RR* X. RR* ) i^i ( B X. B ) ) = ( B X. B ) )
1917, 18sylib 208 . . . . . 6 |- ( B C_ RR* -> ( ( RR* X. RR* ) i^i ( B X. B ) ) = ( B X. B ) )
2019difeq1d 3727 . . . . 5 |- ( B C_ RR* -> ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < ) = ( ( B X. B ) \ `' < ) )
2115, 20syl5req 2669 . . . 4 |- ( B C_ RR* -> ( ( B X. B ) \ `' < ) = ( <_ i^i ( B X. B ) ) )
22 isoeq3 6569 . . . 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 17 . . 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 736 . 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 6581 . . 3 |- ( F Isom < , < ( A , B ) <-> F Isom `' < , `' < ( A , B ) )
26 eqid 2622 . . . 4 |- ( ( A X. A ) \ `' < ) = ( ( A X. A ) \ `' < )
27 eqid 2622 . . . 4 |- ( ( B X. B ) \ `' < ) = ( ( B X. B ) \ `' < )
2826, 27isocnv3 6582 . . 3 |- ( F Isom `' < , `' < ( A , B ) <-> F Isom ( ( A X. A ) \ `' < ) , ( ( B X. B ) \ `' < ) ( A , B ) )
2925, 28bitri 264 . 2 |- ( F Isom < , < ( A , B ) <-> F Isom ( ( A X. A ) \ `' < ) , ( ( B X. B ) \ `' < ) ( A , B ) )
30 isores1 6584 . . 3 |- ( F Isom <_ , <_ ( A , B ) <-> F Isom ( <_ i^i ( A X. A ) ) , <_ ( A , B ) )
31 isores2 6583 . . 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 264 . 2 |- ( F Isom <_ , <_ ( A , B ) <-> F Isom ( <_ i^i ( A X. A ) ) , ( <_ i^i ( B X. B ) ) ( A , B ) )
3324, 29, 323bitr4g 303 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 196   /\ wa 384   = wceq 1483   \ cdif 3571   i^i cin 3573   C_ wss 3574   X. cxp 5112   `'ccnv 5113   Isom wiso 5889   RR*cxr 10073   < clt 10074   <_ cle 10075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-le 10080
This theorem is referenced by:  leisorel  13244  icopnfhmeo  22742  iccpnfhmeo  22744  xrhmeo  22745
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