Theorem leiso 12622

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

Step	Hyp	Ref	Expression
1		df-le 9681	. . . . . . 7 |- <_ = ( ( RR* X. RR* ) \ `' < )
2	1	ineq1i 3630	. . . . . 6 |- ( <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) \ `' < ) i^i ( A X. A ) )
3		indif1 3687	. . . . . 6 |- ( ( ( RR* X. RR* ) \ `' < ) i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ `' < )
4	2, 3	eqtri 2473	. . . . 5 |- ( <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ `' < )
5		xpss12 4940	. . . . . . . 8 |- ( ( A C_ RR* /\ A C_ RR* ) -> ( A X. A ) C_ ( RR* X. RR* ) )
6	5	anidms 651	. . . . . . 7 |- ( A C_ RR* -> ( A X. A ) C_ ( RR* X. RR* ) )
7		dfss1 3637	. . . . . . 7 |- ( ( A X. A ) C_ ( RR* X. RR* ) <-> ( ( RR* X. RR* ) i^i ( A X. A ) ) = ( A X. A ) )
8	6, 7	sylib 200	. . . . . 6 |- ( A C_ RR* -> ( ( RR* X. RR* ) i^i ( A X. A ) ) = ( A X. A ) )
9	8	difeq1d 3550	. . . . 5 |- ( A C_ RR* -> ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ `' < ) = ( ( A X. A ) \ `' < ) )
10	4, 9	syl5req 2498	. . . 4 |- ( A C_ RR* -> ( ( A X. A ) \ `' < ) = ( <_ i^i ( A X. A ) ) )
11		isoeq2 6211	. . . 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 ) ) )
12	10, 11	syl 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 ) ) )
13	1	ineq1i 3630	. . . . . 6 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) )
14		indif1 3687	. . . . . 6 |- ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
15	13, 14	eqtri 2473	. . . . 5 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
16		xpss12 4940	. . . . . . . 8 |- ( ( B C_ RR* /\ B C_ RR* ) -> ( B X. B ) C_ ( RR* X. RR* ) )
17	16	anidms 651	. . . . . . 7 |- ( B C_ RR* -> ( B X. B ) C_ ( RR* X. RR* ) )
18		dfss1 3637	. . . . . . 7 |- ( ( B X. B ) C_ ( RR* X. RR* ) <-> ( ( RR* X. RR* ) i^i ( B X. B ) ) = ( B X. B ) )
19	17, 18	sylib 200	. . . . . 6 |- ( B C_ RR* -> ( ( RR* X. RR* ) i^i ( B X. B ) ) = ( B X. B ) )
20	19	difeq1d 3550	. . . . 5 |- ( B C_ RR* -> ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < ) = ( ( B X. B ) \ `' < ) )
21	15, 20	syl5req 2498	. . . 4 |- ( B C_ RR* -> ( ( B X. B ) \ `' < ) = ( <_ i^i ( B X. B ) ) )
22		isoeq3 6212	. . . 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 ) ) )
23	21, 22	syl 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 ) ) )
24	12, 23	sylan9bb 706	. 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 6222	. . 3 |- ( F Isom < , < ( A , B ) <-> F Isom `' < , `' < ( A , B ) )
26		eqid 2451	. . . 4 |- ( ( A X. A ) \ `' < ) = ( ( A X. A ) \ `' < )
27		eqid 2451	. . . 4 |- ( ( B X. B ) \ `' < ) = ( ( B X. B ) \ `' < )
28	26, 27	isocnv3 6223	. . 3 |- ( F Isom `' < , `' < ( A , B ) <-> F Isom ( ( A X. A ) \ `' < ) , ( ( B X. B ) \ `' < ) ( A , B ) )
29	25, 28	bitri 253	. 2 |- ( F Isom < , < ( A , B ) <-> F Isom ( ( A X. A ) \ `' < ) , ( ( B X. B ) \ `' < ) ( A , B ) )
30		isores1 6225	. . 3 |- ( F Isom <_ , <_ ( A , B ) <-> F Isom ( <_ i^i ( A X. A ) ) , <_ ( A , B ) )
31		isores2 6224	. . 3 |- ( F Isom ( <_ i^i ( A X. A ) ) , <_ ( A , B ) <-> F Isom ( <_ i^i ( A X. A ) ) , ( <_ i^i ( B X. B ) ) ( A , B ) )
32	30, 31	bitri 253	. 2 |- ( F Isom <_ , <_ ( A , B ) <-> F Isom ( <_ i^i ( A X. A ) ) , ( <_ i^i ( B X. B ) ) ( A , B ) )
33	24, 29, 32	3bitr4g 292	1 |- ( ( A C_ RR* /\ B C_ RR* ) -> ( F Isom < , < ( A , B ) <-> F Isom <_ , <_ ( A , B ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   \ cdif 3401   i^i cin 3403   C_ wss 3404   X. cxp 4832   `'ccnv 4833   Isom wiso 5583   RR*cxr 9674   < clt 9675   <_ cle 9676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-le 9681

This theorem is referenced by:  leisorel  12623  icopnfhmeo  21971  iccpnfhmeo  21973  xrhmeo  21974