Theorem leiso 12204

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 9416	. . . . . . 7 |- <_ = ( ( RR* X. RR* ) \ `' < )
2	1	ineq1i 3543	. . . . . 6 |- ( <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) \ `' < ) i^i ( A X. A ) )
3		indif1 3589	. . . . . 6 |- ( ( ( RR* X. RR* ) \ `' < ) i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ `' < )
4	2, 3	eqtri 2458	. . . . 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 645	. . . . . . 7 |- ( A C_ RR* -> ( A X. A ) C_ ( RR* X. RR* ) )
7		dfss1 3550	. . . . . . 7 |- ( ( A X. A ) C_ ( RR* X. RR* ) <-> ( ( RR* X. RR* ) i^i ( A X. A ) ) = ( A X. A ) )
8	6, 7	sylib 196	. . . . . 6 |- ( A C_ RR* -> ( ( RR* X. RR* ) i^i ( A X. A ) ) = ( A X. A ) )
9	8	difeq1d 3468	. . . . 5 |- ( A C_ RR* -> ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ `' < ) = ( ( A X. A ) \ `' < ) )
10	4, 9	syl5req 2483	. . . 4 |- ( A C_ RR* -> ( ( A X. A ) \ `' < ) = ( <_ i^i ( A X. A ) ) )
11		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 ) ) )
12	10, 11	syl 16	. . 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 3543	. . . . . 6 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) )
14		indif1 3589	. . . . . 6 |- ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
15	13, 14	eqtri 2458	. . . . 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 645	. . . . . . 7 |- ( B C_ RR* -> ( B X. B ) C_ ( RR* X. RR* ) )
18		dfss1 3550	. . . . . . 7 |- ( ( B X. B ) C_ ( RR* X. RR* ) <-> ( ( RR* X. RR* ) i^i ( B X. B ) ) = ( B X. B ) )
19	17, 18	sylib 196	. . . . . 6 |- ( B C_ RR* -> ( ( RR* X. RR* ) i^i ( B X. B ) ) = ( B X. B ) )
20	19	difeq1d 3468	. . . . 5 |- ( B C_ RR* -> ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < ) = ( ( B X. B ) \ `' < ) )
21	15, 20	syl5req 2483	. . . 4 |- ( B C_ RR* -> ( ( B X. B ) \ `' < ) = ( <_ i^i ( B X. B ) ) )
22		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 ) ) )
23	21, 22	syl 16	. . 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 699	. 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 6017	. . 3 |- ( F Isom < , < ( A , B ) <-> F Isom `' < , `' < ( A , B ) )
26		eqid 2438	. . . 4 |- ( ( A X. A ) \ `' < ) = ( ( A X. A ) \ `' < )
27		eqid 2438	. . . 4 |- ( ( B X. B ) \ `' < ) = ( ( B X. B ) \ `' < )
28	26, 27	isocnv3 6018	. . 3 |- ( F Isom `' < , `' < ( A , B ) <-> F Isom ( ( A X. A ) \ `' < ) , ( ( B X. B ) \ `' < ) ( A , B ) )
29	25, 28	bitri 249	. 2 |- ( F Isom < , < ( A , B ) <-> F Isom ( ( A X. A ) \ `' < ) , ( ( B X. B ) \ `' < ) ( A , B ) )
30		isores1 6020	. . 3 |- ( F Isom <_ , <_ ( A , B ) <-> F Isom ( <_ i^i ( A X. A ) ) , <_ ( A , B ) )
31		isores2 6019	. . 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 249	. 2 |- ( F Isom <_ , <_ ( A , B ) <-> F Isom ( <_ i^i ( A X. A ) ) , ( <_ i^i ( B X. B ) ) ( A , B ) )
33	24, 29, 32	3bitr4g 288	1 |- ( ( A C_ RR* /\ B C_ RR* ) -> ( F Isom < , < ( A , B ) <-> F Isom <_ , <_ ( A , B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   \ cdif 3320   i^i cin 3322   C_ wss 3323   X. cxp 4833   `'ccnv 4834   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-le 9416

This theorem is referenced by:  leisorel  12205  icopnfhmeo  20495  iccpnfhmeo  20497  xrhmeo  20498