Theorem leiso 12474

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 9634	. . . . . . 7 |- <_ = ( ( RR* X. RR* ) \ `' < )
2	1	ineq1i 3696	. . . . . 6 |- ( <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) \ `' < ) i^i ( A X. A ) )
3		indif1 3742	. . . . . 6 |- ( ( ( RR* X. RR* ) \ `' < ) i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ `' < )
4	2, 3	eqtri 2496	. . . . 5 |- ( <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ `' < )
5		xpss12 5108	. . . . . . . 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 3703	. . . . . . 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 3621	. . . . 5 |- ( A C_ RR* -> ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ `' < ) = ( ( A X. A ) \ `' < ) )
10	4, 9	syl5req 2521	. . . 4 |- ( A C_ RR* -> ( ( A X. A ) \ `' < ) = ( <_ i^i ( A X. A ) ) )
11		isoeq2 6204	. . . 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 3696	. . . . . 6 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) )
14		indif1 3742	. . . . . 6 |- ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
15	13, 14	eqtri 2496	. . . . 5 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
16		xpss12 5108	. . . . . . . 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 3703	. . . . . . 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 3621	. . . . 5 |- ( B C_ RR* -> ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < ) = ( ( B X. B ) \ `' < ) )
21	15, 20	syl5req 2521	. . . 4 |- ( B C_ RR* -> ( ( B X. B ) \ `' < ) = ( <_ i^i ( B X. B ) ) )
22		isoeq3 6205	. . . 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 6215	. . 3 |- ( F Isom < , < ( A , B ) <-> F Isom `' < , `' < ( A , B ) )
26		eqid 2467	. . . 4 |- ( ( A X. A ) \ `' < ) = ( ( A X. A ) \ `' < )
27		eqid 2467	. . . 4 |- ( ( B X. B ) \ `' < ) = ( ( B X. B ) \ `' < )
28	26, 27	isocnv3 6216	. . 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 6218	. . 3 |- ( F Isom <_ , <_ ( A , B ) <-> F Isom ( <_ i^i ( A X. A ) ) , <_ ( A , B ) )
31		isores2 6217	. . 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 1379   \ cdif 3473   i^i cin 3475   C_ wss 3476   X. cxp 4997   `'ccnv 4998   Isom wiso 5589   RR*cxr 9627   < clt 9628   <_ cle 9629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-le 9634

This theorem is referenced by:  leisorel  12475  icopnfhmeo  21206  iccpnfhmeo  21208  xrhmeo  21209