Theorem gtiso 29478

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( ( A X. A ) \ < ) = ( ( A X. A ) \ < )
2		eqid 2622	. . . . 5 |- ( ( B X. B ) \ `' < ) = ( ( B X. B ) \ `' < )
3	1, 2	isocnv3 6582	. . . 4 |- ( F Isom < , `' < ( A , B ) <-> F Isom ( ( A X. A ) \ < ) , ( ( B X. B ) \ `' < ) ( A , B ) )
4	3	a1i 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 10080	. . . . . . . . . 10 |- <_ = ( ( RR* X. RR* ) \ `' < )
6	5	cnveqi 5297	. . . . . . . . 9 |- `' <_ = `' ( ( RR* X. RR* ) \ `' < )
7		cnvdif 5539	. . . . . . . . 9 |- `' ( ( RR* X. RR* ) \ `' < ) = ( `' ( RR* X. RR* ) \ `' `' < )
8		cnvxp 5551	. . . . . . . . . 10 |- `' ( RR* X. RR* ) = ( RR* X. RR* )
9		ltrel 10100	. . . . . . . . . . 11 |- Rel <
10		dfrel2 5583	. . . . . . . . . . 11 |- ( Rel < <-> `' `' < = < )
11	9, 10	mpbi 220	. . . . . . . . . 10 |- `' `' < = <
12	8, 11	difeq12i 3726	. . . . . . . . 9 |- ( `' ( RR* X. RR* ) \ `' `' < ) = ( ( RR* X. RR* ) \ < )
13	6, 7, 12	3eqtri 2648	. . . . . . . 8 |- `' <_ = ( ( RR* X. RR* ) \ < )
14	13	ineq1i 3810	. . . . . . 7 |- ( `' <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) \ < ) i^i ( A X. A ) )
15		indif1 3871	. . . . . . 7 |- ( ( ( RR* X. RR* ) \ < ) i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < )
16	14, 15	eqtri 2644	. . . . . 6 |- ( `' <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < )
17		xpss12 5225	. . . . . . . . 9 |- ( ( A C_ RR* /\ A C_ RR* ) -> ( A X. A ) C_ ( RR* X. RR* ) )
18	17	anidms 677	. . . . . . . 8 |- ( A C_ RR* -> ( A X. A ) C_ ( RR* X. RR* ) )
19		sseqin2 3817	. . . . . . . 8 |- ( ( A X. A ) C_ ( RR* X. RR* ) <-> ( ( RR* X. RR* ) i^i ( A X. A ) ) = ( A X. A ) )
20	18, 19	sylib 208	. . . . . . 7 |- ( A C_ RR* -> ( ( RR* X. RR* ) i^i ( A X. A ) ) = ( A X. A ) )
21	20	difeq1d 3727	. . . . . 6 |- ( A C_ RR* -> ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < ) = ( ( A X. A ) \ < ) )
22	16, 21	syl5req 2669	. . . . 5 |- ( A C_ RR* -> ( ( A X. A ) \ < ) = ( `' <_ i^i ( A X. A ) ) )
23	22	adantr 481	. . . 4 |- ( ( A C_ RR* /\ B C_ RR* ) -> ( ( A X. A ) \ < ) = ( `' <_ i^i ( A X. A ) ) )
24		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 ) ) )
25	23, 24	syl 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 ) ) )
26	5	ineq1i 3810	. . . . . . 7 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) )
27		indif1 3871	. . . . . . 7 |- ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
28	26, 27	eqtri 2644	. . . . . 6 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
29		xpss12 5225	. . . . . . . . 9 |- ( ( B C_ RR* /\ B C_ RR* ) -> ( B X. B ) C_ ( RR* X. RR* ) )
30	29	anidms 677	. . . . . . . 8 |- ( B C_ RR* -> ( B X. B ) C_ ( RR* X. RR* ) )
31		sseqin2 3817	. . . . . . . 8 |- ( ( B X. B ) C_ ( RR* X. RR* ) <-> ( ( RR* X. RR* ) i^i ( B X. B ) ) = ( B X. B ) )
32	30, 31	sylib 208	. . . . . . 7 |- ( B C_ RR* -> ( ( RR* X. RR* ) i^i ( B X. B ) ) = ( B X. B ) )
33	32	difeq1d 3727	. . . . . 6 |- ( B C_ RR* -> ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < ) = ( ( B X. B ) \ `' < ) )
34	28, 33	syl5req 2669	. . . . 5 |- ( B C_ RR* -> ( ( B X. B ) \ `' < ) = ( <_ i^i ( B X. B ) ) )
35	34	adantl 482	. . . 4 |- ( ( A C_ RR* /\ B C_ RR* ) -> ( ( B X. B ) \ `' < ) = ( <_ i^i ( B X. B ) ) )
36		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 ) ) )
37	35, 36	syl 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 ) ) )
38	4, 25, 37	3bitrd 294	. 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 6581	. . 3 |- ( F Isom `' <_ , <_ ( A , B ) <-> F Isom `' `' <_ , `' <_ ( A , B ) )
40		isores2 6583	. . . 4 |- ( F Isom `' <_ , <_ ( A , B ) <-> F Isom `' <_ , ( <_ i^i ( B X. B ) ) ( A , B ) )
41		isores1 6584	. . . 4 |- ( F Isom `' <_ , ( <_ i^i ( B X. B ) ) ( A , B ) <-> F Isom ( `' <_ i^i ( A X. A ) ) , ( <_ i^i ( B X. B ) ) ( A , B ) )
42	40, 41	bitri 264	. . 3 |- ( F Isom `' <_ , <_ ( A , B ) <-> F Isom ( `' <_ i^i ( A X. A ) ) , ( <_ i^i ( B X. B ) ) ( A , B ) )
43		lerel 10102	. . . . 5 |- Rel <_
44		dfrel2 5583	. . . . 5 |- ( Rel <_ <-> `' `' <_ = <_ )
45	43, 44	mpbi 220	. . . 4 |- `' `' <_ = <_
46		isoeq2 6568	. . . 4 |- ( `' `' <_ = <_ -> ( F Isom `' `' <_ , `' <_ ( A , B ) <-> F Isom <_ , `' <_ ( A , B ) ) )
47	45, 46	ax-mp 5	. . 3 |- ( F Isom `' `' <_ , `' <_ ( A , B ) <-> F Isom <_ , `' <_ ( A , B ) )
48	39, 42, 47	3bitr3ri 291	. 2 |- ( F Isom <_ , `' <_ ( A , B ) <-> F Isom ( `' <_ i^i ( A X. A ) ) , ( <_ i^i ( B X. B ) ) ( A , B ) )
49	38, 48	syl6bbr 278	1 |- ( ( A C_ RR* /\ B C_ RR* ) -> ( F Isom < , `' < ( A , B ) <-> F Isom <_ , `' <_ ( A , B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   \ cdif 3571   i^i cin 3573   C_ wss 3574   X. cxp 5112   `'ccnv 5113   Rel wrel 5119   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-xr 10078  df-ltxr 10079  df-le 10080

This theorem is referenced by:  xrge0iifhmeo  29982