Theorem gtiso 28333

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 2462	. . . . 5 |- ( ( A X. A ) \ < ) = ( ( A X. A ) \ < )
2		eqid 2462	. . . . 5 |- ( ( B X. B ) \ `' < ) = ( ( B X. B ) \ `' < )
3	1, 2	isocnv3 6253	. . . 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 9712	. . . . . . . . . 10 |- <_ = ( ( RR* X. RR* ) \ `' < )
6	5	cnveqi 5031	. . . . . . . . 9 |- `' <_ = `' ( ( RR* X. RR* ) \ `' < )
7		cnvdif 5264	. . . . . . . . 9 |- `' ( ( RR* X. RR* ) \ `' < ) = ( `' ( RR* X. RR* ) \ `' `' < )
8		cnvxp 5276	. . . . . . . . . 10 |- `' ( RR* X. RR* ) = ( RR* X. RR* )
9		ltrel 9727	. . . . . . . . . . 11 |- Rel <
10		dfrel2 5308	. . . . . . . . . . 11 |- ( Rel < <-> `' `' < = < )
11	9, 10	mpbi 213	. . . . . . . . . 10 |- `' `' < = <
12	8, 11	difeq12i 3561	. . . . . . . . 9 |- ( `' ( RR* X. RR* ) \ `' `' < ) = ( ( RR* X. RR* ) \ < )
13	6, 7, 12	3eqtri 2488	. . . . . . . 8 |- `' <_ = ( ( RR* X. RR* ) \ < )
14	13	ineq1i 3642	. . . . . . 7 |- ( `' <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) \ < ) i^i ( A X. A ) )
15		indif1 3699	. . . . . . 7 |- ( ( ( RR* X. RR* ) \ < ) i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < )
16	14, 15	eqtri 2484	. . . . . 6 |- ( `' <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < )
17		xpss12 4962	. . . . . . . . 9 |- ( ( A C_ RR* /\ A C_ RR* ) -> ( A X. A ) C_ ( RR* X. RR* ) )
18	17	anidms 655	. . . . . . . 8 |- ( A C_ RR* -> ( A X. A ) C_ ( RR* X. RR* ) )
19		dfss1 3649	. . . . . . . 8 |- ( ( A X. A ) C_ ( RR* X. RR* ) <-> ( ( RR* X. RR* ) i^i ( A X. A ) ) = ( A X. A ) )
20	18, 19	sylib 201	. . . . . . 7 |- ( A C_ RR* -> ( ( RR* X. RR* ) i^i ( A X. A ) ) = ( A X. A ) )
21	20	difeq1d 3562	. . . . . 6 |- ( A C_ RR* -> ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < ) = ( ( A X. A ) \ < ) )
22	16, 21	syl5req 2509	. . . . 5 |- ( A C_ RR* -> ( ( A X. A ) \ < ) = ( `' <_ i^i ( A X. A ) ) )
23	22	adantr 471	. . . 4 |- ( ( A C_ RR* /\ B C_ RR* ) -> ( ( A X. A ) \ < ) = ( `' <_ i^i ( A X. A ) ) )
24		isoeq2 6241	. . . 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 3642	. . . . . . 7 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) )
27		indif1 3699	. . . . . . 7 |- ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
28	26, 27	eqtri 2484	. . . . . 6 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
29		xpss12 4962	. . . . . . . . 9 |- ( ( B C_ RR* /\ B C_ RR* ) -> ( B X. B ) C_ ( RR* X. RR* ) )
30	29	anidms 655	. . . . . . . 8 |- ( B C_ RR* -> ( B X. B ) C_ ( RR* X. RR* ) )
31		dfss1 3649	. . . . . . . 8 |- ( ( B X. B ) C_ ( RR* X. RR* ) <-> ( ( RR* X. RR* ) i^i ( B X. B ) ) = ( B X. B ) )
32	30, 31	sylib 201	. . . . . . 7 |- ( B C_ RR* -> ( ( RR* X. RR* ) i^i ( B X. B ) ) = ( B X. B ) )
33	32	difeq1d 3562	. . . . . 6 |- ( B C_ RR* -> ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < ) = ( ( B X. B ) \ `' < ) )
34	28, 33	syl5req 2509	. . . . 5 |- ( B C_ RR* -> ( ( B X. B ) \ `' < ) = ( <_ i^i ( B X. B ) ) )
35	34	adantl 472	. . . 4 |- ( ( A C_ RR* /\ B C_ RR* ) -> ( ( B X. B ) \ `' < ) = ( <_ i^i ( B X. B ) ) )
36		isoeq3 6242	. . . 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 287	. 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 6252	. . 3 |- ( F Isom `' <_ , <_ ( A , B ) <-> F Isom `' `' <_ , `' <_ ( A , B ) )
40		isores2 6254	. . . 4 |- ( F Isom `' <_ , <_ ( A , B ) <-> F Isom `' <_ , ( <_ i^i ( B X. B ) ) ( A , B ) )
41		isores1 6255	. . . 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 257	. . 3 |- ( F Isom `' <_ , <_ ( A , B ) <-> F Isom ( `' <_ i^i ( A X. A ) ) , ( <_ i^i ( B X. B ) ) ( A , B ) )
43		lerel 9729	. . . . 5 |- Rel <_
44		dfrel2 5308	. . . . 5 |- ( Rel <_ <-> `' `' <_ = <_ )
45	43, 44	mpbi 213	. . . 4 |- `' `' <_ = <_
46		isoeq2 6241	. . . 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 284	. 2 |- ( F Isom <_ , `' <_ ( A , B ) <-> F Isom ( `' <_ i^i ( A X. A ) ) , ( <_ i^i ( B X. B ) ) ( A , B ) )
49	38, 48	syl6bbr 271	1 |- ( ( A C_ RR* /\ B C_ RR* ) -> ( F Isom < , `' < ( A , B ) <-> F Isom <_ , `' <_ ( A , B ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   \ cdif 3413   i^i cin 3415   C_ wss 3416   X. cxp 4854   `'ccnv 4855   Rel wrel 4861   Isom wiso 5606   RR*cxr 9705   < clt 9706   <_ cle 9707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-isom 5614  df-xr 9710  df-ltxr 9711  df-le 9712

This theorem is referenced by:  xrge0iifhmeo  28793