Theorem gtiso 27384

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 2441	. . . . 5 |- ( ( A X. A ) \ < ) = ( ( A X. A ) \ < )
2		eqid 2441	. . . . 5 |- ( ( B X. B ) \ `' < ) = ( ( B X. B ) \ `' < )
3	1, 2	isocnv3 6209	. . . 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 9632	. . . . . . . . . 10 |- <_ = ( ( RR* X. RR* ) \ `' < )
6	5	cnveqi 5163	. . . . . . . . 9 |- `' <_ = `' ( ( RR* X. RR* ) \ `' < )
7		cnvdif 5398	. . . . . . . . 9 |- `' ( ( RR* X. RR* ) \ `' < ) = ( `' ( RR* X. RR* ) \ `' `' < )
8		cnvxp 5410	. . . . . . . . . 10 |- `' ( RR* X. RR* ) = ( RR* X. RR* )
9		ltrel 9647	. . . . . . . . . . 11 |- Rel <
10		dfrel2 5443	. . . . . . . . . . 11 |- ( Rel < <-> `' `' < = < )
11	9, 10	mpbi 208	. . . . . . . . . 10 |- `' `' < = <
12	8, 11	difeq12i 3602	. . . . . . . . 9 |- ( `' ( RR* X. RR* ) \ `' `' < ) = ( ( RR* X. RR* ) \ < )
13	6, 7, 12	3eqtri 2474	. . . . . . . 8 |- `' <_ = ( ( RR* X. RR* ) \ < )
14	13	ineq1i 3678	. . . . . . 7 |- ( `' <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) \ < ) i^i ( A X. A ) )
15		indif1 3724	. . . . . . 7 |- ( ( ( RR* X. RR* ) \ < ) i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < )
16	14, 15	eqtri 2470	. . . . . 6 |- ( `' <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < )
17		xpss12 5094	. . . . . . . . 9 |- ( ( A C_ RR* /\ A C_ RR* ) -> ( A X. A ) C_ ( RR* X. RR* ) )
18	17	anidms 645	. . . . . . . 8 |- ( A C_ RR* -> ( A X. A ) C_ ( RR* X. RR* ) )
19		dfss1 3685	. . . . . . . 8 |- ( ( A X. A ) C_ ( RR* X. RR* ) <-> ( ( RR* X. RR* ) i^i ( A X. A ) ) = ( A X. A ) )
20	18, 19	sylib 196	. . . . . . 7 |- ( A C_ RR* -> ( ( RR* X. RR* ) i^i ( A X. A ) ) = ( A X. A ) )
21	20	difeq1d 3603	. . . . . 6 |- ( A C_ RR* -> ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < ) = ( ( A X. A ) \ < ) )
22	16, 21	syl5req 2495	. . . . 5 |- ( A C_ RR* -> ( ( A X. A ) \ < ) = ( `' <_ i^i ( A X. A ) ) )
23	22	adantr 465	. . . 4 |- ( ( A C_ RR* /\ B C_ RR* ) -> ( ( A X. A ) \ < ) = ( `' <_ i^i ( A X. A ) ) )
24		isoeq2 6197	. . . 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 16	. . 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 3678	. . . . . . 7 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) )
27		indif1 3724	. . . . . . 7 |- ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
28	26, 27	eqtri 2470	. . . . . 6 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
29		xpss12 5094	. . . . . . . . 9 |- ( ( B C_ RR* /\ B C_ RR* ) -> ( B X. B ) C_ ( RR* X. RR* ) )
30	29	anidms 645	. . . . . . . 8 |- ( B C_ RR* -> ( B X. B ) C_ ( RR* X. RR* ) )
31		dfss1 3685	. . . . . . . 8 |- ( ( B X. B ) C_ ( RR* X. RR* ) <-> ( ( RR* X. RR* ) i^i ( B X. B ) ) = ( B X. B ) )
32	30, 31	sylib 196	. . . . . . 7 |- ( B C_ RR* -> ( ( RR* X. RR* ) i^i ( B X. B ) ) = ( B X. B ) )
33	32	difeq1d 3603	. . . . . 6 |- ( B C_ RR* -> ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < ) = ( ( B X. B ) \ `' < ) )
34	28, 33	syl5req 2495	. . . . 5 |- ( B C_ RR* -> ( ( B X. B ) \ `' < ) = ( <_ i^i ( B X. B ) ) )
35	34	adantl 466	. . . 4 |- ( ( A C_ RR* /\ B C_ RR* ) -> ( ( B X. B ) \ `' < ) = ( <_ i^i ( B X. B ) ) )
36		isoeq3 6198	. . . 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 16	. . 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 279	. 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 6208	. . 3 |- ( F Isom `' <_ , <_ ( A , B ) <-> F Isom `' `' <_ , `' <_ ( A , B ) )
40		isores2 6210	. . . 4 |- ( F Isom `' <_ , <_ ( A , B ) <-> F Isom `' <_ , ( <_ i^i ( B X. B ) ) ( A , B ) )
41		isores1 6211	. . . 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 249	. . 3 |- ( F Isom `' <_ , <_ ( A , B ) <-> F Isom ( `' <_ i^i ( A X. A ) ) , ( <_ i^i ( B X. B ) ) ( A , B ) )
43		lerel 9649	. . . . 5 |- Rel <_
44		dfrel2 5443	. . . . 5 |- ( Rel <_ <-> `' `' <_ = <_ )
45	43, 44	mpbi 208	. . . 4 |- `' `' <_ = <_
46		isoeq2 6197	. . . 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 276	. 2 |- ( F Isom <_ , `' <_ ( A , B ) <-> F Isom ( `' <_ i^i ( A X. A ) ) , ( <_ i^i ( B X. B ) ) ( A , B ) )
49	38, 48	syl6bbr 263	1 |- ( ( A C_ RR* /\ B C_ RR* ) -> ( F Isom < , `' < ( A , B ) <-> F Isom <_ , `' <_ ( A , B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1381   \ cdif 3455   i^i cin 3457   C_ wss 3458   X. cxp 4983   `'ccnv 4984   Rel wrel 4990   Isom wiso 5575   RR*cxr 9625   < clt 9626   <_ cle 9627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-xr 9630  df-ltxr 9631  df-le 9632

This theorem is referenced by:  xrge0iifhmeo  27784