Theorem gtiso 26148

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 2454	. . . . 5 |- ( ( A X. A ) \ < ) = ( ( A X. A ) \ < )
2		eqid 2454	. . . . 5 |- ( ( B X. B ) \ `' < ) = ( ( B X. B ) \ `' < )
3	1, 2	isocnv3 6133	. . . 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 9536	. . . . . . . . . 10 |- <_ = ( ( RR* X. RR* ) \ `' < )
6	5	cnveqi 5123	. . . . . . . . 9 |- `' <_ = `' ( ( RR* X. RR* ) \ `' < )
7		cnvdif 5352	. . . . . . . . 9 |- `' ( ( RR* X. RR* ) \ `' < ) = ( `' ( RR* X. RR* ) \ `' `' < )
8		cnvxp 5364	. . . . . . . . . 10 |- `' ( RR* X. RR* ) = ( RR* X. RR* )
9		ltrel 9551	. . . . . . . . . . 11 |- Rel <
10		dfrel2 5397	. . . . . . . . . . 11 |- ( Rel < <-> `' `' < = < )
11	9, 10	mpbi 208	. . . . . . . . . 10 |- `' `' < = <
12	8, 11	difeq12i 3581	. . . . . . . . 9 |- ( `' ( RR* X. RR* ) \ `' `' < ) = ( ( RR* X. RR* ) \ < )
13	6, 7, 12	3eqtri 2487	. . . . . . . 8 |- `' <_ = ( ( RR* X. RR* ) \ < )
14	13	ineq1i 3657	. . . . . . 7 |- ( `' <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) \ < ) i^i ( A X. A ) )
15		indif1 3703	. . . . . . 7 |- ( ( ( RR* X. RR* ) \ < ) i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < )
16	14, 15	eqtri 2483	. . . . . 6 |- ( `' <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < )
17		xpss12 5054	. . . . . . . . 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 3664	. . . . . . . 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 3582	. . . . . 6 |- ( A C_ RR* -> ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < ) = ( ( A X. A ) \ < ) )
22	16, 21	syl5req 2508	. . . . 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 6121	. . . 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 3657	. . . . . . 7 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) )
27		indif1 3703	. . . . . . 7 |- ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
28	26, 27	eqtri 2483	. . . . . 6 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
29		xpss12 5054	. . . . . . . . 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 3664	. . . . . . . 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 3582	. . . . . 6 |- ( B C_ RR* -> ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < ) = ( ( B X. B ) \ `' < ) )
34	28, 33	syl5req 2508	. . . . 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 6122	. . . 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 6132	. . 3 |- ( F Isom `' <_ , <_ ( A , B ) <-> F Isom `' `' <_ , `' <_ ( A , B ) )
40		isores2 6134	. . . 4 |- ( F Isom `' <_ , <_ ( A , B ) <-> F Isom `' <_ , ( <_ i^i ( B X. B ) ) ( A , B ) )
41		isores1 6135	. . . 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 9553	. . . . 5 |- Rel <_
44		dfrel2 5397	. . . . 5 |- ( Rel <_ <-> `' `' <_ = <_ )
45	43, 44	mpbi 208	. . . 4 |- `' `' <_ = <_
46		isoeq2 6121	. . . 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 1370   \ cdif 3434   i^i cin 3436   C_ wss 3437   X. cxp 4947   `'ccnv 4948   Rel wrel 4954   Isom wiso 5528   RR*cxr 9529   < clt 9530   <_ cle 9531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-xr 9534  df-ltxr 9535  df-le 9536

This theorem is referenced by:  xrge0iifhmeo  26512