Theorem gtiso 28162

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 2420	. . . . 5 |- ( ( A X. A ) \ < ) = ( ( A X. A ) \ < )
2		eqid 2420	. . . . 5 |- ( ( B X. B ) \ `' < ) = ( ( B X. B ) \ `' < )
3	1, 2	isocnv3 6229	. . . 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 9670	. . . . . . . . . 10 |- <_ = ( ( RR* X. RR* ) \ `' < )
6	5	cnveqi 5020	. . . . . . . . 9 |- `' <_ = `' ( ( RR* X. RR* ) \ `' < )
7		cnvdif 5253	. . . . . . . . 9 |- `' ( ( RR* X. RR* ) \ `' < ) = ( `' ( RR* X. RR* ) \ `' `' < )
8		cnvxp 5265	. . . . . . . . . 10 |- `' ( RR* X. RR* ) = ( RR* X. RR* )
9		ltrel 9685	. . . . . . . . . . 11 |- Rel <
10		dfrel2 5297	. . . . . . . . . . 11 |- ( Rel < <-> `' `' < = < )
11	9, 10	mpbi 211	. . . . . . . . . 10 |- `' `' < = <
12	8, 11	difeq12i 3578	. . . . . . . . 9 |- ( `' ( RR* X. RR* ) \ `' `' < ) = ( ( RR* X. RR* ) \ < )
13	6, 7, 12	3eqtri 2453	. . . . . . . 8 |- `' <_ = ( ( RR* X. RR* ) \ < )
14	13	ineq1i 3657	. . . . . . 7 |- ( `' <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) \ < ) i^i ( A X. A ) )
15		indif1 3714	. . . . . . 7 |- ( ( ( RR* X. RR* ) \ < ) i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < )
16	14, 15	eqtri 2449	. . . . . 6 |- ( `' <_ i^i ( A X. A ) ) = ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < )
17		xpss12 4951	. . . . . . . . 9 |- ( ( A C_ RR* /\ A C_ RR* ) -> ( A X. A ) C_ ( RR* X. RR* ) )
18	17	anidms 649	. . . . . . . 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 199	. . . . . . 7 |- ( A C_ RR* -> ( ( RR* X. RR* ) i^i ( A X. A ) ) = ( A X. A ) )
21	20	difeq1d 3579	. . . . . 6 |- ( A C_ RR* -> ( ( ( RR* X. RR* ) i^i ( A X. A ) ) \ < ) = ( ( A X. A ) \ < ) )
22	16, 21	syl5req 2474	. . . . 5 |- ( A C_ RR* -> ( ( A X. A ) \ < ) = ( `' <_ i^i ( A X. A ) ) )
23	22	adantr 466	. . . 4 |- ( ( A C_ RR* /\ B C_ RR* ) -> ( ( A X. A ) \ < ) = ( `' <_ i^i ( A X. A ) ) )
24		isoeq2 6217	. . . 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 3657	. . . . . . 7 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) )
27		indif1 3714	. . . . . . 7 |- ( ( ( RR* X. RR* ) \ `' < ) i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
28	26, 27	eqtri 2449	. . . . . 6 |- ( <_ i^i ( B X. B ) ) = ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < )
29		xpss12 4951	. . . . . . . . 9 |- ( ( B C_ RR* /\ B C_ RR* ) -> ( B X. B ) C_ ( RR* X. RR* ) )
30	29	anidms 649	. . . . . . . 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 199	. . . . . . 7 |- ( B C_ RR* -> ( ( RR* X. RR* ) i^i ( B X. B ) ) = ( B X. B ) )
33	32	difeq1d 3579	. . . . . 6 |- ( B C_ RR* -> ( ( ( RR* X. RR* ) i^i ( B X. B ) ) \ `' < ) = ( ( B X. B ) \ `' < ) )
34	28, 33	syl5req 2474	. . . . 5 |- ( B C_ RR* -> ( ( B X. B ) \ `' < ) = ( <_ i^i ( B X. B ) ) )
35	34	adantl 467	. . . 4 |- ( ( A C_ RR* /\ B C_ RR* ) -> ( ( B X. B ) \ `' < ) = ( <_ i^i ( B X. B ) ) )
36		isoeq3 6218	. . . 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 282	. 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 6228	. . 3 |- ( F Isom `' <_ , <_ ( A , B ) <-> F Isom `' `' <_ , `' <_ ( A , B ) )
40		isores2 6230	. . . 4 |- ( F Isom `' <_ , <_ ( A , B ) <-> F Isom `' <_ , ( <_ i^i ( B X. B ) ) ( A , B ) )
41		isores1 6231	. . . 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 252	. . 3 |- ( F Isom `' <_ , <_ ( A , B ) <-> F Isom ( `' <_ i^i ( A X. A ) ) , ( <_ i^i ( B X. B ) ) ( A , B ) )
43		lerel 9687	. . . . 5 |- Rel <_
44		dfrel2 5297	. . . . 5 |- ( Rel <_ <-> `' `' <_ = <_ )
45	43, 44	mpbi 211	. . . 4 |- `' `' <_ = <_
46		isoeq2 6217	. . . 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 279	. 2 |- ( F Isom <_ , `' <_ ( A , B ) <-> F Isom ( `' <_ i^i ( A X. A ) ) , ( <_ i^i ( B X. B ) ) ( A , B ) )
49	38, 48	syl6bbr 266	1 |- ( ( A C_ RR* /\ B C_ RR* ) -> ( F Isom < , `' < ( A , B ) <-> F Isom <_ , `' <_ ( A , B ) ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   \ cdif 3430   i^i cin 3432   C_ wss 3433   X. cxp 4843   `'ccnv 4844   Rel wrel 4850   Isom wiso 5593   RR*cxr 9663   < clt 9664   <_ cle 9665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-xr 9668  df-ltxr 9669  df-le 9670

This theorem is referenced by:  xrge0iifhmeo  28622