Theorem dvcnvre 23782

Description: The derivative rule for inverse functions. If F is a continuous and differentiable bijective function from X to Y which never has derivative 0, then `' F is also differentiable, and its derivative is the reciprocal of the derivative of F. (Contributed by Mario Carneiro, 24-Feb-2015.)

Hypotheses

Ref	Expression
dvcnvre.f	|- ( ph -> F e. ( X -cn-> RR ) )
dvcnvre.d	|- ( ph -> dom ( RR _D F ) = X )
dvcnvre.z	|- ( ph -> -. 0 e. ran ( RR _D F ) )
dvcnvre.1	|- ( ph -> F : X -1-1-onto-> Y )

Assertion

Ref	Expression
dvcnvre	|- ( ph -> ( RR _D `' F ) = ( x e. Y |-> ( 1 / ( ( RR _D F ) ` ( `' F ` x ) ) ) ) )

Distinct variable groups:   x, F   ph, x   x, X   x, Y

Proof of Theorem dvcnvre

Dummy variables y r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
2	1	tgioo2 22606	. 2 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
3		reelprrecn 10028	. . 3 |- RR e. { RR , CC }
4	3	a1i 11	. 2 |- ( ph -> RR e. { RR , CC } )
5		retop 22565	. . . . 5 |- ( topGen ` ran (,) ) e. Top
6		dvcnvre.1	. . . . . . 7 |- ( ph -> F : X -1-1-onto-> Y )
7		f1ofo 6144	. . . . . . 7 |- ( F : X -1-1-onto-> Y -> F : X -onto-> Y )
8		forn 6118	. . . . . . 7 |- ( F : X -onto-> Y -> ran F = Y )
9	6, 7, 8	3syl 18	. . . . . 6 |- ( ph -> ran F = Y )
10		dvcnvre.f	. . . . . . 7 |- ( ph -> F e. ( X -cn-> RR ) )
11		cncff 22696	. . . . . . 7 |- ( F e. ( X -cn-> RR ) -> F : X --> RR )
12		frn 6053	. . . . . . 7 |- ( F : X --> RR -> ran F C_ RR )
13	10, 11, 12	3syl 18	. . . . . 6 |- ( ph -> ran F C_ RR )
14	9, 13	eqsstr3d 3640	. . . . 5 |- ( ph -> Y C_ RR )
15		uniretop 22566	. . . . . 6 |- RR = U. ( topGen ` ran (,) )
16	15	ntrss2 20861	. . . . 5 |- ( ( ( topGen ` ran (,) ) e. Top /\ Y C_ RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` Y ) C_ Y )
17	5, 14, 16	sylancr 695	. . . 4 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` Y ) C_ Y )
18		f1ocnvfv2 6533	. . . . . . . 8 |- ( ( F : X -1-1-onto-> Y /\ x e. Y ) -> ( F ` ( `' F ` x ) ) = x )
19	6, 18	sylan 488	. . . . . . 7 |- ( ( ph /\ x e. Y ) -> ( F ` ( `' F ` x ) ) = x )
20		f1ocnv 6149	. . . . . . . . . . . 12 |- ( F : X -1-1-onto-> Y -> `' F : Y -1-1-onto-> X )
21		f1of 6137	. . . . . . . . . . . 12 |- ( `' F : Y -1-1-onto-> X -> `' F : Y --> X )
22	6, 20, 21	3syl 18	. . . . . . . . . . 11 |- ( ph -> `' F : Y --> X )
23	22	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ x e. Y ) -> ( `' F ` x ) e. X )
24		dvcnvre.d	. . . . . . . . . . . . . . 15 |- ( ph -> dom ( RR _D F ) = X )
25		dvbsss 23666	. . . . . . . . . . . . . . . 16 |- dom ( RR _D F ) C_ RR
26	25	a1i 11	. . . . . . . . . . . . . . 15 |- ( ph -> dom ( RR _D F ) C_ RR )
27	24, 26	eqsstr3d 3640	. . . . . . . . . . . . . 14 |- ( ph -> X C_ RR )
28	15	ntrss2 20861	. . . . . . . . . . . . . 14 |- ( ( ( topGen ` ran (,) ) e. Top /\ X C_ RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` X ) C_ X )
29	5, 27, 28	sylancr 695	. . . . . . . . . . . . 13 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` X ) C_ X )
30		ax-resscn 9993	. . . . . . . . . . . . . . . 16 |- RR C_ CC
31	30	a1i 11	. . . . . . . . . . . . . . 15 |- ( ph -> RR C_ CC )
32	10, 11	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> F : X --> RR )
33		fss 6056	. . . . . . . . . . . . . . . 16 |- ( ( F : X --> RR /\ RR C_ CC ) -> F : X --> CC )
34	32, 30, 33	sylancl 694	. . . . . . . . . . . . . . 15 |- ( ph -> F : X --> CC )
35	31, 34, 27, 2, 1	dvbssntr 23664	. . . . . . . . . . . . . 14 |- ( ph -> dom ( RR _D F ) C_ ( ( int ` ( topGen ` ran (,) ) ) ` X ) )
36	24, 35	eqsstr3d 3640	. . . . . . . . . . . . 13 |- ( ph -> X C_ ( ( int ` ( topGen ` ran (,) ) ) ` X ) )
37	29, 36	eqssd 3620	. . . . . . . . . . . 12 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` X ) = X )
38	15	isopn3 20870	. . . . . . . . . . . . 13 |- ( ( ( topGen ` ran (,) ) e. Top /\ X C_ RR ) -> ( X e. ( topGen ` ran (,) ) <-> ( ( int ` ( topGen ` ran (,) ) ) ` X ) = X ) )
39	5, 27, 38	sylancr 695	. . . . . . . . . . . 12 |- ( ph -> ( X e. ( topGen ` ran (,) ) <-> ( ( int ` ( topGen ` ran (,) ) ) ` X ) = X ) )
40	37, 39	mpbird 247	. . . . . . . . . . 11 |- ( ph -> X e. ( topGen ` ran (,) ) )
41		eqid 2622	. . . . . . . . . . . . 13 |- ( ( abs o. - ) |` ( RR X. RR ) ) = ( ( abs o. - ) |` ( RR X. RR ) )
42	41	rexmet 22594	. . . . . . . . . . . 12 |- ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR )
43		eqid 2622	. . . . . . . . . . . . . 14 |- ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
44	41, 43	tgioo 22599	. . . . . . . . . . . . 13 |- ( topGen ` ran (,) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
45	44	mopni2 22298	. . . . . . . . . . . 12 |- ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ X e. ( topGen ` ran (,) ) /\ ( `' F ` x ) e. X ) -> E. r e. RR+ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X )
46	42, 45	mp3an1 1411	. . . . . . . . . . 11 |- ( ( X e. ( topGen ` ran (,) ) /\ ( `' F ` x ) e. X ) -> E. r e. RR+ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X )
47	40, 46	sylan 488	. . . . . . . . . 10 |- ( ( ph /\ ( `' F ` x ) e. X ) -> E. r e. RR+ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X )
48	23, 47	syldan 487	. . . . . . . . 9 |- ( ( ph /\ x e. Y ) -> E. r e. RR+ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X )
49	10	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> F e. ( X -cn-> RR ) )
50	24	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> dom ( RR _D F ) = X )
51		dvcnvre.z	. . . . . . . . . . 11 |- ( ph -> -. 0 e. ran ( RR _D F ) )
52	51	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> -. 0 e. ran ( RR _D F ) )
53	6	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> F : X -1-1-onto-> Y )
54	23	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> ( `' F ` x ) e. X )
55		rphalfcl 11858	. . . . . . . . . . 11 |- ( r e. RR+ -> ( r / 2 ) e. RR+ )
56	55	ad2antrl 764	. . . . . . . . . 10 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> ( r / 2 ) e. RR+ )
57	27	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> X C_ RR )
58	57, 54	sseldd 3604	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> ( `' F ` x ) e. RR )
59	56	rpred 11872	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> ( r / 2 ) e. RR )
60	58, 59	resubcld 10458	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> ( ( `' F ` x ) - ( r / 2 ) ) e. RR )
61	58, 59	readdcld 10069	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> ( ( `' F ` x ) + ( r / 2 ) ) e. RR )
62		elicc2 12238	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( `' F ` x ) - ( r / 2 ) ) e. RR /\ ( ( `' F ` x ) + ( r / 2 ) ) e. RR ) -> ( y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) <-> ( y e. RR /\ ( ( `' F ` x ) - ( r / 2 ) ) <_ y /\ y <_ ( ( `' F ` x ) + ( r / 2 ) ) ) ) )
63	60, 61, 62	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> ( y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) <-> ( y e. RR /\ ( ( `' F ` x ) - ( r / 2 ) ) <_ y /\ y <_ ( ( `' F ` x ) + ( r / 2 ) ) ) ) )
64	63	biimpa 501	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( y e. RR /\ ( ( `' F ` x ) - ( r / 2 ) ) <_ y /\ y <_ ( ( `' F ` x ) + ( r / 2 ) ) ) )
65	64	simp1d 1073	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> y e. RR )
66	58	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( `' F ` x ) e. RR )
67		simplrl 800	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> r e. RR+ )
68	67	rpred 11872	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> r e. RR )
69	66, 68	resubcld 10458	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( ( `' F ` x ) - r ) e. RR )
70	60	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( ( `' F ` x ) - ( r / 2 ) ) e. RR )
71	67, 55	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( r / 2 ) e. RR+ )
72	71	rpred 11872	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( r / 2 ) e. RR )
73		rphalflt 11860	. . . . . . . . . . . . . . . . . 18 |- ( r e. RR+ -> ( r / 2 ) < r )
74	67, 73	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( r / 2 ) < r )
75	72, 68, 66, 74	ltsub2dd 10640	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( ( `' F ` x ) - r ) < ( ( `' F ` x ) - ( r / 2 ) ) )
76	64	simp2d 1074	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( ( `' F ` x ) - ( r / 2 ) ) <_ y )
77	69, 70, 65, 75, 76	ltletrd 10197	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( ( `' F ` x ) - r ) < y )
78	61	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( ( `' F ` x ) + ( r / 2 ) ) e. RR )
79	66, 68	readdcld 10069	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( ( `' F ` x ) + r ) e. RR )
80	64	simp3d 1075	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> y <_ ( ( `' F ` x ) + ( r / 2 ) ) )
81	72, 68, 66, 74	ltadd2dd 10196	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( ( `' F ` x ) + ( r / 2 ) ) < ( ( `' F ` x ) + r ) )
82	65, 78, 79, 80, 81	lelttrd 10195	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> y < ( ( `' F ` x ) + r ) )
83	69	rexrd 10089	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( ( `' F ` x ) - r ) e. RR* )
84	79	rexrd 10089	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( ( `' F ` x ) + r ) e. RR* )
85		elioo2 12216	. . . . . . . . . . . . . . . 16 |- ( ( ( ( `' F ` x ) - r ) e. RR* /\ ( ( `' F ` x ) + r ) e. RR* ) -> ( y e. ( ( ( `' F ` x ) - r ) (,) ( ( `' F ` x ) + r ) ) <-> ( y e. RR /\ ( ( `' F ` x ) - r ) < y /\ y < ( ( `' F ` x ) + r ) ) ) )
86	83, 84, 85	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> ( y e. ( ( ( `' F ` x ) - r ) (,) ( ( `' F ` x ) + r ) ) <-> ( y e. RR /\ ( ( `' F ` x ) - r ) < y /\ y < ( ( `' F ` x ) + r ) ) ) )
87	65, 77, 82, 86	mpbir3and 1245	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) /\ y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) ) -> y e. ( ( ( `' F ` x ) - r ) (,) ( ( `' F ` x ) + r ) ) )
88	87	ex 450	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> ( y e. ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) -> y e. ( ( ( `' F ` x ) - r ) (,) ( ( `' F ` x ) + r ) ) ) )
89	88	ssrdv 3609	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) C_ ( ( ( `' F ` x ) - r ) (,) ( ( `' F ` x ) + r ) ) )
90		rpre 11839	. . . . . . . . . . . . . 14 |- ( r e. RR+ -> r e. RR )
91	90	ad2antrl 764	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> r e. RR )
92	41	bl2ioo 22595	. . . . . . . . . . . . 13 |- ( ( ( `' F ` x ) e. RR /\ r e. RR ) -> ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) = ( ( ( `' F ` x ) - r ) (,) ( ( `' F ` x ) + r ) ) )
93	58, 91, 92	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) = ( ( ( `' F ` x ) - r ) (,) ( ( `' F ` x ) + r ) ) )
94	89, 93	sseqtr4d 3642	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) C_ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) )
95		simprr 796	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X )
96	94, 95	sstrd 3613	. . . . . . . . . 10 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> ( ( ( `' F ` x ) - ( r / 2 ) ) [,] ( ( `' F ` x ) + ( r / 2 ) ) ) C_ X )
97		eqid 2622	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
98		eqid 2622	. . . . . . . . . 10 |- ( ( TopOpen ` ℂfld ) ↾t X ) = ( ( TopOpen ` ℂfld ) ↾t X )
99		eqid 2622	. . . . . . . . . 10 |- ( ( TopOpen ` ℂfld ) ↾t Y ) = ( ( TopOpen ` ℂfld ) ↾t Y )
100	49, 50, 52, 53, 54, 56, 96, 97, 1, 98, 99	dvcnvrelem2 23781	. . . . . . . . 9 |- ( ( ( ph /\ x e. Y ) /\ ( r e. RR+ /\ ( ( `' F ` x ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ X ) ) -> ( ( F ` ( `' F ` x ) ) e. ( ( int ` ( topGen ` ran (,) ) ) ` Y ) /\ `' F e. ( ( ( ( TopOpen ` ℂfld ) ↾t Y ) CnP ( ( TopOpen ` ℂfld ) ↾t X ) ) ` ( F ` ( `' F ` x ) ) ) ) )
101	48, 100	rexlimddv 3035	. . . . . . . 8 |- ( ( ph /\ x e. Y ) -> ( ( F ` ( `' F ` x ) ) e. ( ( int ` ( topGen ` ran (,) ) ) ` Y ) /\ `' F e. ( ( ( ( TopOpen ` ℂfld ) ↾t Y ) CnP ( ( TopOpen ` ℂfld ) ↾t X ) ) ` ( F ` ( `' F ` x ) ) ) ) )
102	101	simpld 475	. . . . . . 7 |- ( ( ph /\ x e. Y ) -> ( F ` ( `' F ` x ) ) e. ( ( int ` ( topGen ` ran (,) ) ) ` Y ) )
103	19, 102	eqeltrrd 2702	. . . . . 6 |- ( ( ph /\ x e. Y ) -> x e. ( ( int ` ( topGen ` ran (,) ) ) ` Y ) )
104	103	ex 450	. . . . 5 |- ( ph -> ( x e. Y -> x e. ( ( int ` ( topGen ` ran (,) ) ) ` Y ) ) )
105	104	ssrdv 3609	. . . 4 |- ( ph -> Y C_ ( ( int ` ( topGen ` ran (,) ) ) ` Y ) )
106	17, 105	eqssd 3620	. . 3 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` Y ) = Y )
107	15	isopn3 20870	. . . 4 |- ( ( ( topGen ` ran (,) ) e. Top /\ Y C_ RR ) -> ( Y e. ( topGen ` ran (,) ) <-> ( ( int ` ( topGen ` ran (,) ) ) ` Y ) = Y ) )
108	5, 14, 107	sylancr 695	. . 3 |- ( ph -> ( Y e. ( topGen ` ran (,) ) <-> ( ( int ` ( topGen ` ran (,) ) ) ` Y ) = Y ) )
109	106, 108	mpbird 247	. 2 |- ( ph -> Y e. ( topGen ` ran (,) ) )
110	101	simprd 479	. . . . . 6 |- ( ( ph /\ x e. Y ) -> `' F e. ( ( ( ( TopOpen ` ℂfld ) ↾t Y ) CnP ( ( TopOpen ` ℂfld ) ↾t X ) ) ` ( F ` ( `' F ` x ) ) ) )
111	19	fveq2d 6195	. . . . . 6 |- ( ( ph /\ x e. Y ) -> ( ( ( ( TopOpen ` ℂfld ) ↾t Y ) CnP ( ( TopOpen ` ℂfld ) ↾t X ) ) ` ( F ` ( `' F ` x ) ) ) = ( ( ( ( TopOpen ` ℂfld ) ↾t Y ) CnP ( ( TopOpen ` ℂfld ) ↾t X ) ) ` x ) )
112	110, 111	eleqtrd 2703	. . . . 5 |- ( ( ph /\ x e. Y ) -> `' F e. ( ( ( ( TopOpen ` ℂfld ) ↾t Y ) CnP ( ( TopOpen ` ℂfld ) ↾t X ) ) ` x ) )
113	112	ralrimiva 2966	. . . 4 |- ( ph -> A. x e. Y `' F e. ( ( ( ( TopOpen ` ℂfld ) ↾t Y ) CnP ( ( TopOpen ` ℂfld ) ↾t X ) ) ` x ) )
114	1	cnfldtopon 22586	. . . . . 6 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
115	14, 30	syl6ss 3615	. . . . . 6 |- ( ph -> Y C_ CC )
116		resttopon 20965	. . . . . 6 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ Y C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t Y ) e. (TopOn ` Y ) )
117	114, 115, 116	sylancr 695	. . . . 5 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t Y ) e. (TopOn ` Y ) )
118	27, 30	syl6ss 3615	. . . . . 6 |- ( ph -> X C_ CC )
119		resttopon 20965	. . . . . 6 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ X C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t X ) e. (TopOn ` X ) )
120	114, 118, 119	sylancr 695	. . . . 5 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t X ) e. (TopOn ` X ) )
121		cncnp 21084	. . . . 5 |- ( ( ( ( TopOpen ` ℂfld ) ↾t Y ) e. (TopOn ` Y ) /\ ( ( TopOpen ` ℂfld ) ↾t X ) e. (TopOn ` X ) ) -> ( `' F e. ( ( ( TopOpen ` ℂfld ) ↾t Y ) Cn ( ( TopOpen ` ℂfld ) ↾t X ) ) <-> ( `' F : Y --> X /\ A. x e. Y `' F e. ( ( ( ( TopOpen ` ℂfld ) ↾t Y ) CnP ( ( TopOpen ` ℂfld ) ↾t X ) ) ` x ) ) ) )
122	117, 120, 121	syl2anc 693	. . . 4 |- ( ph -> ( `' F e. ( ( ( TopOpen ` ℂfld ) ↾t Y ) Cn ( ( TopOpen ` ℂfld ) ↾t X ) ) <-> ( `' F : Y --> X /\ A. x e. Y `' F e. ( ( ( ( TopOpen ` ℂfld ) ↾t Y ) CnP ( ( TopOpen ` ℂfld ) ↾t X ) ) ` x ) ) ) )
123	22, 113, 122	mpbir2and 957	. . 3 |- ( ph -> `' F e. ( ( ( TopOpen ` ℂfld ) ↾t Y ) Cn ( ( TopOpen ` ℂfld ) ↾t X ) ) )
124	1, 99, 98	cncfcn 22712	. . . 4 |- ( ( Y C_ CC /\ X C_ CC ) -> ( Y -cn-> X ) = ( ( ( TopOpen ` ℂfld ) ↾t Y ) Cn ( ( TopOpen ` ℂfld ) ↾t X ) ) )
125	115, 118, 124	syl2anc 693	. . 3 |- ( ph -> ( Y -cn-> X ) = ( ( ( TopOpen ` ℂfld ) ↾t Y ) Cn ( ( TopOpen ` ℂfld ) ↾t X ) ) )
126	123, 125	eleqtrrd 2704	. 2 |- ( ph -> `' F e. ( Y -cn-> X ) )
127	1, 2, 4, 109, 6, 126, 24, 51	dvcnv 23740	1 |- ( ph -> ( RR _D `' F ) = ( x e. Y |-> ( 1 / ( ( RR _D F ) ` ( `' F ` x ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   -->wf 5884   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   RR+crp 11832   (,)cioo 12175   [,]cicc 12178   abscabs 13974   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098   *Metcxmt 19731   ballcbl 19733   MetOpencmopn 19736  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   intcnt 20821   Cn ccn 21028   CnP ccnp 21029   -cn->ccncf 22679   _D cdv 23627

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  dvrelog  24383