Theorem tpr2rico 28714

Description: For any point of an open set of the usual topology on ( RR X. RR ) there is an open square which contains that point and is entirely in the open set. This is square is actually a ball by the ( l ^ +oo ) norm X. (Contributed by Thierry Arnoux, 21-Sep-2017.)

Hypotheses

Ref	Expression
tpr2rico.0	|- J = ( topGen ` ran (,) )
tpr2rico.1	|- G = ( u e. RR , v e. RR |-> ( u + ( _i x. v ) ) )
tpr2rico.2	|- B = ran ( x e. ran (,) , y e. ran (,) |-> ( x X. y ) )

Assertion

Ref	Expression
tpr2rico	|- ( ( A e. ( J tX J ) /\ X e. A ) -> E. r e. B ( X e. r /\ r C_ A ) )

Distinct variable groups:   v, u, x, y   x, r, A   B, r   x, G   x, J   x, X   y, r, X

Allowed substitution hints:   A( y, v, u)   B( x, y, v, u)   G( y, v, u, r)   J( y, v, u, r)   X( v, u)

Proof of Theorem tpr2rico

Dummy variables z m d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ioo 11640	. . . . . . . . . 10 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
2	1	ixxf 11646	. . . . . . . . 9 |- (,) : ( RR* X. RR* ) --> ~P RR*
3		ffn 5743	. . . . . . . . 9 |- ( (,) : ( RR* X. RR* ) --> ~P RR* -> (,) Fn ( RR* X. RR* ) )
4	2, 3	mp1i 13	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> (,) Fn ( RR* X. RR* ) )
5		elssuni 4245	. . . . . . . . . . . . . 14 |- ( A e. ( J tX J ) -> A C_ U. ( J tX J ) )
6		tpr2rico.0	. . . . . . . . . . . . . . . 16 |- J = ( topGen ` ran (,) )
7		retop 21769	. . . . . . . . . . . . . . . 16 |- ( topGen ` ran (,) ) e. Top
8	6, 7	eqeltri 2506	. . . . . . . . . . . . . . 15 |- J e. Top
9		uniretop 21770	. . . . . . . . . . . . . . . 16 |- RR = U. ( topGen ` ran (,) )
10	6	unieqi 4225	. . . . . . . . . . . . . . . 16 |- U. J = U. ( topGen ` ran (,) )
11	9, 10	eqtr4i 2454	. . . . . . . . . . . . . . 15 |- RR = U. J
12	8, 8, 11, 11	txunii 20595	. . . . . . . . . . . . . 14 |- ( RR X. RR ) = U. ( J tX J )
13	5, 12	syl6sseqr 3511	. . . . . . . . . . . . 13 |- ( A e. ( J tX J ) -> A C_ ( RR X. RR ) )
14	13	ad2antrr 730	. . . . . . . . . . . 12 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> A C_ ( RR X. RR ) )
15		simplr 760	. . . . . . . . . . . 12 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> X e. A )
16	14, 15	sseldd 3465	. . . . . . . . . . 11 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> X e. ( RR X. RR ) )
17		xp1st 6834	. . . . . . . . . . 11 |- ( X e. ( RR X. RR ) -> ( 1st ` X ) e. RR )
18	16, 17	syl 17	. . . . . . . . . 10 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 1st ` X ) e. RR )
19		simpr 462	. . . . . . . . . . . 12 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> d e. RR+ )
20	19	rpred 11342	. . . . . . . . . . 11 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> d e. RR )
21	20	rehalfcld 10860	. . . . . . . . . 10 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( d / 2 ) e. RR )
22	18, 21	resubcld 10048	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) - ( d / 2 ) ) e. RR )
23	22	rexrd 9691	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) - ( d / 2 ) ) e. RR* )
24	18, 21	readdcld 9671	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) + ( d / 2 ) ) e. RR )
25	24	rexrd 9691	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) + ( d / 2 ) ) e. RR* )
26		fnovrn 6455	. . . . . . . 8 |- ( ( (,) Fn ( RR* X. RR* ) /\ ( ( 1st ` X ) - ( d / 2 ) ) e. RR* /\ ( ( 1st ` X ) + ( d / 2 ) ) e. RR* ) -> ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) e. ran (,) )
27	4, 23, 25, 26	syl3anc 1264	. . . . . . 7 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) e. ran (,) )
28		xp2nd 6835	. . . . . . . . . . 11 |- ( X e. ( RR X. RR ) -> ( 2nd ` X ) e. RR )
29	16, 28	syl 17	. . . . . . . . . 10 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 2nd ` X ) e. RR )
30	29, 21	resubcld 10048	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) - ( d / 2 ) ) e. RR )
31	30	rexrd 9691	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) - ( d / 2 ) ) e. RR* )
32	29, 21	readdcld 9671	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) + ( d / 2 ) ) e. RR )
33	32	rexrd 9691	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) + ( d / 2 ) ) e. RR* )
34		fnovrn 6455	. . . . . . . 8 |- ( ( (,) Fn ( RR* X. RR* ) /\ ( ( 2nd ` X ) - ( d / 2 ) ) e. RR* /\ ( ( 2nd ` X ) + ( d / 2 ) ) e. RR* ) -> ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) e. ran (,) )
35	4, 31, 33, 34	syl3anc 1264	. . . . . . 7 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) e. ran (,) )
36		eqidd 2423	. . . . . . 7 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) = ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) )
37		xpeq1 4864	. . . . . . . . 9 |- ( x = ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) -> ( x X. y ) = ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. y ) )
38	37	eqeq2d 2436	. . . . . . . 8 |- ( x = ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) -> ( ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) = ( x X. y ) <-> ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) = ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. y ) ) )
39		xpeq2 4865	. . . . . . . . 9 |- ( y = ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) -> ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. y ) = ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) )
40	39	eqeq2d 2436	. . . . . . . 8 |- ( y = ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) -> ( ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) = ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. y ) <-> ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) = ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) ) )
41	38, 40	rspc2ev 3193	. . . . . . 7 |- ( ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) e. ran (,) /\ ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) e. ran (,) /\ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) = ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) ) -> E. x e. ran (,) E. y e. ran (,) ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) = ( x X. y ) )
42	27, 35, 36, 41	syl3anc 1264	. . . . . 6 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> E. x e. ran (,) E. y e. ran (,) ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) = ( x X. y ) )
43		eqid 2422	. . . . . . 7 |- ( x e. ran (,) , y e. ran (,) |-> ( x X. y ) ) = ( x e. ran (,) , y e. ran (,) |-> ( x X. y ) )
44		vex 3084	. . . . . . . 8 |- x e. _V
45		vex 3084	. . . . . . . 8 |- y e. _V
46	44, 45	xpex 6606	. . . . . . 7 |- ( x X. y ) e. _V
47	43, 46	elrnmpt2 6420	. . . . . 6 |- ( ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) e. ran ( x e. ran (,) , y e. ran (,) |-> ( x X. y ) ) <-> E. x e. ran (,) E. y e. ran (,) ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) = ( x X. y ) )
48	42, 47	sylibr 215	. . . . 5 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) e. ran ( x e. ran (,) , y e. ran (,) |-> ( x X. y ) ) )
49		tpr2rico.2	. . . . 5 |- B = ran ( x e. ran (,) , y e. ran (,) |-> ( x X. y ) )
50	48, 49	syl6eleqr 2521	. . . 4 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) e. B )
51	50	ralrimiva 2839	. . 3 |- ( ( A e. ( J tX J ) /\ X e. A ) -> A. d e. RR+ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) e. B )
52		xpss 4957	. . . . . . 7 |- ( RR X. RR ) C_ ( _V X. _V )
53	52, 16	sseldi 3462	. . . . . 6 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> X e. ( _V X. _V ) )
54	18	rexrd 9691	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 1st ` X ) e. RR* )
55	19	rphalfcld 11354	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( d / 2 ) e. RR+ )
56	18, 55	ltsubrpd 11371	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) - ( d / 2 ) ) < ( 1st ` X ) )
57	18, 55	ltaddrpd 11372	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 1st ` X ) < ( ( 1st ` X ) + ( d / 2 ) ) )
58		elioo1 11677	. . . . . . . . 9 |- ( ( ( ( 1st ` X ) - ( d / 2 ) ) e. RR* /\ ( ( 1st ` X ) + ( d / 2 ) ) e. RR* ) -> ( ( 1st ` X ) e. ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) <-> ( ( 1st ` X ) e. RR* /\ ( ( 1st ` X ) - ( d / 2 ) ) < ( 1st ` X ) /\ ( 1st ` X ) < ( ( 1st ` X ) + ( d / 2 ) ) ) ) )
59	23, 25, 58	syl2anc 665	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) e. ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) <-> ( ( 1st ` X ) e. RR* /\ ( ( 1st ` X ) - ( d / 2 ) ) < ( 1st ` X ) /\ ( 1st ` X ) < ( ( 1st ` X ) + ( d / 2 ) ) ) ) )
60	54, 56, 57, 59	mpbir3and 1188	. . . . . . 7 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 1st ` X ) e. ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) )
61	29	rexrd 9691	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 2nd ` X ) e. RR* )
62	29, 55	ltsubrpd 11371	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) - ( d / 2 ) ) < ( 2nd ` X ) )
63	29, 55	ltaddrpd 11372	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 2nd ` X ) < ( ( 2nd ` X ) + ( d / 2 ) ) )
64		elioo1 11677	. . . . . . . . 9 |- ( ( ( ( 2nd ` X ) - ( d / 2 ) ) e. RR* /\ ( ( 2nd ` X ) + ( d / 2 ) ) e. RR* ) -> ( ( 2nd ` X ) e. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) <-> ( ( 2nd ` X ) e. RR* /\ ( ( 2nd ` X ) - ( d / 2 ) ) < ( 2nd ` X ) /\ ( 2nd ` X ) < ( ( 2nd ` X ) + ( d / 2 ) ) ) ) )
65	31, 33, 64	syl2anc 665	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) e. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) <-> ( ( 2nd ` X ) e. RR* /\ ( ( 2nd ` X ) - ( d / 2 ) ) < ( 2nd ` X ) /\ ( 2nd ` X ) < ( ( 2nd ` X ) + ( d / 2 ) ) ) ) )
66	61, 62, 63, 65	mpbir3and 1188	. . . . . . 7 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 2nd ` X ) e. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) )
67	60, 66	jca 534	. . . . . 6 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) e. ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) /\ ( 2nd ` X ) e. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) )
68		elxp7 6837	. . . . . 6 |- ( X e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) <-> ( X e. ( _V X. _V ) /\ ( ( 1st ` X ) e. ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) /\ ( 2nd ` X ) e. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) ) )
69	53, 67, 68	sylanbrc 668	. . . . 5 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> X e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) )
70	69	ralrimiva 2839	. . . 4 |- ( ( A e. ( J tX J ) /\ X e. A ) -> A. d e. RR+ X e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) )
71		mnfle 11436	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 1st ` X ) - ( d / 2 ) ) e. RR* -> -oo <_ ( ( 1st ` X ) - ( d / 2 ) ) )
72	23, 71	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> -oo <_ ( ( 1st ` X ) - ( d / 2 ) ) )
73		pnfge 11433	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 1st ` X ) + ( d / 2 ) ) e. RR* -> ( ( 1st ` X ) + ( d / 2 ) ) <_ +oo )
74	25, 73	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) + ( d / 2 ) ) <_ +oo )
75		mnfxr 11415	. . . . . . . . . . . . . . . . . 18 |- -oo e. RR*
76		pnfxr 11413	. . . . . . . . . . . . . . . . . 18 |- +oo e. RR*
77		ioossioo 11727	. . . . . . . . . . . . . . . . . 18 |- ( ( ( -oo e. RR* /\ +oo e. RR* ) /\ ( -oo <_ ( ( 1st ` X ) - ( d / 2 ) ) /\ ( ( 1st ` X ) + ( d / 2 ) ) <_ +oo ) ) -> ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) C_ ( -oo (,) +oo ) )
78	75, 76, 77	mpanl12 686	. . . . . . . . . . . . . . . . 17 |- ( ( -oo <_ ( ( 1st ` X ) - ( d / 2 ) ) /\ ( ( 1st ` X ) + ( d / 2 ) ) <_ +oo ) -> ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) C_ ( -oo (,) +oo ) )
79	72, 74, 78	syl2anc 665	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) C_ ( -oo (,) +oo ) )
80		ioomax 11710	. . . . . . . . . . . . . . . 16 |- ( -oo (,) +oo ) = RR
81	79, 80	syl6sseq 3510	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) C_ RR )
82		mnfle 11436	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 2nd ` X ) - ( d / 2 ) ) e. RR* -> -oo <_ ( ( 2nd ` X ) - ( d / 2 ) ) )
83	31, 82	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> -oo <_ ( ( 2nd ` X ) - ( d / 2 ) ) )
84		pnfge 11433	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 2nd ` X ) + ( d / 2 ) ) e. RR* -> ( ( 2nd ` X ) + ( d / 2 ) ) <_ +oo )
85	33, 84	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) + ( d / 2 ) ) <_ +oo )
86		ioossioo 11727	. . . . . . . . . . . . . . . . . 18 |- ( ( ( -oo e. RR* /\ +oo e. RR* ) /\ ( -oo <_ ( ( 2nd ` X ) - ( d / 2 ) ) /\ ( ( 2nd ` X ) + ( d / 2 ) ) <_ +oo ) ) -> ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) C_ ( -oo (,) +oo ) )
87	75, 76, 86	mpanl12 686	. . . . . . . . . . . . . . . . 17 |- ( ( -oo <_ ( ( 2nd ` X ) - ( d / 2 ) ) /\ ( ( 2nd ` X ) + ( d / 2 ) ) <_ +oo ) -> ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) C_ ( -oo (,) +oo ) )
88	83, 85, 87	syl2anc 665	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) C_ ( -oo (,) +oo ) )
89	88, 80	syl6sseq 3510	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) C_ RR )
90		xpss12 4956	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) C_ RR /\ ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) C_ RR ) -> ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ ( RR X. RR ) )
91	81, 89, 90	syl2anc 665	. . . . . . . . . . . . . 14 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ ( RR X. RR ) )
92	91	sselda 3464	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) ) -> x e. ( RR X. RR ) )
93	92	expcom 436	. . . . . . . . . . . 12 |- ( x e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) -> ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> x e. ( RR X. RR ) ) )
94	93	ancld 555	. . . . . . . . . . 11 |- ( x e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) -> ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) ) )
95	94	imdistanri 695	. . . . . . . . . 10 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) ) -> ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) /\ x e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) ) )
96	13	adantr 466	. . . . . . . . . . . . . . . 16 |- ( ( A e. ( J tX J ) /\ ( X e. A /\ d e. RR+ /\ x e. ( RR X. RR ) ) ) -> A C_ ( RR X. RR ) )
97		simpr1 1011	. . . . . . . . . . . . . . . 16 |- ( ( A e. ( J tX J ) /\ ( X e. A /\ d e. RR+ /\ x e. ( RR X. RR ) ) ) -> X e. A )
98	96, 97	sseldd 3465	. . . . . . . . . . . . . . 15 |- ( ( A e. ( J tX J ) /\ ( X e. A /\ d e. RR+ /\ x e. ( RR X. RR ) ) ) -> X e. ( RR X. RR ) )
99	98	3anassrs 1228	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> X e. ( RR X. RR ) )
100		simpr 462	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> x e. ( RR X. RR ) )
101		simplr 760	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> d e. RR+ )
102	101	rphalfcld 11354	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( d / 2 ) e. RR+ )
103		tpr2rico.1	. . . . . . . . . . . . . . 15 |- G = ( u e. RR , v e. RR |-> ( u + ( _i x. v ) ) )
104	103	cnre2csqima 28713	. . . . . . . . . . . . . 14 |- ( ( X e. ( RR X. RR ) /\ x e. ( RR X. RR ) /\ ( d / 2 ) e. RR+ ) -> ( x e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) -> ( ( abs ` ( Re ` ( ( G ` x ) - ( G ` X ) ) ) ) < ( d / 2 ) /\ ( abs ` ( Im ` ( ( G ` x ) - ( G ` X ) ) ) ) < ( d / 2 ) ) ) )
105	99, 100, 102, 104	syl3anc 1264	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( x e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) -> ( ( abs ` ( Re ` ( ( G ` x ) - ( G ` X ) ) ) ) < ( d / 2 ) /\ ( abs ` ( Im ` ( ( G ` x ) - ( G ` X ) ) ) ) < ( d / 2 ) ) ) )
106		eqid 2422	. . . . . . . . . . . . . . . . . . . . 21 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
107	103, 6, 106	cnrehmeo 21968	. . . . . . . . . . . . . . . . . . . 20 |- G e. ( ( J tX J ) Homeo ( TopOpen ` ℂfld ) )
108	106	cnfldtopon 21790	. . . . . . . . . . . . . . . . . . . . . 22 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
109	108	toponunii 19934	. . . . . . . . . . . . . . . . . . . . 21 |- CC = U. ( TopOpen ` ℂfld )
110	12, 109	hmeof1o 20766	. . . . . . . . . . . . . . . . . . . 20 |- ( G e. ( ( J tX J ) Homeo ( TopOpen ` ℂfld ) ) -> G : ( RR X. RR ) -1-1-onto-> CC )
111		f1of 5828	. . . . . . . . . . . . . . . . . . . 20 |- ( G : ( RR X. RR ) -1-1-onto-> CC -> G : ( RR X. RR ) --> CC )
112	107, 110, 111	mp2b 10	. . . . . . . . . . . . . . . . . . 19 |- G : ( RR X. RR ) --> CC
113	112	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> G : ( RR X. RR ) --> CC )
114	113, 99	ffvelrnd 6035	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( G ` X ) e. CC )
115	112	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> G : ( RR X. RR ) --> CC )
116	115	ffvelrnda 6034	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( G ` x ) e. CC )
117		sqsscirc2 28711	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( G ` X ) e. CC /\ ( G ` x ) e. CC ) /\ d e. RR+ ) -> ( ( ( abs ` ( Re ` ( ( G ` x ) - ( G ` X ) ) ) ) < ( d / 2 ) /\ ( abs ` ( Im ` ( ( G ` x ) - ( G ` X ) ) ) ) < ( d / 2 ) ) -> ( abs ` ( ( G ` x ) - ( G ` X ) ) ) < d ) )
118	114, 116, 101, 117	syl21anc 1263	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( ( ( abs ` ( Re ` ( ( G ` x ) - ( G ` X ) ) ) ) < ( d / 2 ) /\ ( abs ` ( Im ` ( ( G ` x ) - ( G ` X ) ) ) ) < ( d / 2 ) ) -> ( abs ` ( ( G ` x ) - ( G ` X ) ) ) < d ) )
119	118	imp 430	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) /\ ( ( abs ` ( Re ` ( ( G ` x ) - ( G ` X ) ) ) ) < ( d / 2 ) /\ ( abs ` ( Im ` ( ( G ` x ) - ( G ` X ) ) ) ) < ( d / 2 ) ) ) -> ( abs ` ( ( G ` x ) - ( G ` X ) ) ) < d )
120	101	rpxrd 11343	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> d e. RR* )
121	120	adantr 466	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) /\ ( abs ` ( ( G ` x ) - ( G ` X ) ) ) < d ) -> d e. RR* )
122		cnxmet 21780	. . . . . . . . . . . . . . . . 17 |- ( abs o. - ) e. ( *Met ` CC )
123	121, 122	jctil 539	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) /\ ( abs ` ( ( G ` x ) - ( G ` X ) ) ) < d ) -> ( ( abs o. - ) e. ( *Met ` CC ) /\ d e. RR* ) )
124	114	adantr 466	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) /\ ( abs ` ( ( G ` x ) - ( G ` X ) ) ) < d ) -> ( G ` X ) e. CC )
125	116	adantr 466	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) /\ ( abs ` ( ( G ` x ) - ( G ` X ) ) ) < d ) -> ( G ` x ) e. CC )
126	124, 125	jca 534	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) /\ ( abs ` ( ( G ` x ) - ( G ` X ) ) ) < d ) -> ( ( G ` X ) e. CC /\ ( G ` x ) e. CC ) )
127		eqid 2422	. . . . . . . . . . . . . . . . . . 19 |- ( abs o. - ) = ( abs o. - )
128	127	cnmetdval 21778	. . . . . . . . . . . . . . . . . 18 |- ( ( ( G ` x ) e. CC /\ ( G ` X ) e. CC ) -> ( ( G ` x ) ( abs o. - ) ( G ` X ) ) = ( abs ` ( ( G ` x ) - ( G ` X ) ) ) )
129	125, 124, 128	syl2anc 665	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) /\ ( abs ` ( ( G ` x ) - ( G ` X ) ) ) < d ) -> ( ( G ` x ) ( abs o. - ) ( G ` X ) ) = ( abs ` ( ( G ` x ) - ( G ` X ) ) ) )
130		simpr 462	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) /\ ( abs ` ( ( G ` x ) - ( G ` X ) ) ) < d ) -> ( abs ` ( ( G ` x ) - ( G ` X ) ) ) < d )
131	129, 130	eqbrtrd 4441	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) /\ ( abs ` ( ( G ` x ) - ( G ` X ) ) ) < d ) -> ( ( G ` x ) ( abs o. - ) ( G ` X ) ) < d )
132		elbl3 21394	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( abs o. - ) e. ( *Met ` CC ) /\ d e. RR* ) /\ ( ( G ` X ) e. CC /\ ( G ` x ) e. CC ) ) -> ( ( G ` x ) e. ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) <-> ( ( G ` x ) ( abs o. - ) ( G ` X ) ) < d ) )
133	132	biimpar 487	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( abs o. - ) e. ( *Met ` CC ) /\ d e. RR* ) /\ ( ( G ` X ) e. CC /\ ( G ` x ) e. CC ) ) /\ ( ( G ` x ) ( abs o. - ) ( G ` X ) ) < d ) -> ( G ` x ) e. ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) )
134	123, 126, 131, 133	syl21anc 1263	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) /\ ( abs ` ( ( G ` x ) - ( G ` X ) ) ) < d ) -> ( G ` x ) e. ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) )
135	119, 134	syldan 472	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) /\ ( ( abs ` ( Re ` ( ( G ` x ) - ( G ` X ) ) ) ) < ( d / 2 ) /\ ( abs ` ( Im ` ( ( G ` x ) - ( G ` X ) ) ) ) < ( d / 2 ) ) ) -> ( G ` x ) e. ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) )
136	135	ex 435	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( ( ( abs ` ( Re ` ( ( G ` x ) - ( G ` X ) ) ) ) < ( d / 2 ) /\ ( abs ` ( Im ` ( ( G ` x ) - ( G ` X ) ) ) ) < ( d / 2 ) ) -> ( G ` x ) e. ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) )
137	105, 136	syld 45	. . . . . . . . . . . 12 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( x e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) -> ( G ` x ) e. ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) )
138		f1ocnv 5840	. . . . . . . . . . . . . . 15 |- ( G : ( RR X. RR ) -1-1-onto-> CC -> `' G : CC -1-1-onto-> ( RR X. RR ) )
139	107, 110, 138	mp2b 10	. . . . . . . . . . . . . 14 |- `' G : CC -1-1-onto-> ( RR X. RR )
140		f1ofun 5830	. . . . . . . . . . . . . 14 |- ( `' G : CC -1-1-onto-> ( RR X. RR ) -> Fun `' G )
141	139, 140	ax-mp 5	. . . . . . . . . . . . 13 |- Fun `' G
142		f1odm 5832	. . . . . . . . . . . . . . 15 |- ( `' G : CC -1-1-onto-> ( RR X. RR ) -> dom `' G = CC )
143	139, 142	ax-mp 5	. . . . . . . . . . . . . 14 |- dom `' G = CC
144	116, 143	syl6eleqr 2521	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( G ` x ) e. dom `' G )
145		funfvima 6152	. . . . . . . . . . . . 13 |- ( ( Fun `' G /\ ( G ` x ) e. dom `' G ) -> ( ( G ` x ) e. ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) -> ( `' G ` ( G ` x ) ) e. ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) ) )
146	141, 144, 145	sylancr 667	. . . . . . . . . . . 12 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( ( G ` x ) e. ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) -> ( `' G ` ( G ` x ) ) e. ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) ) )
147	107, 110	mp1i 13	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> G : ( RR X. RR ) -1-1-onto-> CC )
148		f1ocnvfv1 6187	. . . . . . . . . . . . . . 15 |- ( ( G : ( RR X. RR ) -1-1-onto-> CC /\ x e. ( RR X. RR ) ) -> ( `' G ` ( G ` x ) ) = x )
149	147, 100, 148	syl2anc 665	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( `' G ` ( G ` x ) ) = x )
150	149	eleq1d 2491	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( ( `' G ` ( G ` x ) ) e. ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) <-> x e. ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) ) )
151	150	biimpd 210	. . . . . . . . . . . 12 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( ( `' G ` ( G ` x ) ) e. ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) -> x e. ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) ) )
152	137, 146, 151	3syld 57	. . . . . . . . . . 11 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( x e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) -> x e. ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) ) )
153	152	imp 430	. . . . . . . . . 10 |- ( ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) /\ x e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) ) -> x e. ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) )
154	95, 153	syl 17	. . . . . . . . 9 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) ) -> x e. ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) )
155	154	ex 435	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( x e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) -> x e. ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) ) )
156	155	ssrdv 3470	. . . . . . 7 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) )
157	156	ralrimiva 2839	. . . . . 6 |- ( ( A e. ( J tX J ) /\ X e. A ) -> A. d e. RR+ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) )
158	103	mpt2fun 6409	. . . . . . . . . 10 |- Fun G
159	158	a1i 11	. . . . . . . . 9 |- ( ( A e. ( J tX J ) /\ X e. A ) -> Fun G )
160	13	sselda 3464	. . . . . . . . . 10 |- ( ( A e. ( J tX J ) /\ X e. A ) -> X e. ( RR X. RR ) )
161		f1odm 5832	. . . . . . . . . . 11 |- ( G : ( RR X. RR ) -1-1-onto-> CC -> dom G = ( RR X. RR ) )
162	107, 110, 161	mp2b 10	. . . . . . . . . 10 |- dom G = ( RR X. RR )
163	160, 162	syl6eleqr 2521	. . . . . . . . 9 |- ( ( A e. ( J tX J ) /\ X e. A ) -> X e. dom G )
164		simpr 462	. . . . . . . . 9 |- ( ( A e. ( J tX J ) /\ X e. A ) -> X e. A )
165		funfvima 6152	. . . . . . . . . 10 |- ( ( Fun G /\ X e. dom G ) -> ( X e. A -> ( G ` X ) e. ( G " A ) ) )
166	165	imp 430	. . . . . . . . 9 |- ( ( ( Fun G /\ X e. dom G ) /\ X e. A ) -> ( G ` X ) e. ( G " A ) )
167	159, 163, 164, 166	syl21anc 1263	. . . . . . . 8 |- ( ( A e. ( J tX J ) /\ X e. A ) -> ( G ` X ) e. ( G " A ) )
168		hmeoima 20767	. . . . . . . . . . 11 |- ( ( G e. ( ( J tX J ) Homeo ( TopOpen ` ℂfld ) ) /\ A e. ( J tX J ) ) -> ( G " A ) e. ( TopOpen ` ℂfld ) )
169	107, 168	mpan 674	. . . . . . . . . 10 |- ( A e. ( J tX J ) -> ( G " A ) e. ( TopOpen ` ℂfld ) )
170	106	cnfldtopn 21789	. . . . . . . . . . . . 13 |- ( TopOpen ` ℂfld ) = ( MetOpen ` ( abs o. - ) )
171	170	elmopn2 21447	. . . . . . . . . . . 12 |- ( ( abs o. - ) e. ( *Met ` CC ) -> ( ( G " A ) e. ( TopOpen ` ℂfld ) <-> ( ( G " A ) C_ CC /\ A. m e. ( G " A ) E. d e. RR+ ( m ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) ) ) )
172	122, 171	ax-mp 5	. . . . . . . . . . 11 |- ( ( G " A ) e. ( TopOpen ` ℂfld ) <-> ( ( G " A ) C_ CC /\ A. m e. ( G " A ) E. d e. RR+ ( m ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) ) )
173	172	simprbi 465	. . . . . . . . . 10 |- ( ( G " A ) e. ( TopOpen ` ℂfld ) -> A. m e. ( G " A ) E. d e. RR+ ( m ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) )
174	169, 173	syl 17	. . . . . . . . 9 |- ( A e. ( J tX J ) -> A. m e. ( G " A ) E. d e. RR+ ( m ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) )
175	174	adantr 466	. . . . . . . 8 |- ( ( A e. ( J tX J ) /\ X e. A ) -> A. m e. ( G " A ) E. d e. RR+ ( m ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) )
176		oveq1 6309	. . . . . . . . . . 11 |- ( m = ( G ` X ) -> ( m ( ball ` ( abs o. - ) ) d ) = ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) )
177	176	sseq1d 3491	. . . . . . . . . 10 |- ( m = ( G ` X ) -> ( ( m ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) <-> ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) ) )
178	177	rexbidv 2939	. . . . . . . . 9 |- ( m = ( G ` X ) -> ( E. d e. RR+ ( m ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) <-> E. d e. RR+ ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) ) )
179	178	rspcva 3180	. . . . . . . 8 |- ( ( ( G ` X ) e. ( G " A ) /\ A. m e. ( G " A ) E. d e. RR+ ( m ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) ) -> E. d e. RR+ ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) )
180	167, 175, 179	syl2anc 665	. . . . . . 7 |- ( ( A e. ( J tX J ) /\ X e. A ) -> E. d e. RR+ ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) )
181		imass2 5220	. . . . . . . . . 10 |- ( ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) -> ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ ( `' G " ( G " A ) ) )
182		f1of1 5827	. . . . . . . . . . . . 13 |- ( G : ( RR X. RR ) -1-1-onto-> CC -> G : ( RR X. RR ) -1-1-> CC )
183	107, 110, 182	mp2b 10	. . . . . . . . . . . 12 |- G : ( RR X. RR ) -1-1-> CC
184		f1imacnv 5844	. . . . . . . . . . . 12 |- ( ( G : ( RR X. RR ) -1-1-> CC /\ A C_ ( RR X. RR ) ) -> ( `' G " ( G " A ) ) = A )
185	183, 13, 184	sylancr 667	. . . . . . . . . . 11 |- ( A e. ( J tX J ) -> ( `' G " ( G " A ) ) = A )
186	185	sseq2d 3492	. . . . . . . . . 10 |- ( A e. ( J tX J ) -> ( ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ ( `' G " ( G " A ) ) <-> ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ A ) )
187	181, 186	syl5ib 222	. . . . . . . . 9 |- ( A e. ( J tX J ) -> ( ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) -> ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ A ) )
188	187	reximdv 2899	. . . . . . . 8 |- ( A e. ( J tX J ) -> ( E. d e. RR+ ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) -> E. d e. RR+ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ A ) )
189	188	adantr 466	. . . . . . 7 |- ( ( A e. ( J tX J ) /\ X e. A ) -> ( E. d e. RR+ ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) -> E. d e. RR+ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ A ) )
190	180, 189	mpd 15	. . . . . 6 |- ( ( A e. ( J tX J ) /\ X e. A ) -> E. d e. RR+ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ A )
191		r19.29 2963	. . . . . 6 |- ( ( A. d e. RR+ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) /\ E. d e. RR+ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ A ) -> E. d e. RR+ ( ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) /\ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ A ) )
192	157, 190, 191	syl2anc 665	. . . . 5 |- ( ( A e. ( J tX J ) /\ X e. A ) -> E. d e. RR+ ( ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) /\ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ A ) )
193		sstr 3472	. . . . . 6 |- ( ( ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) /\ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ A ) -> ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ A )
194	193	reximi 2893	. . . . 5 |- ( E. d e. RR+ ( ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) /\ ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ A ) -> E. d e. RR+ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ A )
195	192, 194	syl 17	. . . 4 |- ( ( A e. ( J tX J ) /\ X e. A ) -> E. d e. RR+ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ A )
196		r19.29 2963	. . . 4 |- ( ( A. d e. RR+ X e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) /\ E. d e. RR+ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ A ) -> E. d e. RR+ ( X e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) /\ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ A ) )
197	70, 195, 196	syl2anc 665	. . 3 |- ( ( A e. ( J tX J ) /\ X e. A ) -> E. d e. RR+ ( X e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) /\ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ A ) )
198		r19.29 2963	. . 3 |- ( ( A. d e. RR+ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) e. B /\ E. d e. RR+ ( X e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) /\ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ A ) ) -> E. d e. RR+ ( ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) e. B /\ ( X e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) /\ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ A ) ) )
199	51, 197, 198	syl2anc 665	. 2 |- ( ( A e. ( J tX J ) /\ X e. A ) -> E. d e. RR+ ( ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) e. B /\ ( X e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) /\ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ A ) ) )
200		eleq2 2495	. . . . 5 |- ( r = ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) -> ( X e. r <-> X e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) ) )
201		sseq1 3485	. . . . 5 |- ( r = ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) -> ( r C_ A <-> ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ A ) )
202	200, 201	anbi12d 715	. . . 4 |- ( r = ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) -> ( ( X e. r /\ r C_ A ) <-> ( X e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) /\ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ A ) ) )
203	202	rspcev 3182	. . 3 |- ( ( ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) e. B /\ ( X e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) /\ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ A ) ) -> E. r e. B ( X e. r /\ r C_ A ) )
204	203	rexlimivw 2914	. 2 |- ( E. d e. RR+ ( ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) e. B /\ ( X e. ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) /\ ( ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) X. ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) ) C_ A ) ) -> E. r e. B ( X e. r /\ r C_ A ) )
205	199, 204	syl 17	1 |- ( ( A e. ( J tX J ) /\ X e. A ) -> E. r e. B ( X e. r /\ r C_ A ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1868   A.wral 2775   E.wrex 2776   _Vcvv 3081   C_ wss 3436   ~Pcpw 3979   U.cuni 4216   class class class wbr 4420   X. cxp 4848   `'ccnv 4849   dom cdm 4850   ran crn 4851   "cima 4853   o. ccom 4854   Fun wfun 5592   Fn wfn 5593   -->wf 5594   -1-1->wf1 5595   -1-1-onto->wf1o 5597   ` cfv 5598  (class class class)co 6302   |-> cmpt2 6304   1stc1st 6802   2ndc2nd 6803   CCcc 9538   RRcr 9539   _ici 9542   + caddc 9543   x. cmul 9545   +oocpnf 9673   -oocmnf 9674   RR*cxr 9675   < clt 9676   <_ cle 9677   - cmin 9861   / cdiv 10270   2c2 10660   RR+crp 11303   (,)cioo 11636   Recre 13149   Imcim 13150   abscabs 13286   TopOpenctopn 15308   topGenctg 15324   *Metcxmt 18943   ballcbl 18945  ℂ_fldccnfld 18958   Topctop 19904   tX ctx 20562   Homeochmeo 20755

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 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619  ax-mulf 9620

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-fi 7928  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xneg 11410  df-xadd 11411  df-xmul 11412  df-ioo 11640  df-icc 11643  df-fz 11786  df-fzo 11917  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-hom 15202  df-cco 15203  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-pt 15331  df-prds 15334  df-xrs 15388  df-qtop 15394  df-imas 15395  df-xps 15398  df-mre 15480  df-mrc 15481  df-acs 15483  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-mulg 16664  df-cntz 16959  df-cmn 17420  df-psmet 18950  df-xmet 18951  df-met 18952  df-bl 18953  df-mopn 18954  df-cnfld 18959  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-cn 20230  df-cnp 20231  df-tx 20564  df-hmeo 20757  df-xms 21322  df-ms 21323  df-tms 21324  df-cncf 21897

This theorem is referenced by:  dya2iocnei  29100