Theorem tpr2rico 28047

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 11558	. . . . . . . . . 10 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
2	1	ixxf 11564	. . . . . . . . 9 |- (,) : ( RR* X. RR* ) --> ~P RR*
3		ffn 5737	. . . . . . . . 9 |- ( (,) : ( RR* X. RR* ) --> ~P RR* -> (,) Fn ( RR* X. RR* ) )
4	2, 3	mp1i 12	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> (,) Fn ( RR* X. RR* ) )
5		elssuni 4281	. . . . . . . . . . . . . 14 |- ( A e. ( J tX J ) -> A C_ U. ( J tX J ) )
6		tpr2rico.0	. . . . . . . . . . . . . . . 16 |- J = ( topGen ` ran (,) )
7		retop 21393	. . . . . . . . . . . . . . . 16 |- ( topGen ` ran (,) ) e. Top
8	6, 7	eqeltri 2541	. . . . . . . . . . . . . . 15 |- J e. Top
9		uniretop 21394	. . . . . . . . . . . . . . . 16 |- RR = U. ( topGen ` ran (,) )
10	6	unieqi 4260	. . . . . . . . . . . . . . . 16 |- U. J = U. ( topGen ` ran (,) )
11	9, 10	eqtr4i 2489	. . . . . . . . . . . . . . 15 |- RR = U. J
12	8, 8, 11, 11	txunii 20219	. . . . . . . . . . . . . 14 |- ( RR X. RR ) = U. ( J tX J )
13	5, 12	syl6sseqr 3546	. . . . . . . . . . . . 13 |- ( A e. ( J tX J ) -> A C_ ( RR X. RR ) )
14	13	ad2antrr 725	. . . . . . . . . . . 12 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> A C_ ( RR X. RR ) )
15		simplr 755	. . . . . . . . . . . 12 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> X e. A )
16	14, 15	sseldd 3500	. . . . . . . . . . 11 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> X e. ( RR X. RR ) )
17		xp1st 6829	. . . . . . . . . . 11 |- ( X e. ( RR X. RR ) -> ( 1st ` X ) e. RR )
18	16, 17	syl 16	. . . . . . . . . 10 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 1st ` X ) e. RR )
19		simpr 461	. . . . . . . . . . . 12 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> d e. RR+ )
20	19	rpred 11281	. . . . . . . . . . 11 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> d e. RR )
21	20	rehalfcld 10806	. . . . . . . . . 10 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( d / 2 ) e. RR )
22	18, 21	resubcld 10008	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) - ( d / 2 ) ) e. RR )
23	22	rexrd 9660	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) - ( d / 2 ) ) e. RR* )
24	18, 21	readdcld 9640	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) + ( d / 2 ) ) e. RR )
25	24	rexrd 9660	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) + ( d / 2 ) ) e. RR* )
26		fnovrn 6449	. . . . . . . 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 1228	. . . . . . 7 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) e. ran (,) )
28		xp2nd 6830	. . . . . . . . . . 11 |- ( X e. ( RR X. RR ) -> ( 2nd ` X ) e. RR )
29	16, 28	syl 16	. . . . . . . . . 10 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 2nd ` X ) e. RR )
30	29, 21	resubcld 10008	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) - ( d / 2 ) ) e. RR )
31	30	rexrd 9660	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) - ( d / 2 ) ) e. RR* )
32	29, 21	readdcld 9640	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) + ( d / 2 ) ) e. RR )
33	32	rexrd 9660	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) + ( d / 2 ) ) e. RR* )
34		fnovrn 6449	. . . . . . . 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 1228	. . . . . . 7 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) e. ran (,) )
36		eqidd 2458	. . . . . . 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 5022	. . . . . . . . 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 2471	. . . . . . . 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 5023	. . . . . . . . 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 2471	. . . . . . . 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 3221	. . . . . . 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 1228	. . . . . 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 2457	. . . . . . 7 |- ( x e. ran (,) , y e. ran (,) |-> ( x X. y ) ) = ( x e. ran (,) , y e. ran (,) |-> ( x X. y ) )
44		vex 3112	. . . . . . . 8 |- x e. _V
45		vex 3112	. . . . . . . 8 |- y e. _V
46	44, 45	xpex 6603	. . . . . . 7 |- ( x X. y ) e. _V
47	43, 46	elrnmpt2 6414	. . . . . 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 212	. . . . 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 2556	. . . 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 2871	. . 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 5118	. . . . . . 7 |- ( RR X. RR ) C_ ( _V X. _V )
53	52, 16	sseldi 3497	. . . . . 6 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> X e. ( _V X. _V ) )
54	18	rexrd 9660	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 1st ` X ) e. RR* )
55	19	rphalfcld 11293	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( d / 2 ) e. RR+ )
56	18, 55	ltsubrpd 11309	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) - ( d / 2 ) ) < ( 1st ` X ) )
57	18, 55	ltaddrpd 11310	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 1st ` X ) < ( ( 1st ` X ) + ( d / 2 ) ) )
58		elioo1 11594	. . . . . . . . 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 661	. . . . . . . 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 1179	. . . . . . 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 9660	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 2nd ` X ) e. RR* )
62	29, 55	ltsubrpd 11309	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) - ( d / 2 ) ) < ( 2nd ` X ) )
63	29, 55	ltaddrpd 11310	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 2nd ` X ) < ( ( 2nd ` X ) + ( d / 2 ) ) )
64		elioo1 11594	. . . . . . . . 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 661	. . . . . . . 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 1179	. . . . . . 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 532	. . . . . 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 6832	. . . . . 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 664	. . . . 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 2871	. . . 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 11367	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 1st ` X ) - ( d / 2 ) ) e. RR* -> -oo <_ ( ( 1st ` X ) - ( d / 2 ) ) )
72	23, 71	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> -oo <_ ( ( 1st ` X ) - ( d / 2 ) ) )
73		pnfge 11364	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 1st ` X ) + ( d / 2 ) ) e. RR* -> ( ( 1st ` X ) + ( d / 2 ) ) <_ +oo )
74	25, 73	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) + ( d / 2 ) ) <_ +oo )
75		mnfxr 11348	. . . . . . . . . . . . . . . . . 18 |- -oo e. RR*
76		pnfxr 11346	. . . . . . . . . . . . . . . . . 18 |- +oo e. RR*
77		ioossioo 11641	. . . . . . . . . . . . . . . . . 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 682	. . . . . . . . . . . . . . . . 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 661	. . . . . . . . . . . . . . . 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 11624	. . . . . . . . . . . . . . . 16 |- ( -oo (,) +oo ) = RR
81	79, 80	syl6sseq 3545	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) C_ RR )
82		mnfle 11367	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 2nd ` X ) - ( d / 2 ) ) e. RR* -> -oo <_ ( ( 2nd ` X ) - ( d / 2 ) ) )
83	31, 82	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> -oo <_ ( ( 2nd ` X ) - ( d / 2 ) ) )
84		pnfge 11364	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 2nd ` X ) + ( d / 2 ) ) e. RR* -> ( ( 2nd ` X ) + ( d / 2 ) ) <_ +oo )
85	33, 84	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) + ( d / 2 ) ) <_ +oo )
86		ioossioo 11641	. . . . . . . . . . . . . . . . . 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 682	. . . . . . . . . . . . . . . . 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 661	. . . . . . . . . . . . . . . 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 3545	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) C_ RR )
90		xpss12 5117	. . . . . . . . . . . . . . 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 661	. . . . . . . . . . . . . 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 3499	. . . . . . . . . . . . 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 435	. . . . . . . . . . . 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 553	. . . . . . . . . . 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 691	. . . . . . . . . 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 465	. . . . . . . . . . . . . . . 16 |- ( ( A e. ( J tX J ) /\ ( X e. A /\ d e. RR+ /\ x e. ( RR X. RR ) ) ) -> A C_ ( RR X. RR ) )
97		simpr1 1002	. . . . . . . . . . . . . . . 16 |- ( ( A e. ( J tX J ) /\ ( X e. A /\ d e. RR+ /\ x e. ( RR X. RR ) ) ) -> X e. A )
98	96, 97	sseldd 3500	. . . . . . . . . . . . . . 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 1218	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> X e. ( RR X. RR ) )
100		simpr 461	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> x e. ( RR X. RR ) )
101		simplr 755	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> d e. RR+ )
102	101	rphalfcld 11293	. . . . . . . . . . . . . 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 28046	. . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . 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 2457	. . . . . . . . . . . . . . . . . . . . 21 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
107	103, 6, 106	cnrehmeo 21578	. . . . . . . . . . . . . . . . . . . 20 |- G e. ( ( J tX J ) Homeo ( TopOpen ` ℂfld ) )
108	106	cnfldtopon 21415	. . . . . . . . . . . . . . . . . . . . . 22 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
109	108	toponunii 19559	. . . . . . . . . . . . . . . . . . . . 21 |- CC = U. ( TopOpen ` ℂfld )
110	12, 109	hmeof1o 20390	. . . . . . . . . . . . . . . . . . . 20 |- ( G e. ( ( J tX J ) Homeo ( TopOpen ` ℂfld ) ) -> G : ( RR X. RR ) -1-1-onto-> CC )
111		f1of 5822	. . . . . . . . . . . . . . . . . . . 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 6033	. . . . . . . . . . . . . . . . 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 6032	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( G ` x ) e. CC )
117		sqsscirc2 28044	. . . . . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . . . 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 429	. . . . . . . . . . . . . . 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 11282	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> d e. RR* )
121	120	adantr 465	. . . . . . . . . . . . . . . . 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 21405	. . . . . . . . . . . . . . . . 17 |- ( abs o. - ) e. ( *Met ` CC )
123	121, 122	jctil 537	. . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . . . 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 532	. . . . . . . . . . . . . . . 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 2457	. . . . . . . . . . . . . . . . . . 19 |- ( abs o. - ) = ( abs o. - )
128	127	cnmetdval 21403	. . . . . . . . . . . . . . . . . 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 661	. . . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . . 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 4476	. . . . . . . . . . . . . . . 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 21020	. . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . . 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 470	. . . . . . . . . . . . . 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 434	. . . . . . . . . . . . 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 44	. . . . . . . . . . . 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 5834	. . . . . . . . . . . . . . 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 5824	. . . . . . . . . . . . . 14 |- ( `' G : CC -1-1-onto-> ( RR X. RR ) -> Fun `' G )
141	139, 140	ax-mp 5	. . . . . . . . . . . . 13 |- Fun `' G
142		f1odm 5826	. . . . . . . . . . . . . . 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 2556	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( G ` x ) e. dom `' G )
145		funfvima 6148	. . . . . . . . . . . . 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 663	. . . . . . . . . . . 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 12	. . . . . . . . . . . . . . 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 6183	. . . . . . . . . . . . . . 15 |- ( ( G : ( RR X. RR ) -1-1-onto-> CC /\ x e. ( RR X. RR ) ) -> ( `' G ` ( G ` x ) ) = x )
149	147, 100, 148	syl2anc 661	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( `' G ` ( G ` x ) ) = x )
150	149	eleq1d 2526	. . . . . . . . . . . . 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 207	. . . . . . . . . . . 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 55	. . . . . . . . . . 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 429	. . . . . . . . . 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 16	. . . . . . . . 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 434	. . . . . . . 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 3505	. . . . . . 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 2871	. . . . . 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 6403	. . . . . . . . . 10 |- Fun G
159	158	a1i 11	. . . . . . . . 9 |- ( ( A e. ( J tX J ) /\ X e. A ) -> Fun G )
160	13	sselda 3499	. . . . . . . . . 10 |- ( ( A e. ( J tX J ) /\ X e. A ) -> X e. ( RR X. RR ) )
161		f1odm 5826	. . . . . . . . . . 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 2556	. . . . . . . . 9 |- ( ( A e. ( J tX J ) /\ X e. A ) -> X e. dom G )
164		simpr 461	. . . . . . . . 9 |- ( ( A e. ( J tX J ) /\ X e. A ) -> X e. A )
165		funfvima 6148	. . . . . . . . . 10 |- ( ( Fun G /\ X e. dom G ) -> ( X e. A -> ( G ` X ) e. ( G " A ) ) )
166	165	imp 429	. . . . . . . . 9 |- ( ( ( Fun G /\ X e. dom G ) /\ X e. A ) -> ( G ` X ) e. ( G " A ) )
167	159, 163, 164, 166	syl21anc 1227	. . . . . . . 8 |- ( ( A e. ( J tX J ) /\ X e. A ) -> ( G ` X ) e. ( G " A ) )
168		hmeoima 20391	. . . . . . . . . . 11 |- ( ( G e. ( ( J tX J ) Homeo ( TopOpen ` ℂfld ) ) /\ A e. ( J tX J ) ) -> ( G " A ) e. ( TopOpen ` ℂfld ) )
169	107, 168	mpan 670	. . . . . . . . . 10 |- ( A e. ( J tX J ) -> ( G " A ) e. ( TopOpen ` ℂfld ) )
170	106	cnfldtopn 21414	. . . . . . . . . . . . 13 |- ( TopOpen ` ℂfld ) = ( MetOpen ` ( abs o. - ) )
171	170	elmopn2 21073	. . . . . . . . . . . 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 464	. . . . . . . . . 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 16	. . . . . . . . 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 465	. . . . . . . 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 6303	. . . . . . . . . . 11 |- ( m = ( G ` X ) -> ( m ( ball ` ( abs o. - ) ) d ) = ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) )
177	176	sseq1d 3526	. . . . . . . . . 10 |- ( m = ( G ` X ) -> ( ( m ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) <-> ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) ) )
178	177	rexbidv 2968	. . . . . . . . 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 3208	. . . . . . . 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 661	. . . . . . 7 |- ( ( A e. ( J tX J ) /\ X e. A ) -> E. d e. RR+ ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) )
181		imass2 5382	. . . . . . . . . 10 |- ( ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) -> ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ ( `' G " ( G " A ) ) )
182		f1of1 5821	. . . . . . . . . . . . 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 5838	. . . . . . . . . . . 12 |- ( ( G : ( RR X. RR ) -1-1-> CC /\ A C_ ( RR X. RR ) ) -> ( `' G " ( G " A ) ) = A )
185	183, 13, 184	sylancr 663	. . . . . . . . . . 11 |- ( A e. ( J tX J ) -> ( `' G " ( G " A ) ) = A )
186	185	sseq2d 3527	. . . . . . . . . 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 219	. . . . . . . . 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 2931	. . . . . . . 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 465	. . . . . . 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 2992	. . . . . 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 661	. . . . 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 3507	. . . . . 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 2925	. . . . 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 16	. . . 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 2992	. . . 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 661	. . 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 2992	. . 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 661	. 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 2530	. . . . 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 3520	. . . . 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 710	. . . 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 3210	. . 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 2946	. 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 16	1 |- ( ( A e. ( J tX J ) /\ X e. A ) -> E. r e. B ( X e. r /\ r C_ A ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109   C_ wss 3471   ~Pcpw 4015   U.cuni 4251   class class class wbr 4456   X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011   o. ccom 5012   Fun wfun 5588   Fn wfn 5589   -->wf 5590   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798   CCcc 9507   RRcr 9508   _ici 9511   + caddc 9512   x. cmul 9514   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644   < clt 9645   <_ cle 9646   - cmin 9824   / cdiv 10227   2c2 10606   RR+crp 11245   (,)cioo 11554   Recre 12941   Imcim 12942   abscabs 13078   TopOpenctopn 14838   topGenctg 14854   *Metcxmt 18529   ballcbl 18531  ℂ_fldccnfld 18546   Topctop 19520   tX ctx 20186   Homeochmeo 20379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-icc 11561  df-fz 11698  df-fzo 11821  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-hom 14735  df-cco 14736  df-rest 14839  df-topn 14840  df-0g 14858  df-gsum 14859  df-topgen 14860  df-pt 14861  df-prds 14864  df-xrs 14918  df-qtop 14923  df-imas 14924  df-xps 14926  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-mulg 16186  df-cntz 16481  df-cmn 16926  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-cnfld 18547  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-cn 19854  df-cnp 19855  df-tx 20188  df-hmeo 20381  df-xms 20948  df-ms 20949  df-tms 20950  df-cncf 21507

This theorem is referenced by:  dya2iocnei  28414