Theorem tpr2rico 28347

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 11586	. . . . . . . . . 10 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
2	1	ixxf 11592	. . . . . . . . 9 |- (,) : ( RR* X. RR* ) --> ~P RR*
3		ffn 5714	. . . . . . . . 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 4220	. . . . . . . . . . . . . 14 |- ( A e. ( J tX J ) -> A C_ U. ( J tX J ) )
6		tpr2rico.0	. . . . . . . . . . . . . . . 16 |- J = ( topGen ` ran (,) )
7		retop 21560	. . . . . . . . . . . . . . . 16 |- ( topGen ` ran (,) ) e. Top
8	6, 7	eqeltri 2486	. . . . . . . . . . . . . . 15 |- J e. Top
9		uniretop 21561	. . . . . . . . . . . . . . . 16 |- RR = U. ( topGen ` ran (,) )
10	6	unieqi 4200	. . . . . . . . . . . . . . . 16 |- U. J = U. ( topGen ` ran (,) )
11	9, 10	eqtr4i 2434	. . . . . . . . . . . . . . 15 |- RR = U. J
12	8, 8, 11, 11	txunii 20386	. . . . . . . . . . . . . 14 |- ( RR X. RR ) = U. ( J tX J )
13	5, 12	syl6sseqr 3489	. . . . . . . . . . . . 13 |- ( A e. ( J tX J ) -> A C_ ( RR X. RR ) )
14	13	ad2antrr 724	. . . . . . . . . . . 12 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> A C_ ( RR X. RR ) )
15		simplr 754	. . . . . . . . . . . 12 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> X e. A )
16	14, 15	sseldd 3443	. . . . . . . . . . 11 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> X e. ( RR X. RR ) )
17		xp1st 6814	. . . . . . . . . . 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 459	. . . . . . . . . . . 12 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> d e. RR+ )
20	19	rpred 11304	. . . . . . . . . . 11 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> d e. RR )
21	20	rehalfcld 10826	. . . . . . . . . 10 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( d / 2 ) e. RR )
22	18, 21	resubcld 10028	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) - ( d / 2 ) ) e. RR )
23	22	rexrd 9673	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) - ( d / 2 ) ) e. RR* )
24	18, 21	readdcld 9653	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) + ( d / 2 ) ) e. RR )
25	24	rexrd 9673	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) + ( d / 2 ) ) e. RR* )
26		fnovrn 6431	. . . . . . . 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 1230	. . . . . . 7 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) e. ran (,) )
28		xp2nd 6815	. . . . . . . . . . 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 10028	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) - ( d / 2 ) ) e. RR )
31	30	rexrd 9673	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) - ( d / 2 ) ) e. RR* )
32	29, 21	readdcld 9653	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) + ( d / 2 ) ) e. RR )
33	32	rexrd 9673	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) + ( d / 2 ) ) e. RR* )
34		fnovrn 6431	. . . . . . . 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 1230	. . . . . . 7 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) e. ran (,) )
36		eqidd 2403	. . . . . . 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 4837	. . . . . . . . 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 2416	. . . . . . . 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 4838	. . . . . . . . 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 2416	. . . . . . . 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 3171	. . . . . . 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 1230	. . . . . 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 2402	. . . . . . 7 |- ( x e. ran (,) , y e. ran (,) |-> ( x X. y ) ) = ( x e. ran (,) , y e. ran (,) |-> ( x X. y ) )
44		vex 3062	. . . . . . . 8 |- x e. _V
45		vex 3062	. . . . . . . 8 |- y e. _V
46	44, 45	xpex 6586	. . . . . . 7 |- ( x X. y ) e. _V
47	43, 46	elrnmpt2 6396	. . . . . 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 2501	. . . 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 2818	. . 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 4930	. . . . . . 7 |- ( RR X. RR ) C_ ( _V X. _V )
53	52, 16	sseldi 3440	. . . . . 6 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> X e. ( _V X. _V ) )
54	18	rexrd 9673	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 1st ` X ) e. RR* )
55	19	rphalfcld 11316	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( d / 2 ) e. RR+ )
56	18, 55	ltsubrpd 11332	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) - ( d / 2 ) ) < ( 1st ` X ) )
57	18, 55	ltaddrpd 11333	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 1st ` X ) < ( ( 1st ` X ) + ( d / 2 ) ) )
58		elioo1 11622	. . . . . . . . 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 659	. . . . . . . 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 1180	. . . . . . 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 9673	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 2nd ` X ) e. RR* )
62	29, 55	ltsubrpd 11332	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) - ( d / 2 ) ) < ( 2nd ` X ) )
63	29, 55	ltaddrpd 11333	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 2nd ` X ) < ( ( 2nd ` X ) + ( d / 2 ) ) )
64		elioo1 11622	. . . . . . . . 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 659	. . . . . . . 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 1180	. . . . . . 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 530	. . . . . 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 6817	. . . . . 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 662	. . . . 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 2818	. . . 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 11395	. . . . . . . . . . . . . . . . . 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 11392	. . . . . . . . . . . . . . . . . 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 11376	. . . . . . . . . . . . . . . . . 18 |- -oo e. RR*
76		pnfxr 11374	. . . . . . . . . . . . . . . . . 18 |- +oo e. RR*
77		ioossioo 11670	. . . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . . 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 659	. . . . . . . . . . . . . . . 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 11653	. . . . . . . . . . . . . . . 16 |- ( -oo (,) +oo ) = RR
81	79, 80	syl6sseq 3488	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) C_ RR )
82		mnfle 11395	. . . . . . . . . . . . . . . . . 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 11392	. . . . . . . . . . . . . . . . . 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 11670	. . . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . . 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 659	. . . . . . . . . . . . . . . 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 3488	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) C_ RR )
90		xpss12 4929	. . . . . . . . . . . . . . 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 659	. . . . . . . . . . . . . 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 3442	. . . . . . . . . . . . 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 433	. . . . . . . . . . . 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 551	. . . . . . . . . . 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 689	. . . . . . . . . 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 463	. . . . . . . . . . . . . . . 16 |- ( ( A e. ( J tX J ) /\ ( X e. A /\ d e. RR+ /\ x e. ( RR X. RR ) ) ) -> A C_ ( RR X. RR ) )
97		simpr1 1003	. . . . . . . . . . . . . . . 16 |- ( ( A e. ( J tX J ) /\ ( X e. A /\ d e. RR+ /\ x e. ( RR X. RR ) ) ) -> X e. A )
98	96, 97	sseldd 3443	. . . . . . . . . . . . . . 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 1220	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> X e. ( RR X. RR ) )
100		simpr 459	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> x e. ( RR X. RR ) )
101		simplr 754	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> d e. RR+ )
102	101	rphalfcld 11316	. . . . . . . . . . . . . 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 28346	. . . . . . . . . . . . . 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 1230	. . . . . . . . . . . . 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 2402	. . . . . . . . . . . . . . . . . . . . 21 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
107	103, 6, 106	cnrehmeo 21745	. . . . . . . . . . . . . . . . . . . 20 |- G e. ( ( J tX J ) Homeo ( TopOpen ` ℂfld ) )
108	106	cnfldtopon 21582	. . . . . . . . . . . . . . . . . . . . . 22 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
109	108	toponunii 19725	. . . . . . . . . . . . . . . . . . . . 21 |- CC = U. ( TopOpen ` ℂfld )
110	12, 109	hmeof1o 20557	. . . . . . . . . . . . . . . . . . . 20 |- ( G e. ( ( J tX J ) Homeo ( TopOpen ` ℂfld ) ) -> G : ( RR X. RR ) -1-1-onto-> CC )
111		f1of 5799	. . . . . . . . . . . . . . . . . . . 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 6010	. . . . . . . . . . . . . . . . 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 6009	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( G ` x ) e. CC )
117		sqsscirc2 28344	. . . . . . . . . . . . . . . . 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 1229	. . . . . . . . . . . . . . . 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 427	. . . . . . . . . . . . . . 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 11305	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> d e. RR* )
121	120	adantr 463	. . . . . . . . . . . . . . . . 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 21572	. . . . . . . . . . . . . . . . 17 |- ( abs o. - ) e. ( *Met ` CC )
123	121, 122	jctil 535	. . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . . . 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 530	. . . . . . . . . . . . . . . 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 2402	. . . . . . . . . . . . . . . . . . 19 |- ( abs o. - ) = ( abs o. - )
128	127	cnmetdval 21570	. . . . . . . . . . . . . . . . . 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 659	. . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . 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 4415	. . . . . . . . . . . . . . . 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 21187	. . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . 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 1229	. . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . 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 432	. . . . . . . . . . . . 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 42	. . . . . . . . . . . 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 5811	. . . . . . . . . . . . . . 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 5801	. . . . . . . . . . . . . 14 |- ( `' G : CC -1-1-onto-> ( RR X. RR ) -> Fun `' G )
141	139, 140	ax-mp 5	. . . . . . . . . . . . 13 |- Fun `' G
142		f1odm 5803	. . . . . . . . . . . . . . 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 2501	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( G ` x ) e. dom `' G )
145		funfvima 6128	. . . . . . . . . . . . 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 661	. . . . . . . . . . . 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 6163	. . . . . . . . . . . . . . 15 |- ( ( G : ( RR X. RR ) -1-1-onto-> CC /\ x e. ( RR X. RR ) ) -> ( `' G ` ( G ` x ) ) = x )
149	147, 100, 148	syl2anc 659	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( `' G ` ( G ` x ) ) = x )
150	149	eleq1d 2471	. . . . . . . . . . . . 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 54	. . . . . . . . . . 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 427	. . . . . . . . . 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 432	. . . . . . . 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 3448	. . . . . . 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 2818	. . . . . 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 6385	. . . . . . . . . 10 |- Fun G
159	158	a1i 11	. . . . . . . . 9 |- ( ( A e. ( J tX J ) /\ X e. A ) -> Fun G )
160	13	sselda 3442	. . . . . . . . . 10 |- ( ( A e. ( J tX J ) /\ X e. A ) -> X e. ( RR X. RR ) )
161		f1odm 5803	. . . . . . . . . . 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 2501	. . . . . . . . 9 |- ( ( A e. ( J tX J ) /\ X e. A ) -> X e. dom G )
164		simpr 459	. . . . . . . . 9 |- ( ( A e. ( J tX J ) /\ X e. A ) -> X e. A )
165		funfvima 6128	. . . . . . . . . 10 |- ( ( Fun G /\ X e. dom G ) -> ( X e. A -> ( G ` X ) e. ( G " A ) ) )
166	165	imp 427	. . . . . . . . 9 |- ( ( ( Fun G /\ X e. dom G ) /\ X e. A ) -> ( G ` X ) e. ( G " A ) )
167	159, 163, 164, 166	syl21anc 1229	. . . . . . . 8 |- ( ( A e. ( J tX J ) /\ X e. A ) -> ( G ` X ) e. ( G " A ) )
168		hmeoima 20558	. . . . . . . . . . 11 |- ( ( G e. ( ( J tX J ) Homeo ( TopOpen ` ℂfld ) ) /\ A e. ( J tX J ) ) -> ( G " A ) e. ( TopOpen ` ℂfld ) )
169	107, 168	mpan 668	. . . . . . . . . 10 |- ( A e. ( J tX J ) -> ( G " A ) e. ( TopOpen ` ℂfld ) )
170	106	cnfldtopn 21581	. . . . . . . . . . . . 13 |- ( TopOpen ` ℂfld ) = ( MetOpen ` ( abs o. - ) )
171	170	elmopn2 21240	. . . . . . . . . . . 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 462	. . . . . . . . . 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 463	. . . . . . . 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 6285	. . . . . . . . . . 11 |- ( m = ( G ` X ) -> ( m ( ball ` ( abs o. - ) ) d ) = ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) )
177	176	sseq1d 3469	. . . . . . . . . 10 |- ( m = ( G ` X ) -> ( ( m ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) <-> ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) ) )
178	177	rexbidv 2918	. . . . . . . . 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 3158	. . . . . . . 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 659	. . . . . . 7 |- ( ( A e. ( J tX J ) /\ X e. A ) -> E. d e. RR+ ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) )
181		imass2 5192	. . . . . . . . . 10 |- ( ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) -> ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ ( `' G " ( G " A ) ) )
182		f1of1 5798	. . . . . . . . . . . . 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 5815	. . . . . . . . . . . 12 |- ( ( G : ( RR X. RR ) -1-1-> CC /\ A C_ ( RR X. RR ) ) -> ( `' G " ( G " A ) ) = A )
185	183, 13, 184	sylancr 661	. . . . . . . . . . 11 |- ( A e. ( J tX J ) -> ( `' G " ( G " A ) ) = A )
186	185	sseq2d 3470	. . . . . . . . . 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 2878	. . . . . . . 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 463	. . . . . . 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 2942	. . . . . 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 659	. . . . 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 3450	. . . . . 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 2872	. . . . 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 2942	. . . 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 659	. . 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 2942	. . 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 659	. 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 2475	. . . . 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 3463	. . . . 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 709	. . . 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 3160	. . 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 2893	. 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 184   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   A.wral 2754   E.wrex 2755   _Vcvv 3059   C_ wss 3414   ~Pcpw 3955   U.cuni 4191   class class class wbr 4395   X. cxp 4821   `'ccnv 4822   dom cdm 4823   ran crn 4824   "cima 4826   o. ccom 4827   Fun wfun 5563   Fn wfn 5564   -->wf 5565   -1-1->wf1 5566   -1-1-onto->wf1o 5568   ` cfv 5569  (class class class)co 6278   |-> cmpt2 6280   1stc1st 6782   2ndc2nd 6783   CCcc 9520   RRcr 9521   _ici 9524   + caddc 9525   x. cmul 9527   +oocpnf 9655   -oocmnf 9656   RR*cxr 9657   < clt 9658   <_ cle 9659   - cmin 9841   / cdiv 10247   2c2 10626   RR+crp 11265   (,)cioo 11582   Recre 13079   Imcim 13080   abscabs 13216   TopOpenctopn 15036   topGenctg 15052   *Metcxmt 18723   ballcbl 18725  ℂ_fldccnfld 18740   Topctop 19686   tX ctx 20353   Homeochmeo 20546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-icc 11589  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cn 20021  df-cnp 20022  df-tx 20355  df-hmeo 20548  df-xms 21115  df-ms 21116  df-tms 21117  df-cncf 21674

This theorem is referenced by:  dya2iocnei  28730