Theorem tpr2rico 26294

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 11296	. . . . . . . . . 10 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
2	1	ixxf 11302	. . . . . . . . 9 |- (,) : ( RR* X. RR* ) --> ~P RR*
3		ffn 5554	. . . . . . . . 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 4116	. . . . . . . . . . . . . 14 |- ( A e. ( J tX J ) -> A C_ U. ( J tX J ) )
6		tpr2rico.0	. . . . . . . . . . . . . . . 16 |- J = ( topGen ` ran (,) )
7		retop 20315	. . . . . . . . . . . . . . . 16 |- ( topGen ` ran (,) ) e. Top
8	6, 7	eqeltri 2508	. . . . . . . . . . . . . . 15 |- J e. Top
9		uniretop 20316	. . . . . . . . . . . . . . . 16 |- RR = U. ( topGen ` ran (,) )
10	6	unieqi 4095	. . . . . . . . . . . . . . . 16 |- U. J = U. ( topGen ` ran (,) )
11	9, 10	eqtr4i 2461	. . . . . . . . . . . . . . 15 |- RR = U. J
12	8, 8, 11, 11	txunii 19141	. . . . . . . . . . . . . 14 |- ( RR X. RR ) = U. ( J tX J )
13	5, 12	syl6sseqr 3398	. . . . . . . . . . . . 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 754	. . . . . . . . . . . 12 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> X e. A )
16	14, 15	sseldd 3352	. . . . . . . . . . 11 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> X e. ( RR X. RR ) )
17		xp1st 6601	. . . . . . . . . . 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 11019	. . . . . . . . . . 11 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> d e. RR )
21	20	rehalfcld 10563	. . . . . . . . . 10 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( d / 2 ) e. RR )
22	18, 21	resubcld 9768	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) - ( d / 2 ) ) e. RR )
23	22	rexrd 9425	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) - ( d / 2 ) ) e. RR* )
24	18, 21	readdcld 9405	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) + ( d / 2 ) ) e. RR )
25	24	rexrd 9425	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) + ( d / 2 ) ) e. RR* )
26		fnovrn 6233	. . . . . . . 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 1218	. . . . . . 7 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) e. ran (,) )
28		xp2nd 6602	. . . . . . . . . . 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 9768	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) - ( d / 2 ) ) e. RR )
31	30	rexrd 9425	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) - ( d / 2 ) ) e. RR* )
32	29, 21	readdcld 9405	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) + ( d / 2 ) ) e. RR )
33	32	rexrd 9425	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) + ( d / 2 ) ) e. RR* )
34		fnovrn 6233	. . . . . . . 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 1218	. . . . . . 7 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) e. ran (,) )
36		eqidd 2439	. . . . . . 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 4849	. . . . . . . . 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 2449	. . . . . . . 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 4850	. . . . . . . . 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 2449	. . . . . . . 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 3076	. . . . . . 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 1218	. . . . . 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 2438	. . . . . . 7 |- ( x e. ran (,) , y e. ran (,) |-> ( x X. y ) ) = ( x e. ran (,) , y e. ran (,) |-> ( x X. y ) )
44		vex 2970	. . . . . . . 8 |- x e. _V
45		vex 2970	. . . . . . . 8 |- y e. _V
46	44, 45	xpex 6503	. . . . . . 7 |- ( x X. y ) e. _V
47	43, 46	elrnmpt2 6198	. . . . . 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 2529	. . . 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 2794	. . 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 4941	. . . . . . 7 |- ( RR X. RR ) C_ ( _V X. _V )
53	52, 16	sseldi 3349	. . . . . 6 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> X e. ( _V X. _V ) )
54	18	rexrd 9425	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 1st ` X ) e. RR* )
55	19	rphalfcld 11031	. . . . . . . . 9 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( d / 2 ) e. RR+ )
56	18, 55	ltsubrpd 11047	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 1st ` X ) - ( d / 2 ) ) < ( 1st ` X ) )
57	18, 55	ltaddrpd 11048	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 1st ` X ) < ( ( 1st ` X ) + ( d / 2 ) ) )
58		elioo1 11332	. . . . . . . . 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 1171	. . . . . . 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 9425	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 2nd ` X ) e. RR* )
62	29, 55	ltsubrpd 11047	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( 2nd ` X ) - ( d / 2 ) ) < ( 2nd ` X ) )
63	29, 55	ltaddrpd 11048	. . . . . . . 8 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( 2nd ` X ) < ( ( 2nd ` X ) + ( d / 2 ) ) )
64		elioo1 11332	. . . . . . . . 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 1171	. . . . . . 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 6604	. . . . . 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 2794	. . . 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 11105	. . . . . . . . . . . . . . . . . 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 11102	. . . . . . . . . . . . . . . . . 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 11086	. . . . . . . . . . . . . . . . . 18 |- -oo e. RR*
76		pnfxr 11084	. . . . . . . . . . . . . . . . . 18 |- +oo e. RR*
77		ioossioo 26014	. . . . . . . . . . . . . . . . . 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 11362	. . . . . . . . . . . . . . . 16 |- ( -oo (,) +oo ) = RR
81	79, 80	syl6sseq 3397	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 1st ` X ) - ( d / 2 ) ) (,) ( ( 1st ` X ) + ( d / 2 ) ) ) C_ RR )
82		mnfle 11105	. . . . . . . . . . . . . . . . . 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 11102	. . . . . . . . . . . . . . . . . 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 26014	. . . . . . . . . . . . . . . . . 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 3397	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) -> ( ( ( 2nd ` X ) - ( d / 2 ) ) (,) ( ( 2nd ` X ) + ( d / 2 ) ) ) C_ RR )
90		xpss12 4940	. . . . . . . . . . . . . . 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 3351	. . . . . . . . . . . . 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 994	. . . . . . . . . . . . . . . 16 |- ( ( A e. ( J tX J ) /\ ( X e. A /\ d e. RR+ /\ x e. ( RR X. RR ) ) ) -> X e. A )
98	96, 97	sseldd 3352	. . . . . . . . . . . . . . 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 1209	. . . . . . . . . . . . . 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 754	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> d e. RR+ )
102	101	rphalfcld 11031	. . . . . . . . . . . . . 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 26293	. . . . . . . . . . . . . 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 1218	. . . . . . . . . . . . 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 2438	. . . . . . . . . . . . . . . . . . . . 21 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
107	103, 6, 106	cnrehmeo 20500	. . . . . . . . . . . . . . . . . . . 20 |- G e. ( ( J tX J ) Homeo ( TopOpen ` ℂfld ) )
108	106	cnfldtopon 20337	. . . . . . . . . . . . . . . . . . . . . 22 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
109	108	toponunii 18512	. . . . . . . . . . . . . . . . . . . . 21 |- CC = U. ( TopOpen ` ℂfld )
110	12, 109	hmeof1o 19312	. . . . . . . . . . . . . . . . . . . 20 |- ( G e. ( ( J tX J ) Homeo ( TopOpen ` ℂfld ) ) -> G : ( RR X. RR ) -1-1-onto-> CC )
111		f1of 5636	. . . . . . . . . . . . . . . . . . . 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 5839	. . . . . . . . . . . . . . . . 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 5838	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( G ` x ) e. CC )
117		sqsscirc2 26291	. . . . . . . . . . . . . . . . 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 1217	. . . . . . . . . . . . . . . 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 11020	. . . . . . . . . . . . . . . . . 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 20327	. . . . . . . . . . . . . . . . 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 2438	. . . . . . . . . . . . . . . . . . 19 |- ( abs o. - ) = ( abs o. - )
128	127	cnmetdval 20325	. . . . . . . . . . . . . . . . . 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 4307	. . . . . . . . . . . . . . . 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 19942	. . . . . . . . . . . . . . . . 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 1217	. . . . . . . . . . . . . . 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 5648	. . . . . . . . . . . . . . 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 5638	. . . . . . . . . . . . . 14 |- ( `' G : CC -1-1-onto-> ( RR X. RR ) -> Fun `' G )
141	139, 140	ax-mp 5	. . . . . . . . . . . . 13 |- Fun `' G
142		f1odm 5640	. . . . . . . . . . . . . . 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 2529	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ( J tX J ) /\ X e. A ) /\ d e. RR+ ) /\ x e. ( RR X. RR ) ) -> ( G ` x ) e. dom `' G )
145		funfvima 5947	. . . . . . . . . . . . 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 5978	. . . . . . . . . . . . . . 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 2504	. . . . . . . . . . . . 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 3357	. . . . . . 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 2794	. . . . . 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 6187	. . . . . . . . . 10 |- Fun G
159	158	a1i 11	. . . . . . . . 9 |- ( ( A e. ( J tX J ) /\ X e. A ) -> Fun G )
160	13	sselda 3351	. . . . . . . . . 10 |- ( ( A e. ( J tX J ) /\ X e. A ) -> X e. ( RR X. RR ) )
161		f1odm 5640	. . . . . . . . . . 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 2529	. . . . . . . . 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 5947	. . . . . . . . . 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 1217	. . . . . . . 8 |- ( ( A e. ( J tX J ) /\ X e. A ) -> ( G ` X ) e. ( G " A ) )
168		hmeoima 19313	. . . . . . . . . . 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 20336	. . . . . . . . . . . . 13 |- ( TopOpen ` ℂfld ) = ( MetOpen ` ( abs o. - ) )
171	170	elmopn2 19995	. . . . . . . . . . . 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 6093	. . . . . . . . . . 11 |- ( m = ( G ` X ) -> ( m ( ball ` ( abs o. - ) ) d ) = ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) )
177	176	sseq1d 3378	. . . . . . . . . 10 |- ( m = ( G ` X ) -> ( ( m ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) <-> ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) ) )
178	177	rexbidv 2731	. . . . . . . . 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 3066	. . . . . . . 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 5199	. . . . . . . . . 10 |- ( ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) C_ ( G " A ) -> ( `' G " ( ( G ` X ) ( ball ` ( abs o. - ) ) d ) ) C_ ( `' G " ( G " A ) ) )
182		f1of1 5635	. . . . . . . . . . . . 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 5652	. . . . . . . . . . . 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 3379	. . . . . . . . . 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 2822	. . . . . . . 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 2852	. . . . . 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 3359	. . . . . 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 2818	. . . . 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 2852	. . . 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 2852	. . 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 2499	. . . . 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 3372	. . . . 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 3068	. . 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 2832	. 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 965   = wceq 1369   e. wcel 1756   A.wral 2710   E.wrex 2711   _Vcvv 2967   C_ wss 3323   ~Pcpw 3855   U.cuni 4086   class class class wbr 4287   X. cxp 4833   `'ccnv 4834   dom cdm 4835   ran crn 4836   "cima 4838   o. ccom 4839   Fun wfun 5407   Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086   e. cmpt2 6088   1stc1st 6570   2ndc2nd 6571   CCcc 9272   RRcr 9273   _ici 9276   + caddc 9277   x. cmul 9279   +oocpnf 9407   -oocmnf 9408   RR*cxr 9409   < clt 9410   <_ cle 9411   - cmin 9587   / cdiv 9985   2c2 10363   RR+crp 10983   (,)cioo 11292   Recre 12578   Imcim 12579   abscabs 12715   TopOpenctopn 14352   topGenctg 14368   *Metcxmt 17776   ballcbl 17778  ℂ_fldccnfld 17793   Topctop 18473   tX ctx 19108   Homeochmeo 19301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-icc 11299  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cn 18806  df-cnp 18807  df-tx 19110  df-hmeo 19303  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429

This theorem is referenced by:  dya2iocnei  26649