Theorem limcimolemlt 13386

Description: Lemma for limcimo 13387. (Contributed by Jim Kingdon, 3-Jul-2023.)

Hypotheses

Ref	Expression
limcflf.f	|- ( ph -> F : A --> CC )
limcflf.a	|- ( ph -> A C_ CC )
limcimo.b	|- ( ph -> B e. CC )
limcimo.bc	|- ( ph -> B e. C )
limcimo.bs	|- ( ph -> B e. S )
limcimo.c	|- ( ph -> C e. ( K ↾t S ) )
limcimo.s	|- ( ph -> S e. { RR , CC } )
limcimo.ca	|- ( ph -> { q e. C | q # B } C_ A )
limcflfcntop.k	|- K = ( MetOpen ` ( abs o. - ) )
limcimo.d	|- ( ph -> D e. RR+ )
limcimo.x	|- ( ph -> X e. ( F lim CC B ) )
limcimo.y	|- ( ph -> Y e. ( F lim CC B ) )
limcimo.z	|- ( ph -> A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < D ) -> ( abs ` ( ( F ` z ) - X ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) )
limcimo.g	|- ( ph -> G e. RR+ )
limcimo.w	|- ( ph -> A. w e. A ( ( w # B /\ ( abs ` ( w - B ) ) < G ) -> ( abs ` ( ( F ` w ) - Y ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) )

Assertion

Ref	Expression
limcimolemlt	|- ( ph -> ( abs ` ( X - Y ) ) < ( abs ` ( X - Y ) ) )

Distinct variable groups:   w, A   z, A   B, q   w, B   z, B   C, q   z, D   w, F   z, F   w, G   w, X   z, X   w, Y   z, Y

Allowed substitution hints:   ph( z, w, q)   A( q)   C( z, w)   D( w, q)   S( z, w, q)   F( q)   G( z, q)   K( z, w, q)   X( q)   Y( q)

Proof of Theorem limcimolemlt

Dummy variables a b c r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnxmet 13284	. . . 4 |- ( abs o. - ) e. ( *Met ` CC )
2		ax-resscn 7853	. . . . . . 7 |- RR C_ CC
3		sseq1 3170	. . . . . . 7 |- ( S = RR -> ( S C_ CC <-> RR C_ CC ) )
4	2, 3	mpbiri 167	. . . . . 6 |- ( S = RR -> S C_ CC )
5	4	adantl 275	. . . . 5 |- ( ( ph /\ S = RR ) -> S C_ CC )
6		eqimss 3201	. . . . . 6 |- ( S = CC -> S C_ CC )
7	6	adantl 275	. . . . 5 |- ( ( ph /\ S = CC ) -> S C_ CC )
8		limcimo.s	. . . . . 6 |- ( ph -> S e. { RR , CC } )
9		elpri 3604	. . . . . 6 |- ( S e. { RR , CC } -> ( S = RR \/ S = CC ) )
10	8, 9	syl 14	. . . . 5 |- ( ph -> ( S = RR \/ S = CC ) )
11	5, 7, 10	mpjaodan 793	. . . 4 |- ( ph -> S C_ CC )
12		xmetres2 13132	. . . 4 |- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ S C_ CC ) -> ( ( abs o. - ) |` ( S X. S ) ) e. ( *Met ` S ) )
13	1, 11, 12	sylancr 412	. . 3 |- ( ph -> ( ( abs o. - ) |` ( S X. S ) ) e. ( *Met ` S ) )
14		limcimo.c	. . . 4 |- ( ph -> C e. ( K ↾t S ) )
15		eqid 2170	. . . . . 6 |- ( ( abs o. - ) |` ( S X. S ) ) = ( ( abs o. - ) |` ( S X. S ) )
16		limcflfcntop.k	. . . . . 6 |- K = ( MetOpen ` ( abs o. - ) )
17		eqid 2170	. . . . . 6 |- ( MetOpen ` ( ( abs o. - ) |` ( S X. S ) ) ) = ( MetOpen ` ( ( abs o. - ) |` ( S X. S ) ) )
18	15, 16, 17	metrest 13259	. . . . 5 |- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ S C_ CC ) -> ( K ↾t S ) = ( MetOpen ` ( ( abs o. - ) |` ( S X. S ) ) ) )
19	1, 11, 18	sylancr 412	. . . 4 |- ( ph -> ( K ↾t S ) = ( MetOpen ` ( ( abs o. - ) |` ( S X. S ) ) ) )
20	14, 19	eleqtrd 2249	. . 3 |- ( ph -> C e. ( MetOpen ` ( ( abs o. - ) |` ( S X. S ) ) ) )
21		limcimo.bc	. . 3 |- ( ph -> B e. C )
22		limcimo.d	. . . 4 |- ( ph -> D e. RR+ )
23		limcimo.g	. . . 4 |- ( ph -> G e. RR+ )
24		rpmincl 11188	. . . 4 |- ( ( D e. RR+ /\ G e. RR+ ) -> inf ( { D , G } , RR , < ) e. RR+ )
25	22, 23, 24	syl2anc 409	. . 3 |- ( ph -> inf ( { D , G } , RR , < ) e. RR+ )
26	17	mopni3 13237	. . 3 |- ( ( ( ( ( abs o. - ) |` ( S X. S ) ) e. ( *Met ` S ) /\ C e. ( MetOpen ` ( ( abs o. - ) |` ( S X. S ) ) ) /\ B e. C ) /\ inf ( { D , G } , RR , < ) e. RR+ ) -> E. r e. RR+ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) )
27	13, 20, 21, 25, 26	syl31anc 1236	. 2 |- ( ph -> E. r e. RR+ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) )
28		limcimo.x	. . . . . 6 |- ( ph -> X e. ( F lim CC B ) )
29		limcrcl 13380	. . . . . . . . 9 |- ( X e. ( F lim CC B ) -> ( F : dom F --> CC /\ dom F C_ CC /\ B e. CC ) )
30	28, 29	syl 14	. . . . . . . 8 |- ( ph -> ( F : dom F --> CC /\ dom F C_ CC /\ B e. CC ) )
31	30	simp1d 1004	. . . . . . 7 |- ( ph -> F : dom F --> CC )
32	30	simp2d 1005	. . . . . . 7 |- ( ph -> dom F C_ CC )
33		limcimo.b	. . . . . . 7 |- ( ph -> B e. CC )
34	31, 32, 33	ellimc3ap 13383	. . . . . 6 |- ( ph -> ( X e. ( F lim CC B ) <-> ( X e. CC /\ A. a e. RR+ E. b e. RR+ A. c e. dom F ( ( c # B /\ ( abs ` ( c - B ) ) < b ) -> ( abs ` ( ( F ` c ) - X ) ) < a ) ) ) )
35	28, 34	mpbid 146	. . . . 5 |- ( ph -> ( X e. CC /\ A. a e. RR+ E. b e. RR+ A. c e. dom F ( ( c # B /\ ( abs ` ( c - B ) ) < b ) -> ( abs ` ( ( F ` c ) - X ) ) < a ) ) )
36	35	simpld 111	. . . 4 |- ( ph -> X e. CC )
37	36	adantr 274	. . 3 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> X e. CC )
38		limcimo.y	. . . . . 6 |- ( ph -> Y e. ( F lim CC B ) )
39	31, 32, 33	ellimc3ap 13383	. . . . . 6 |- ( ph -> ( Y e. ( F lim CC B ) <-> ( Y e. CC /\ A. a e. RR+ E. b e. RR+ A. c e. dom F ( ( c # B /\ ( abs ` ( c - B ) ) < b ) -> ( abs ` ( ( F ` c ) - Y ) ) < a ) ) ) )
40	38, 39	mpbid 146	. . . . 5 |- ( ph -> ( Y e. CC /\ A. a e. RR+ E. b e. RR+ A. c e. dom F ( ( c # B /\ ( abs ` ( c - B ) ) < b ) -> ( abs ` ( ( F ` c ) - Y ) ) < a ) ) )
41	40	simpld 111	. . . 4 |- ( ph -> Y e. CC )
42	41	adantr 274	. . 3 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> Y e. CC )
43		limcflf.f	. . . . 5 |- ( ph -> F : A --> CC )
44	43	adantr 274	. . . 4 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> F : A --> CC )
45		breq1 3990	. . . . . 6 |- ( q = ( B + ( r / 2 ) ) -> ( q # B <-> ( B + ( r / 2 ) ) # B ) )
46		simprrr 535	. . . . . . 7 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C )
47		limcimo.bs	. . . . . . . . . . . 12 |- ( ph -> B e. S )
48	47	adantr 274	. . . . . . . . . . 11 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> B e. S )
49	47	ad2antrr 485	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) /\ S = RR ) -> B e. S )
50		simpr 109	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) /\ S = RR ) -> S = RR )
51	49, 50	eleqtrd 2249	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) /\ S = RR ) -> B e. RR )
52		simprl 526	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> r e. RR+ )
53	52	rphalfcld 9653	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( r / 2 ) e. RR+ )
54	53	adantr 274	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) /\ S = RR ) -> ( r / 2 ) e. RR+ )
55	54	rpred 9640	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) /\ S = RR ) -> ( r / 2 ) e. RR )
56	51, 55	readdcld 7936	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) /\ S = RR ) -> ( B + ( r / 2 ) ) e. RR )
57	56, 50	eleqtrrd 2250	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) /\ S = RR ) -> ( B + ( r / 2 ) ) e. S )
58	33	ad2antrr 485	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) /\ S = CC ) -> B e. CC )
59	53	adantr 274	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) /\ S = CC ) -> ( r / 2 ) e. RR+ )
60	59	rpcnd 9642	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) /\ S = CC ) -> ( r / 2 ) e. CC )
61	58, 60	addcld 7926	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) /\ S = CC ) -> ( B + ( r / 2 ) ) e. CC )
62		simpr 109	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) /\ S = CC ) -> S = CC )
63	61, 62	eleqtrrd 2250	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) /\ S = CC ) -> ( B + ( r / 2 ) ) e. S )
64	10	adantr 274	. . . . . . . . . . . 12 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( S = RR \/ S = CC ) )
65	57, 63, 64	mpjaodan 793	. . . . . . . . . . 11 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( B + ( r / 2 ) ) e. S )
66	48, 65	ovresd 5990	. . . . . . . . . 10 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( B ( ( abs o. - ) |` ( S X. S ) ) ( B + ( r / 2 ) ) ) = ( B ( abs o. - ) ( B + ( r / 2 ) ) ) )
67	33	adantr 274	. . . . . . . . . . 11 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> B e. CC )
68	53	rpcnd 9642	. . . . . . . . . . . 12 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( r / 2 ) e. CC )
69	67, 68	addcld 7926	. . . . . . . . . . 11 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( B + ( r / 2 ) ) e. CC )
70		eqid 2170	. . . . . . . . . . . 12 |- ( abs o. - ) = ( abs o. - )
71	70	cnmetdval 13282	. . . . . . . . . . 11 |- ( ( B e. CC /\ ( B + ( r / 2 ) ) e. CC ) -> ( B ( abs o. - ) ( B + ( r / 2 ) ) ) = ( abs ` ( B - ( B + ( r / 2 ) ) ) ) )
72	67, 69, 71	syl2anc 409	. . . . . . . . . 10 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( B ( abs o. - ) ( B + ( r / 2 ) ) ) = ( abs ` ( B - ( B + ( r / 2 ) ) ) ) )
73	67, 67, 68	subsub4d 8248	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( ( B - B ) - ( r / 2 ) ) = ( B - ( B + ( r / 2 ) ) ) )
74	67	subidd 8205	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( B - B ) = 0 )
75	74	oveq1d 5865	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( ( B - B ) - ( r / 2 ) ) = ( 0 - ( r / 2 ) ) )
76	73, 75	eqtr3d 2205	. . . . . . . . . . . . 13 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( B - ( B + ( r / 2 ) ) ) = ( 0 - ( r / 2 ) ) )
77	76	fveq2d 5498	. . . . . . . . . . . 12 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( B - ( B + ( r / 2 ) ) ) ) = ( abs ` ( 0 - ( r / 2 ) ) ) )
78		0cnd 7900	. . . . . . . . . . . . 13 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> 0 e. CC )
79	78, 68	abssubd 11144	. . . . . . . . . . . 12 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( 0 - ( r / 2 ) ) ) = ( abs ` ( ( r / 2 ) - 0 ) ) )
80	77, 79	eqtrd 2203	. . . . . . . . . . 11 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( B - ( B + ( r / 2 ) ) ) ) = ( abs ` ( ( r / 2 ) - 0 ) ) )
81	68	subid1d 8206	. . . . . . . . . . . 12 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( ( r / 2 ) - 0 ) = ( r / 2 ) )
82	81	fveq2d 5498	. . . . . . . . . . 11 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( ( r / 2 ) - 0 ) ) = ( abs ` ( r / 2 ) ) )
83	53	rpred 9640	. . . . . . . . . . . 12 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( r / 2 ) e. RR )
84	53	rpge0d 9644	. . . . . . . . . . . 12 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> 0 <_ ( r / 2 ) )
85	83, 84	absidd 11118	. . . . . . . . . . 11 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( r / 2 ) ) = ( r / 2 ) )
86	80, 82, 85	3eqtrd 2207	. . . . . . . . . 10 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( B - ( B + ( r / 2 ) ) ) ) = ( r / 2 ) )
87	66, 72, 86	3eqtrd 2207	. . . . . . . . 9 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( B ( ( abs o. - ) |` ( S X. S ) ) ( B + ( r / 2 ) ) ) = ( r / 2 ) )
88		rphalflt 9627	. . . . . . . . . 10 |- ( r e. RR+ -> ( r / 2 ) < r )
89	88	ad2antrl 487	. . . . . . . . 9 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( r / 2 ) < r )
90	87, 89	eqbrtrd 4009	. . . . . . . 8 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( B ( ( abs o. - ) |` ( S X. S ) ) ( B + ( r / 2 ) ) ) < r )
91	13	adantr 274	. . . . . . . . 9 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( ( abs o. - ) |` ( S X. S ) ) e. ( *Met ` S ) )
92		rpxr 9605	. . . . . . . . . 10 |- ( r e. RR+ -> r e. RR* )
93	92	ad2antrl 487	. . . . . . . . 9 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> r e. RR* )
94		elbl2 13146	. . . . . . . . 9 |- ( ( ( ( ( abs o. - ) |` ( S X. S ) ) e. ( *Met ` S ) /\ r e. RR* ) /\ ( B e. S /\ ( B + ( r / 2 ) ) e. S ) ) -> ( ( B + ( r / 2 ) ) e. ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) <-> ( B ( ( abs o. - ) |` ( S X. S ) ) ( B + ( r / 2 ) ) ) < r ) )
95	91, 93, 48, 65, 94	syl22anc 1234	. . . . . . . 8 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( ( B + ( r / 2 ) ) e. ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) <-> ( B ( ( abs o. - ) |` ( S X. S ) ) ( B + ( r / 2 ) ) ) < r ) )
96	90, 95	mpbird 166	. . . . . . 7 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( B + ( r / 2 ) ) e. ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) )
97	46, 96	sseldd 3148	. . . . . 6 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( B + ( r / 2 ) ) e. C )
98	53	rpap0d 9646	. . . . . . . 8 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( r / 2 ) # 0 )
99	67, 67	negsubdid 8232	. . . . . . . . . . 11 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> -u ( B - B ) = ( -u B + B ) )
100	74	negeqd 8101	. . . . . . . . . . . 12 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> -u ( B - B ) = -u 0 )
101		neg0 8152	. . . . . . . . . . . 12 |- -u 0 = 0
102	100, 101	eqtrdi 2219	. . . . . . . . . . 11 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> -u ( B - B ) = 0 )
103	99, 102	eqtr3d 2205	. . . . . . . . . 10 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( -u B + B ) = 0 )
104	103	oveq1d 5865	. . . . . . . . 9 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( ( -u B + B ) + ( r / 2 ) ) = ( 0 + ( r / 2 ) ) )
105	67	negcld 8204	. . . . . . . . . 10 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> -u B e. CC )
106	105, 67, 68	addassd 7929	. . . . . . . . 9 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( ( -u B + B ) + ( r / 2 ) ) = ( -u B + ( B + ( r / 2 ) ) ) )
107	68	addid2d 8056	. . . . . . . . 9 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( 0 + ( r / 2 ) ) = ( r / 2 ) )
108	104, 106, 107	3eqtr3d 2211	. . . . . . . 8 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( -u B + ( B + ( r / 2 ) ) ) = ( r / 2 ) )
109	98, 108, 103	3brtr4d 4019	. . . . . . 7 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( -u B + ( B + ( r / 2 ) ) ) # ( -u B + B ) )
110		apadd2 8515	. . . . . . . 8 |- ( ( ( B + ( r / 2 ) ) e. CC /\ B e. CC /\ -u B e. CC ) -> ( ( B + ( r / 2 ) ) # B <-> ( -u B + ( B + ( r / 2 ) ) ) # ( -u B + B ) ) )
111	69, 67, 105, 110	syl3anc 1233	. . . . . . 7 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( ( B + ( r / 2 ) ) # B <-> ( -u B + ( B + ( r / 2 ) ) ) # ( -u B + B ) ) )
112	109, 111	mpbird 166	. . . . . 6 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( B + ( r / 2 ) ) # B )
113	45, 97, 112	elrabd 2888	. . . . 5 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( B + ( r / 2 ) ) e. { q e. C | q # B } )
114		limcimo.ca	. . . . . . 7 |- ( ph -> { q e. C | q # B } C_ A )
115	114	sseld 3146	. . . . . 6 |- ( ph -> ( ( B + ( r / 2 ) ) e. { q e. C | q # B } -> ( B + ( r / 2 ) ) e. A ) )
116	115	adantr 274	. . . . 5 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( ( B + ( r / 2 ) ) e. { q e. C | q # B } -> ( B + ( r / 2 ) ) e. A ) )
117	113, 116	mpd 13	. . . 4 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( B + ( r / 2 ) ) e. A )
118	44, 117	ffvelrnd 5629	. . 3 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( F ` ( B + ( r / 2 ) ) ) e. CC )
119	37, 42	subcld 8217	. . . 4 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( X - Y ) e. CC )
120	119	abscld 11132	. . 3 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( X - Y ) ) e. RR )
121	37, 118	abssubd 11144	. . . 4 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( X - ( F ` ( B + ( r / 2 ) ) ) ) ) = ( abs ` ( ( F ` ( B + ( r / 2 ) ) ) - X ) ) )
122	69, 67	subcld 8217	. . . . . . 7 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( ( B + ( r / 2 ) ) - B ) e. CC )
123	122	abscld 11132	. . . . . 6 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( ( B + ( r / 2 ) ) - B ) ) e. RR )
124	52	rpred 9640	. . . . . 6 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> r e. RR )
125	22	rpred 9640	. . . . . . 7 |- ( ph -> D e. RR )
126	125	adantr 274	. . . . . 6 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> D e. RR )
127	67, 68	pncan2d 8219	. . . . . . . . 9 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( ( B + ( r / 2 ) ) - B ) = ( r / 2 ) )
128	127	fveq2d 5498	. . . . . . . 8 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( ( B + ( r / 2 ) ) - B ) ) = ( abs ` ( r / 2 ) ) )
129	128, 85	eqtrd 2203	. . . . . . 7 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( ( B + ( r / 2 ) ) - B ) ) = ( r / 2 ) )
130	129, 89	eqbrtrd 4009	. . . . . 6 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( ( B + ( r / 2 ) ) - B ) ) < r )
131	23	rpred 9640	. . . . . . . . 9 |- ( ph -> G e. RR )
132	131	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> G e. RR )
133		mincl 11181	. . . . . . . 8 |- ( ( D e. RR /\ G e. RR ) -> inf ( { D , G } , RR , < ) e. RR )
134	126, 132, 133	syl2anc 409	. . . . . . 7 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> inf ( { D , G } , RR , < ) e. RR )
135		simprrl 534	. . . . . . 7 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> r < inf ( { D , G } , RR , < ) )
136		min1inf 11182	. . . . . . . 8 |- ( ( D e. RR /\ G e. RR ) -> inf ( { D , G } , RR , < ) <_ D )
137	126, 132, 136	syl2anc 409	. . . . . . 7 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> inf ( { D , G } , RR , < ) <_ D )
138	124, 134, 126, 135, 137	ltletrd 8329	. . . . . 6 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> r < D )
139	123, 124, 126, 130, 138	lttrd 8032	. . . . 5 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( ( B + ( r / 2 ) ) - B ) ) < D )
140		breq1 3990	. . . . . . . 8 |- ( z = ( B + ( r / 2 ) ) -> ( z # B <-> ( B + ( r / 2 ) ) # B ) )
141		fvoveq1 5873	. . . . . . . . 9 |- ( z = ( B + ( r / 2 ) ) -> ( abs ` ( z - B ) ) = ( abs ` ( ( B + ( r / 2 ) ) - B ) ) )
142	141	breq1d 3997	. . . . . . . 8 |- ( z = ( B + ( r / 2 ) ) -> ( ( abs ` ( z - B ) ) < D <-> ( abs ` ( ( B + ( r / 2 ) ) - B ) ) < D ) )
143	140, 142	anbi12d 470	. . . . . . 7 |- ( z = ( B + ( r / 2 ) ) -> ( ( z # B /\ ( abs ` ( z - B ) ) < D ) <-> ( ( B + ( r / 2 ) ) # B /\ ( abs ` ( ( B + ( r / 2 ) ) - B ) ) < D ) ) )
144	143	imbrov2fvoveq 5875	. . . . . 6 |- ( z = ( B + ( r / 2 ) ) -> ( ( ( z # B /\ ( abs ` ( z - B ) ) < D ) -> ( abs ` ( ( F ` z ) - X ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) <-> ( ( ( B + ( r / 2 ) ) # B /\ ( abs ` ( ( B + ( r / 2 ) ) - B ) ) < D ) -> ( abs ` ( ( F ` ( B + ( r / 2 ) ) ) - X ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) ) )
145		limcimo.z	. . . . . . 7 |- ( ph -> A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < D ) -> ( abs ` ( ( F ` z ) - X ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) )
146	145	adantr 274	. . . . . 6 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < D ) -> ( abs ` ( ( F ` z ) - X ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) )
147	144, 146, 117	rspcdva 2839	. . . . 5 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( ( ( B + ( r / 2 ) ) # B /\ ( abs ` ( ( B + ( r / 2 ) ) - B ) ) < D ) -> ( abs ` ( ( F ` ( B + ( r / 2 ) ) ) - X ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) )
148	112, 139, 147	mp2and 431	. . . 4 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( ( F ` ( B + ( r / 2 ) ) ) - X ) ) < ( ( abs ` ( X - Y ) ) / 2 ) )
149	121, 148	eqbrtrd 4009	. . 3 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( X - ( F ` ( B + ( r / 2 ) ) ) ) ) < ( ( abs ` ( X - Y ) ) / 2 ) )
150		min2inf 11183	. . . . . . 7 |- ( ( D e. RR /\ G e. RR ) -> inf ( { D , G } , RR , < ) <_ G )
151	126, 132, 150	syl2anc 409	. . . . . 6 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> inf ( { D , G } , RR , < ) <_ G )
152	124, 134, 132, 135, 151	ltletrd 8329	. . . . 5 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> r < G )
153	123, 124, 132, 130, 152	lttrd 8032	. . . 4 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( ( B + ( r / 2 ) ) - B ) ) < G )
154		breq1 3990	. . . . . . 7 |- ( w = ( B + ( r / 2 ) ) -> ( w # B <-> ( B + ( r / 2 ) ) # B ) )
155		fvoveq1 5873	. . . . . . . 8 |- ( w = ( B + ( r / 2 ) ) -> ( abs ` ( w - B ) ) = ( abs ` ( ( B + ( r / 2 ) ) - B ) ) )
156	155	breq1d 3997	. . . . . . 7 |- ( w = ( B + ( r / 2 ) ) -> ( ( abs ` ( w - B ) ) < G <-> ( abs ` ( ( B + ( r / 2 ) ) - B ) ) < G ) )
157	154, 156	anbi12d 470	. . . . . 6 |- ( w = ( B + ( r / 2 ) ) -> ( ( w # B /\ ( abs ` ( w - B ) ) < G ) <-> ( ( B + ( r / 2 ) ) # B /\ ( abs ` ( ( B + ( r / 2 ) ) - B ) ) < G ) ) )
158	157	imbrov2fvoveq 5875	. . . . 5 |- ( w = ( B + ( r / 2 ) ) -> ( ( ( w # B /\ ( abs ` ( w - B ) ) < G ) -> ( abs ` ( ( F ` w ) - Y ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) <-> ( ( ( B + ( r / 2 ) ) # B /\ ( abs ` ( ( B + ( r / 2 ) ) - B ) ) < G ) -> ( abs ` ( ( F ` ( B + ( r / 2 ) ) ) - Y ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) ) )
159		limcimo.w	. . . . . 6 |- ( ph -> A. w e. A ( ( w # B /\ ( abs ` ( w - B ) ) < G ) -> ( abs ` ( ( F ` w ) - Y ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) )
160	159	adantr 274	. . . . 5 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> A. w e. A ( ( w # B /\ ( abs ` ( w - B ) ) < G ) -> ( abs ` ( ( F ` w ) - Y ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) )
161	158, 160, 117	rspcdva 2839	. . . 4 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( ( ( B + ( r / 2 ) ) # B /\ ( abs ` ( ( B + ( r / 2 ) ) - B ) ) < G ) -> ( abs ` ( ( F ` ( B + ( r / 2 ) ) ) - Y ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) )
162	112, 153, 161	mp2and 431	. . 3 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( ( F ` ( B + ( r / 2 ) ) ) - Y ) ) < ( ( abs ` ( X - Y ) ) / 2 ) )
163	37, 42, 118, 120, 149, 162	abs3lemd 11152	. 2 |- ( ( ph /\ ( r e. RR+ /\ ( r < inf ( { D , G } , RR , < ) /\ ( B ( ball ` ( ( abs o. - ) |` ( S X. S ) ) ) r ) C_ C ) ) ) -> ( abs ` ( X - Y ) ) < ( abs ` ( X - Y ) ) )
164	27, 163	rexlimddv 2592	1 |- ( ph -> ( abs ` ( X - Y ) ) < ( abs ` ( X - Y ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   C_ wss 3121   {cpr 3582   class class class wbr 3987   X. cxp 4607   dom cdm 4609   |` cres 4611   o. ccom 4613   -->wf 5192   ` cfv 5196  (class class class)co 5850  infcinf 6956   CCcc 7759   RRcr 7760   0cc0 7761   + caddc 7764   RR*cxr 7940   < clt 7941   <_ cle 7942   - cmin 8077   -ucneg 8078   # cap 8487   / cdiv 8576   2c2 8916   RR+crp 9597   abscabs 10948   ↾_t crest 12566   *Metcxmet 12733   ballcbl 12735   MetOpencmopn 12738   lim CC climc 13376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880  ax-caucvg 7881

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-map 6624  df-pm 6625  df-sup 6957  df-inf 6958  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-xneg 9716  df-xadd 9717  df-seqfrec 10389  df-exp 10463  df-cj 10793  df-re 10794  df-im 10795  df-rsqrt 10949  df-abs 10950  df-rest 12568  df-topgen 12587  df-psmet 12740  df-xmet 12741  df-met 12742  df-bl 12743  df-mopn 12744  df-top 12749  df-topon 12762  df-bases 12794  df-limced 13378

This theorem is referenced by:  limcimo  13387