Theorem bcthlem4 22288

Description: Lemma for bcth 22290. Given any open ball ( C ( ball ` D ) R ) as starting point (and in particular, a ball in int ( U. ran M )), the limit point x of the centers of the induced sequence of balls g is outside U. ran M. Note that a set A has empty interior iff every nonempty open set U contains points outside A, i.e. ( U \ A ) =/= (/). (Contributed by Mario Carneiro, 7-Jan-2014.)

Hypotheses

Ref	Expression
bcth.2	|- J = ( MetOpen ` D )
bcthlem.4	|- ( ph -> D e. ( CMet ` X ) )
bcthlem.5	|- F = ( k e. NN , z e. ( X X. RR+ ) |-> { <. x , r >. | ( ( x e. X /\ r e. RR+ ) /\ ( r < ( 1 / k ) /\ ( ( cls ` J ) ` ( x ( ball ` D ) r ) ) C_ ( ( ( ball ` D ) ` z ) \ ( M ` k ) ) ) ) } )
bcthlem.6	|- ( ph -> M : NN --> ( Clsd ` J ) )
bcthlem.7	|- ( ph -> R e. RR+ )
bcthlem.8	|- ( ph -> C e. X )
bcthlem.9	|- ( ph -> g : NN --> ( X X. RR+ ) )
bcthlem.10	|- ( ph -> ( g ` 1 ) = <. C , R >. )
bcthlem.11	|- ( ph -> A. k e. NN ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) )

Assertion

Ref	Expression
bcthlem4	|- ( ph -> ( ( C ( ball ` D ) R ) \ U. ran M ) =/= (/) )

Distinct variable groups:   k, r, x, z   C, r, x   g, k, r, x, z, D   g, F, k, r, x, z   g, J, k, r, x, z   g, M, k, r, x, z   ph, k, r, x, z   x, R   g, X, k, r, x, z

Allowed substitution hints:   ph( g)   C( z, g, k)   R( z, g, k, r)

Proof of Theorem bcthlem4

Dummy variables n m y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bcthlem.4	. . . 4 |- ( ph -> D e. ( CMet ` X ) )
2		cmetmet 22249	. . . . . . 7 |- ( D e. ( CMet ` X ) -> D e. ( Met ` X ) )
3	1, 2	syl 17	. . . . . 6 |- ( ph -> D e. ( Met ` X ) )
4		metxmet 21342	. . . . . 6 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
5	3, 4	syl 17	. . . . 5 |- ( ph -> D e. ( *Met ` X ) )
6		bcthlem.9	. . . . 5 |- ( ph -> g : NN --> ( X X. RR+ ) )
7		bcth.2	. . . . . 6 |- J = ( MetOpen ` D )
8		bcthlem.5	. . . . . 6 |- F = ( k e. NN , z e. ( X X. RR+ ) |-> { <. x , r >. | ( ( x e. X /\ r e. RR+ ) /\ ( r < ( 1 / k ) /\ ( ( cls ` J ) ` ( x ( ball ` D ) r ) ) C_ ( ( ( ball ` D ) ` z ) \ ( M ` k ) ) ) ) } )
9		bcthlem.6	. . . . . 6 |- ( ph -> M : NN --> ( Clsd ` J ) )
10		bcthlem.7	. . . . . 6 |- ( ph -> R e. RR+ )
11		bcthlem.8	. . . . . 6 |- ( ph -> C e. X )
12		bcthlem.10	. . . . . 6 |- ( ph -> ( g ` 1 ) = <. C , R >. )
13		bcthlem.11	. . . . . 6 |- ( ph -> A. k e. NN ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) )
14	7, 1, 8, 9, 10, 11, 6, 12, 13	bcthlem2 22286	. . . . 5 |- ( ph -> A. n e. NN ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) )
15		elrp 11301	. . . . . . . . 9 |- ( r e. RR+ <-> ( r e. RR /\ 0 < r ) )
16		nnrecl 10864	. . . . . . . . 9 |- ( ( r e. RR /\ 0 < r ) -> E. m e. NN ( 1 / m ) < r )
17	15, 16	sylbi 199	. . . . . . . 8 |- ( r e. RR+ -> E. m e. NN ( 1 / m ) < r )
18	17	adantl 468	. . . . . . 7 |- ( ( ph /\ r e. RR+ ) -> E. m e. NN ( 1 / m ) < r )
19		peano2nn 10618	. . . . . . . . . 10 |- ( m e. NN -> ( m + 1 ) e. NN )
20	19	adantl 468	. . . . . . . . 9 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( m + 1 ) e. NN )
21		oveq1 6295	. . . . . . . . . . . . . . . . 17 |- ( k = m -> ( k + 1 ) = ( m + 1 ) )
22	21	fveq2d 5867	. . . . . . . . . . . . . . . 16 |- ( k = m -> ( g ` ( k + 1 ) ) = ( g ` ( m + 1 ) ) )
23		id 22	. . . . . . . . . . . . . . . . 17 |- ( k = m -> k = m )
24		fveq2 5863	. . . . . . . . . . . . . . . . 17 |- ( k = m -> ( g ` k ) = ( g ` m ) )
25	23, 24	oveq12d 6306	. . . . . . . . . . . . . . . 16 |- ( k = m -> ( k F ( g ` k ) ) = ( m F ( g ` m ) ) )
26	22, 25	eleq12d 2522	. . . . . . . . . . . . . . 15 |- ( k = m -> ( ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) <-> ( g ` ( m + 1 ) ) e. ( m F ( g ` m ) ) ) )
27	26	rspccva 3148	. . . . . . . . . . . . . 14 |- ( ( A. k e. NN ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) /\ m e. NN ) -> ( g ` ( m + 1 ) ) e. ( m F ( g ` m ) ) )
28	13, 27	sylan 474	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( g ` ( m + 1 ) ) e. ( m F ( g ` m ) ) )
29	6	ffvelrnda 6020	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( g ` m ) e. ( X X. RR+ ) )
30	7, 1, 8	bcthlem1 22285	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( m e. NN /\ ( g ` m ) e. ( X X. RR+ ) ) ) -> ( ( g ` ( m + 1 ) ) e. ( m F ( g ` m ) ) <-> ( ( g ` ( m + 1 ) ) e. ( X X. RR+ ) /\ ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) /\ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) ) ) )
31	30	expr 619	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( ( g ` m ) e. ( X X. RR+ ) -> ( ( g ` ( m + 1 ) ) e. ( m F ( g ` m ) ) <-> ( ( g ` ( m + 1 ) ) e. ( X X. RR+ ) /\ ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) /\ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) ) ) ) )
32	29, 31	mpd 15	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( ( g ` ( m + 1 ) ) e. ( m F ( g ` m ) ) <-> ( ( g ` ( m + 1 ) ) e. ( X X. RR+ ) /\ ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) /\ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) ) ) )
33	28, 32	mpbid 214	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> ( ( g ` ( m + 1 ) ) e. ( X X. RR+ ) /\ ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) /\ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) ) )
34	33	simp2d 1020	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) )
35	34	adantlr 720	. . . . . . . . . 10 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) )
36	33	simp1d 1019	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( g ` ( m + 1 ) ) e. ( X X. RR+ ) )
37		xp2nd 6821	. . . . . . . . . . . . . 14 |- ( ( g ` ( m + 1 ) ) e. ( X X. RR+ ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR+ )
38	36, 37	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR+ )
39	38	rpred 11338	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR )
40	39	adantlr 720	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR )
41		nnrecre 10643	. . . . . . . . . . . 12 |- ( m e. NN -> ( 1 / m ) e. RR )
42	41	adantl 468	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 1 / m ) e. RR )
43		rpre 11305	. . . . . . . . . . . 12 |- ( r e. RR+ -> r e. RR )
44	43	ad2antlr 732	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> r e. RR )
45		lttr 9707	. . . . . . . . . . 11 |- ( ( ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR /\ ( 1 / m ) e. RR /\ r e. RR ) -> ( ( ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) /\ ( 1 / m ) < r ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) )
46	40, 42, 44, 45	syl3anc 1267	. . . . . . . . . 10 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( ( ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) /\ ( 1 / m ) < r ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) )
47	35, 46	mpand 680	. . . . . . . . 9 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( ( 1 / m ) < r -> ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) )
48		fveq2 5863	. . . . . . . . . . . 12 |- ( n = ( m + 1 ) -> ( g ` n ) = ( g ` ( m + 1 ) ) )
49	48	fveq2d 5867	. . . . . . . . . . 11 |- ( n = ( m + 1 ) -> ( 2nd ` ( g ` n ) ) = ( 2nd ` ( g ` ( m + 1 ) ) ) )
50	49	breq1d 4411	. . . . . . . . . 10 |- ( n = ( m + 1 ) -> ( ( 2nd ` ( g ` n ) ) < r <-> ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) )
51	50	rspcev 3149	. . . . . . . . 9 |- ( ( ( m + 1 ) e. NN /\ ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) -> E. n e. NN ( 2nd ` ( g ` n ) ) < r )
52	20, 47, 51	syl6an 548	. . . . . . . 8 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( ( 1 / m ) < r -> E. n e. NN ( 2nd ` ( g ` n ) ) < r ) )
53	52	rexlimdva 2878	. . . . . . 7 |- ( ( ph /\ r e. RR+ ) -> ( E. m e. NN ( 1 / m ) < r -> E. n e. NN ( 2nd ` ( g ` n ) ) < r ) )
54	18, 53	mpd 15	. . . . . 6 |- ( ( ph /\ r e. RR+ ) -> E. n e. NN ( 2nd ` ( g ` n ) ) < r )
55	54	ralrimiva 2801	. . . . 5 |- ( ph -> A. r e. RR+ E. n e. NN ( 2nd ` ( g ` n ) ) < r )
56	5, 6, 14, 55	caubl 22270	. . . 4 |- ( ph -> ( 1st o. g ) e. ( Cau ` D ) )
57	7	cmetcau 22252	. . . 4 |- ( ( D e. ( CMet ` X ) /\ ( 1st o. g ) e. ( Cau ` D ) ) -> ( 1st o. g ) e. dom ( ~~> t ` J ) )
58	1, 56, 57	syl2anc 666	. . 3 |- ( ph -> ( 1st o. g ) e. dom ( ~~> t ` J ) )
59		fo1st 6810	. . . . . 6 |- 1st : _V -onto-> _V
60		fofun 5792	. . . . . 6 |- ( 1st : _V -onto-> _V -> Fun 1st )
61	59, 60	ax-mp 5	. . . . 5 |- Fun 1st
62		vex 3047	. . . . 5 |- g e. _V
63		cofunexg 6754	. . . . 5 |- ( ( Fun 1st /\ g e. _V ) -> ( 1st o. g ) e. _V )
64	61, 62, 63	mp2an 677	. . . 4 |- ( 1st o. g ) e. _V
65	64	eldm 5031	. . 3 |- ( ( 1st o. g ) e. dom ( ~~> t ` J ) <-> E. x ( 1st o. g ) ( ~~> t ` J ) x )
66	58, 65	sylib 200	. 2 |- ( ph -> E. x ( 1st o. g ) ( ~~> t ` J ) x )
67		1nn 10617	. . . . . 6 |- 1 e. NN
68	7, 1, 8, 9, 10, 11, 6, 12, 13	bcthlem3 22287	. . . . . 6 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ 1 e. NN ) -> x e. ( ( ball ` D ) ` ( g ` 1 ) ) )
69	67, 68	mp3an3 1352	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( ( ball ` D ) ` ( g ` 1 ) ) )
70	12	fveq2d 5867	. . . . . . 7 |- ( ph -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( ( ball ` D ) ` <. C , R >. ) )
71		df-ov 6291	. . . . . . 7 |- ( C ( ball ` D ) R ) = ( ( ball ` D ) ` <. C , R >. )
72	70, 71	syl6eqr 2502	. . . . . 6 |- ( ph -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( C ( ball ` D ) R ) )
73	72	adantr 467	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( C ( ball ` D ) R ) )
74	69, 73	eleqtrd 2530	. . . 4 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( C ( ball ` D ) R ) )
75	7	mopntop 21448	. . . . . . . . . . . . . 14 |- ( D e. ( *Met ` X ) -> J e. Top )
76	5, 75	syl 17	. . . . . . . . . . . . 13 |- ( ph -> J e. Top )
77	76	adantr 467	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> J e. Top )
78	5	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> D e. ( *Met ` X ) )
79		xp1st 6820	. . . . . . . . . . . . . . 15 |- ( ( g ` ( m + 1 ) ) e. ( X X. RR+ ) -> ( 1st ` ( g ` ( m + 1 ) ) ) e. X )
80	36, 79	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( 1st ` ( g ` ( m + 1 ) ) ) e. X )
81	38	rpxrd 11339	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR* )
82		blssm 21426	. . . . . . . . . . . . . 14 |- ( ( D e. ( *Met ` X ) /\ ( 1st ` ( g ` ( m + 1 ) ) ) e. X /\ ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR* ) -> ( ( 1st ` ( g ` ( m + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( m + 1 ) ) ) ) C_ X )
83	78, 80, 81, 82	syl3anc 1267	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( ( 1st ` ( g ` ( m + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( m + 1 ) ) ) ) C_ X )
84		1st2nd2 6827	. . . . . . . . . . . . . . . 16 |- ( ( g ` ( m + 1 ) ) e. ( X X. RR+ ) -> ( g ` ( m + 1 ) ) = <. ( 1st ` ( g ` ( m + 1 ) ) ) , ( 2nd ` ( g ` ( m + 1 ) ) ) >. )
85	36, 84	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ph /\ m e. NN ) -> ( g ` ( m + 1 ) ) = <. ( 1st ` ( g ` ( m + 1 ) ) ) , ( 2nd ` ( g ` ( m + 1 ) ) ) >. )
86	85	fveq2d 5867	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) = ( ( ball ` D ) ` <. ( 1st ` ( g ` ( m + 1 ) ) ) , ( 2nd ` ( g ` ( m + 1 ) ) ) >. ) )
87		df-ov 6291	. . . . . . . . . . . . . 14 |- ( ( 1st ` ( g ` ( m + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( m + 1 ) ) ) ) = ( ( ball ` D ) ` <. ( 1st ` ( g ` ( m + 1 ) ) ) , ( 2nd ` ( g ` ( m + 1 ) ) ) >. )
88	86, 87	syl6reqr 2503	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( ( 1st ` ( g ` ( m + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( m + 1 ) ) ) ) = ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
89	7	mopnuni 21449	. . . . . . . . . . . . . . 15 |- ( D e. ( *Met ` X ) -> X = U. J )
90	5, 89	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> X = U. J )
91	90	adantr 467	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> X = U. J )
92	83, 88, 91	3sstr3d 3473	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ U. J )
93		eqid 2450	. . . . . . . . . . . . 13 |- U. J = U. J
94	93	sscls 20064	. . . . . . . . . . . 12 |- ( ( J e. Top /\ ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ U. J ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) )
95	77, 92, 94	syl2anc 666	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) )
96	33	simp3d 1021	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
97	95, 96	sstrd 3441	. . . . . . . . . 10 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
98	97	3adant2 1026	. . . . . . . . 9 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
99	7, 1, 8, 9, 10, 11, 6, 12, 13	bcthlem3 22287	. . . . . . . . . 10 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ ( m + 1 ) e. NN ) -> x e. ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
100	19, 99	syl3an3 1302	. . . . . . . . 9 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> x e. ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
101	98, 100	sseldd 3432	. . . . . . . 8 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> x e. ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
102	101	eldifbd 3416	. . . . . . 7 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> -. x e. ( M ` m ) )
103	102	3expa 1207	. . . . . 6 |- ( ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) /\ m e. NN ) -> -. x e. ( M ` m ) )
104	103	ralrimiva 2801	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> A. m e. NN -. x e. ( M ` m ) )
105		eluni2 4201	. . . . . . . . 9 |- ( x e. U. ran M <-> E. y e. ran M x e. y )
106		ffn 5726	. . . . . . . . . . 11 |- ( M : NN --> ( Clsd ` J ) -> M Fn NN )
107	9, 106	syl 17	. . . . . . . . . 10 |- ( ph -> M Fn NN )
108		eleq2 2517	. . . . . . . . . . 11 |- ( y = ( M ` m ) -> ( x e. y <-> x e. ( M ` m ) ) )
109	108	rexrn 6022	. . . . . . . . . 10 |- ( M Fn NN -> ( E. y e. ran M x e. y <-> E. m e. NN x e. ( M ` m ) ) )
110	107, 109	syl 17	. . . . . . . . 9 |- ( ph -> ( E. y e. ran M x e. y <-> E. m e. NN x e. ( M ` m ) ) )
111	105, 110	syl5bb 261	. . . . . . . 8 |- ( ph -> ( x e. U. ran M <-> E. m e. NN x e. ( M ` m ) ) )
112	111	notbid 296	. . . . . . 7 |- ( ph -> ( -. x e. U. ran M <-> -. E. m e. NN x e. ( M ` m ) ) )
113		ralnex 2833	. . . . . . 7 |- ( A. m e. NN -. x e. ( M ` m ) <-> -. E. m e. NN x e. ( M ` m ) )
114	112, 113	syl6bbr 267	. . . . . 6 |- ( ph -> ( -. x e. U. ran M <-> A. m e. NN -. x e. ( M ` m ) ) )
115	114	biimpar 488	. . . . 5 |- ( ( ph /\ A. m e. NN -. x e. ( M ` m ) ) -> -. x e. U. ran M )
116	104, 115	syldan 473	. . . 4 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> -. x e. U. ran M )
117	74, 116	eldifd 3414	. . 3 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( ( C ( ball ` D ) R ) \ U. ran M ) )
118		ne0i 3736	. . 3 |- ( x e. ( ( C ( ball ` D ) R ) \ U. ran M ) -> ( ( C ( ball ` D ) R ) \ U. ran M ) =/= (/) )
119	117, 118	syl 17	. 2 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> ( ( C ( ball ` D ) R ) \ U. ran M ) =/= (/) )
120	66, 119	exlimddv 1780	1 |- ( ph -> ( ( C ( ball ` D ) R ) \ U. ran M ) =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   = wceq 1443   E.wex 1662   e. wcel 1886   =/= wne 2621   A.wral 2736   E.wrex 2737   _Vcvv 3044   \ cdif 3400   C_ wss 3403   (/)c0 3730   <.cop 3973   U.cuni 4197   class class class wbr 4401   {copab 4459   X. cxp 4831   dom cdm 4833   ran crn 4834   o. ccom 4837   Fun wfun 5575   Fn wfn 5576   -->wf 5577   -onto->wfo 5579   ` cfv 5581  (class class class)co 6288   |-> cmpt2 6290   1stc1st 6788   2ndc2nd 6789   RRcr 9535   0cc0 9536   1c1 9537   + caddc 9539   RR*cxr 9671   < clt 9672   / cdiv 10266   NNcn 10606   RR+crp 11299   *Metcxmt 18948   Metcme 18949   ballcbl 18950   MetOpencmopn 18953   Topctop 19910   Clsdccld 20024   clsccl 20026   ~~> tclm 20235   Caucca 22216   CMetcms 22217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-sup 7953  df-inf 7954  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ico 11638  df-rest 15314  df-topgen 15335  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-top 19914  df-bases 19915  df-topon 19916  df-cld 20027  df-ntr 20028  df-cls 20029  df-nei 20107  df-lm 20238  df-fil 20854  df-fm 20946  df-flim 20947  df-flf 20948  df-cfil 22218  df-cau 22219  df-cmet 22220

This theorem is referenced by:  bcthlem5  22289