Theorem bcthlem4 23124

Description: Lemma for bcth 23126. 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 23084	. . . . . . 7 |- ( D e. ( CMet ` X ) -> D e. ( Met ` X ) )
3	1, 2	syl 17	. . . . . 6 |- ( ph -> D e. ( Met ` X ) )
4		metxmet 22139	. . . . . 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 23122	. . . . 5 |- ( ph -> A. n e. NN ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) )
15		elrp 11834	. . . . . . . . 9 |- ( r e. RR+ <-> ( r e. RR /\ 0 < r ) )
16		nnrecl 11290	. . . . . . . . 9 |- ( ( r e. RR /\ 0 < r ) -> E. m e. NN ( 1 / m ) < r )
17	15, 16	sylbi 207	. . . . . . . 8 |- ( r e. RR+ -> E. m e. NN ( 1 / m ) < r )
18	17	adantl 482	. . . . . . 7 |- ( ( ph /\ r e. RR+ ) -> E. m e. NN ( 1 / m ) < r )
19		peano2nn 11032	. . . . . . . . . 10 |- ( m e. NN -> ( m + 1 ) e. NN )
20	19	adantl 482	. . . . . . . . 9 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( m + 1 ) e. NN )
21		oveq1 6657	. . . . . . . . . . . . . . . . 17 |- ( k = m -> ( k + 1 ) = ( m + 1 ) )
22	21	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( k = m -> ( g ` ( k + 1 ) ) = ( g ` ( m + 1 ) ) )
23		id 22	. . . . . . . . . . . . . . . . 17 |- ( k = m -> k = m )
24		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( k = m -> ( g ` k ) = ( g ` m ) )
25	23, 24	oveq12d 6668	. . . . . . . . . . . . . . . 16 |- ( k = m -> ( k F ( g ` k ) ) = ( m F ( g ` m ) ) )
26	22, 25	eleq12d 2695	. . . . . . . . . . . . . . 15 |- ( k = m -> ( ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) <-> ( g ` ( m + 1 ) ) e. ( m F ( g ` m ) ) ) )
27	26	rspccva 3308	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( g ` ( m + 1 ) ) e. ( m F ( g ` m ) ) )
29	6	ffvelrnda 6359	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( g ` m ) e. ( X X. RR+ ) )
30	7, 1, 8	bcthlem1 23121	. . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . 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 222	. . . . . . . . . . . 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 1074	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) )
35	34	adantlr 751	. . . . . . . . . 10 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) )
36	33	simp1d 1073	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( g ` ( m + 1 ) ) e. ( X X. RR+ ) )
37		xp2nd 7199	. . . . . . . . . . . . . 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 11872	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR )
40	39	adantlr 751	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR )
41		nnrecre 11057	. . . . . . . . . . . 12 |- ( m e. NN -> ( 1 / m ) e. RR )
42	41	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 1 / m ) e. RR )
43		rpre 11839	. . . . . . . . . . . 12 |- ( r e. RR+ -> r e. RR )
44	43	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> r e. RR )
45		lttr 10114	. . . . . . . . . . 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 1326	. . . . . . . . . 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 711	. . . . . . . . 9 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( ( 1 / m ) < r -> ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) )
48		fveq2 6191	. . . . . . . . . . . 12 |- ( n = ( m + 1 ) -> ( g ` n ) = ( g ` ( m + 1 ) ) )
49	48	fveq2d 6195	. . . . . . . . . . 11 |- ( n = ( m + 1 ) -> ( 2nd ` ( g ` n ) ) = ( 2nd ` ( g ` ( m + 1 ) ) ) )
50	49	breq1d 4663	. . . . . . . . . 10 |- ( n = ( m + 1 ) -> ( ( 2nd ` ( g ` n ) ) < r <-> ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) )
51	50	rspcev 3309	. . . . . . . . 9 |- ( ( ( m + 1 ) e. NN /\ ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) -> E. n e. NN ( 2nd ` ( g ` n ) ) < r )
52	20, 47, 51	syl6an 568	. . . . . . . 8 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( ( 1 / m ) < r -> E. n e. NN ( 2nd ` ( g ` n ) ) < r ) )
53	52	rexlimdva 3031	. . . . . . 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 2966	. . . . 5 |- ( ph -> A. r e. RR+ E. n e. NN ( 2nd ` ( g ` n ) ) < r )
56	5, 6, 14, 55	caubl 23106	. . . 4 |- ( ph -> ( 1st o. g ) e. ( Cau ` D ) )
57	7	cmetcau 23087	. . . 4 |- ( ( D e. ( CMet ` X ) /\ ( 1st o. g ) e. ( Cau ` D ) ) -> ( 1st o. g ) e. dom ( ~~> t ` J ) )
58	1, 56, 57	syl2anc 693	. . 3 |- ( ph -> ( 1st o. g ) e. dom ( ~~> t ` J ) )
59		fo1st 7188	. . . . . 6 |- 1st : _V -onto-> _V
60		fofun 6116	. . . . . 6 |- ( 1st : _V -onto-> _V -> Fun 1st )
61	59, 60	ax-mp 5	. . . . 5 |- Fun 1st
62		vex 3203	. . . . 5 |- g e. _V
63		cofunexg 7130	. . . . 5 |- ( ( Fun 1st /\ g e. _V ) -> ( 1st o. g ) e. _V )
64	61, 62, 63	mp2an 708	. . . 4 |- ( 1st o. g ) e. _V
65	64	eldm 5321	. . 3 |- ( ( 1st o. g ) e. dom ( ~~> t ` J ) <-> E. x ( 1st o. g ) ( ~~> t ` J ) x )
66	58, 65	sylib 208	. 2 |- ( ph -> E. x ( 1st o. g ) ( ~~> t ` J ) x )
67		1nn 11031	. . . . . 6 |- 1 e. NN
68	7, 1, 8, 9, 10, 11, 6, 12, 13	bcthlem3 23123	. . . . . 6 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ 1 e. NN ) -> x e. ( ( ball ` D ) ` ( g ` 1 ) ) )
69	67, 68	mp3an3 1413	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( ( ball ` D ) ` ( g ` 1 ) ) )
70	12	fveq2d 6195	. . . . . . 7 |- ( ph -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( ( ball ` D ) ` <. C , R >. ) )
71		df-ov 6653	. . . . . . 7 |- ( C ( ball ` D ) R ) = ( ( ball ` D ) ` <. C , R >. )
72	70, 71	syl6eqr 2674	. . . . . 6 |- ( ph -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( C ( ball ` D ) R ) )
73	72	adantr 481	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( C ( ball ` D ) R ) )
74	69, 73	eleqtrd 2703	. . . 4 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( C ( ball ` D ) R ) )
75	7	mopntop 22245	. . . . . . . . . . . . . 14 |- ( D e. ( *Met ` X ) -> J e. Top )
76	5, 75	syl 17	. . . . . . . . . . . . 13 |- ( ph -> J e. Top )
77	76	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> J e. Top )
78	5	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> D e. ( *Met ` X ) )
79		xp1st 7198	. . . . . . . . . . . . . . 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 11873	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR* )
82		blssm 22223	. . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( ( 1st ` ( g ` ( m + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( m + 1 ) ) ) ) C_ X )
84		1st2nd2 7205	. . . . . . . . . . . . . . . 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 6195	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) = ( ( ball ` D ) ` <. ( 1st ` ( g ` ( m + 1 ) ) ) , ( 2nd ` ( g ` ( m + 1 ) ) ) >. ) )
87		df-ov 6653	. . . . . . . . . . . . . 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 2675	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( ( 1st ` ( g ` ( m + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( m + 1 ) ) ) ) = ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
89	7	mopnuni 22246	. . . . . . . . . . . . . . 15 |- ( D e. ( *Met ` X ) -> X = U. J )
90	5, 89	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> X = U. J )
91	90	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> X = U. J )
92	83, 88, 91	3sstr3d 3647	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ U. J )
93		eqid 2622	. . . . . . . . . . . . 13 |- U. J = U. J
94	93	sscls 20860	. . . . . . . . . . . 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 693	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) )
96	33	simp3d 1075	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
97	95, 96	sstrd 3613	. . . . . . . . . 10 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
98	97	3adant2 1080	. . . . . . . . 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 23123	. . . . . . . . . 10 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ ( m + 1 ) e. NN ) -> x e. ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
100	19, 99	syl3an3 1361	. . . . . . . . 9 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> x e. ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
101	98, 100	sseldd 3604	. . . . . . . 8 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> x e. ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
102	101	eldifbd 3587	. . . . . . 7 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> -. x e. ( M ` m ) )
103	102	3expa 1265	. . . . . 6 |- ( ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) /\ m e. NN ) -> -. x e. ( M ` m ) )
104	103	ralrimiva 2966	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> A. m e. NN -. x e. ( M ` m ) )
105		eluni2 4440	. . . . . . . . 9 |- ( x e. U. ran M <-> E. y e. ran M x e. y )
106		ffn 6045	. . . . . . . . . . 11 |- ( M : NN --> ( Clsd ` J ) -> M Fn NN )
107	9, 106	syl 17	. . . . . . . . . 10 |- ( ph -> M Fn NN )
108		eleq2 2690	. . . . . . . . . . 11 |- ( y = ( M ` m ) -> ( x e. y <-> x e. ( M ` m ) ) )
109	108	rexrn 6361	. . . . . . . . . 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 272	. . . . . . . 8 |- ( ph -> ( x e. U. ran M <-> E. m e. NN x e. ( M ` m ) ) )
112	111	notbid 308	. . . . . . 7 |- ( ph -> ( -. x e. U. ran M <-> -. E. m e. NN x e. ( M ` m ) ) )
113		ralnex 2992	. . . . . . 7 |- ( A. m e. NN -. x e. ( M ` m ) <-> -. E. m e. NN x e. ( M ` m ) )
114	112, 113	syl6bbr 278	. . . . . 6 |- ( ph -> ( -. x e. U. ran M <-> A. m e. NN -. x e. ( M ` m ) ) )
115	114	biimpar 502	. . . . 5 |- ( ( ph /\ A. m e. NN -. x e. ( M ` m ) ) -> -. x e. U. ran M )
116	104, 115	syldan 487	. . . 4 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> -. x e. U. ran M )
117	74, 116	eldifd 3585	. . 3 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( ( C ( ball ` D ) R ) \ U. ran M ) )
118		ne0i 3921	. . 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 1863	1 |- ( ph -> ( ( C ( ball ` D ) R ) \ U. ran M ) =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   <.cop 4183   U.cuni 4436   class class class wbr 4653   {copab 4712   X. cxp 5112   dom cdm 5114   ran crn 5115   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   / cdiv 10684   NNcn 11020   RR+crp 11832   *Metcxmt 19731   Metcme 19732   ballcbl 19733   MetOpencmopn 19736   Topctop 20698   Clsdccld 20820   clsccl 20822   ~~> tclm 21030   Caucca 23051   CMetcms 23052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lm 21033  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-cfil 23053  df-cau 23054  df-cmet 23055

This theorem is referenced by:  bcthlem5  23125