Theorem bcthlem4 22188

Description: Lemma for bcth 22190. 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 22149	. . . . . . 7 |- ( D e. ( CMet ` X ) -> D e. ( Met ` X ) )
3	1, 2	syl 17	. . . . . 6 |- ( ph -> D e. ( Met ` X ) )
4		metxmet 21280	. . . . . 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 22186	. . . . 5 |- ( ph -> A. n e. NN ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) )
15		elrp 11304	. . . . . . . . 9 |- ( r e. RR+ <-> ( r e. RR /\ 0 < r ) )
16		nnrecl 10867	. . . . . . . . 9 |- ( ( r e. RR /\ 0 < r ) -> E. m e. NN ( 1 / m ) < r )
17	15, 16	sylbi 198	. . . . . . . 8 |- ( r e. RR+ -> E. m e. NN ( 1 / m ) < r )
18	17	adantl 467	. . . . . . 7 |- ( ( ph /\ r e. RR+ ) -> E. m e. NN ( 1 / m ) < r )
19		peano2nn 10621	. . . . . . . . . 10 |- ( m e. NN -> ( m + 1 ) e. NN )
20	19	adantl 467	. . . . . . . . 9 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( m + 1 ) e. NN )
21		oveq1 6312	. . . . . . . . . . . . . . . . 17 |- ( k = m -> ( k + 1 ) = ( m + 1 ) )
22	21	fveq2d 5885	. . . . . . . . . . . . . . . 16 |- ( k = m -> ( g ` ( k + 1 ) ) = ( g ` ( m + 1 ) ) )
23		id 23	. . . . . . . . . . . . . . . . 17 |- ( k = m -> k = m )
24		fveq2 5881	. . . . . . . . . . . . . . . . 17 |- ( k = m -> ( g ` k ) = ( g ` m ) )
25	23, 24	oveq12d 6323	. . . . . . . . . . . . . . . 16 |- ( k = m -> ( k F ( g ` k ) ) = ( m F ( g ` m ) ) )
26	22, 25	eleq12d 2511	. . . . . . . . . . . . . . 15 |- ( k = m -> ( ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) <-> ( g ` ( m + 1 ) ) e. ( m F ( g ` m ) ) ) )
27	26	rspccva 3187	. . . . . . . . . . . . . 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 473	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( g ` ( m + 1 ) ) e. ( m F ( g ` m ) ) )
29	6	ffvelrnda 6037	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( g ` m ) e. ( X X. RR+ ) )
30	7, 1, 8	bcthlem1 22185	. . . . . . . . . . . . . . 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 618	. . . . . . . . . . . . . 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 213	. . . . . . . . . . . 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 1018	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) )
35	34	adantlr 719	. . . . . . . . . 10 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) )
36	33	simp1d 1017	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( g ` ( m + 1 ) ) e. ( X X. RR+ ) )
37		xp2nd 6838	. . . . . . . . . . . . . 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 11341	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR )
40	39	adantlr 719	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR )
41		nnrecre 10646	. . . . . . . . . . . 12 |- ( m e. NN -> ( 1 / m ) e. RR )
42	41	adantl 467	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 1 / m ) e. RR )
43		rpre 11308	. . . . . . . . . . . 12 |- ( r e. RR+ -> r e. RR )
44	43	ad2antlr 731	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> r e. RR )
45		lttr 9709	. . . . . . . . . . 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 1264	. . . . . . . . . 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 679	. . . . . . . . 9 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( ( 1 / m ) < r -> ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) )
48		fveq2 5881	. . . . . . . . . . . 12 |- ( n = ( m + 1 ) -> ( g ` n ) = ( g ` ( m + 1 ) ) )
49	48	fveq2d 5885	. . . . . . . . . . 11 |- ( n = ( m + 1 ) -> ( 2nd ` ( g ` n ) ) = ( 2nd ` ( g ` ( m + 1 ) ) ) )
50	49	breq1d 4436	. . . . . . . . . 10 |- ( n = ( m + 1 ) -> ( ( 2nd ` ( g ` n ) ) < r <-> ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) )
51	50	rspcev 3188	. . . . . . . . 9 |- ( ( ( m + 1 ) e. NN /\ ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) -> E. n e. NN ( 2nd ` ( g ` n ) ) < r )
52	20, 47, 51	syl6an 547	. . . . . . . 8 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( ( 1 / m ) < r -> E. n e. NN ( 2nd ` ( g ` n ) ) < r ) )
53	52	rexlimdva 2924	. . . . . . 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 2846	. . . . 5 |- ( ph -> A. r e. RR+ E. n e. NN ( 2nd ` ( g ` n ) ) < r )
56	5, 6, 14, 55	caubl 22170	. . . 4 |- ( ph -> ( 1st o. g ) e. ( Cau ` D ) )
57	7	cmetcau 22152	. . . 4 |- ( ( D e. ( CMet ` X ) /\ ( 1st o. g ) e. ( Cau ` D ) ) -> ( 1st o. g ) e. dom ( ~~> t ` J ) )
58	1, 56, 57	syl2anc 665	. . 3 |- ( ph -> ( 1st o. g ) e. dom ( ~~> t ` J ) )
59		fo1st 6827	. . . . . 6 |- 1st : _V -onto-> _V
60		fofun 5811	. . . . . 6 |- ( 1st : _V -onto-> _V -> Fun 1st )
61	59, 60	ax-mp 5	. . . . 5 |- Fun 1st
62		vex 3090	. . . . 5 |- g e. _V
63		cofunexg 6771	. . . . 5 |- ( ( Fun 1st /\ g e. _V ) -> ( 1st o. g ) e. _V )
64	61, 62, 63	mp2an 676	. . . 4 |- ( 1st o. g ) e. _V
65	64	eldm 5052	. . 3 |- ( ( 1st o. g ) e. dom ( ~~> t ` J ) <-> E. x ( 1st o. g ) ( ~~> t ` J ) x )
66	58, 65	sylib 199	. 2 |- ( ph -> E. x ( 1st o. g ) ( ~~> t ` J ) x )
67		1nn 10620	. . . . . 6 |- 1 e. NN
68	7, 1, 8, 9, 10, 11, 6, 12, 13	bcthlem3 22187	. . . . . 6 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ 1 e. NN ) -> x e. ( ( ball ` D ) ` ( g ` 1 ) ) )
69	67, 68	mp3an3 1349	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( ( ball ` D ) ` ( g ` 1 ) ) )
70	12	fveq2d 5885	. . . . . . 7 |- ( ph -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( ( ball ` D ) ` <. C , R >. ) )
71		df-ov 6308	. . . . . . 7 |- ( C ( ball ` D ) R ) = ( ( ball ` D ) ` <. C , R >. )
72	70, 71	syl6eqr 2488	. . . . . 6 |- ( ph -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( C ( ball ` D ) R ) )
73	72	adantr 466	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( C ( ball ` D ) R ) )
74	69, 73	eleqtrd 2519	. . . 4 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( C ( ball ` D ) R ) )
75	7	mopntop 21386	. . . . . . . . . . . . . 14 |- ( D e. ( *Met ` X ) -> J e. Top )
76	5, 75	syl 17	. . . . . . . . . . . . 13 |- ( ph -> J e. Top )
77	76	adantr 466	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> J e. Top )
78	5	adantr 466	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> D e. ( *Met ` X ) )
79		xp1st 6837	. . . . . . . . . . . . . . 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 11342	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR* )
82		blssm 21364	. . . . . . . . . . . . . 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 1264	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( ( 1st ` ( g ` ( m + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( m + 1 ) ) ) ) C_ X )
84		1st2nd2 6844	. . . . . . . . . . . . . . . 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 5885	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) = ( ( ball ` D ) ` <. ( 1st ` ( g ` ( m + 1 ) ) ) , ( 2nd ` ( g ` ( m + 1 ) ) ) >. ) )
87		df-ov 6308	. . . . . . . . . . . . . 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 2489	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( ( 1st ` ( g ` ( m + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( m + 1 ) ) ) ) = ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
89	7	mopnuni 21387	. . . . . . . . . . . . . . 15 |- ( D e. ( *Met ` X ) -> X = U. J )
90	5, 89	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> X = U. J )
91	90	adantr 466	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> X = U. J )
92	83, 88, 91	3sstr3d 3512	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ U. J )
93		eqid 2429	. . . . . . . . . . . . 13 |- U. J = U. J
94	93	sscls 20002	. . . . . . . . . . . 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 665	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) )
96	33	simp3d 1019	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
97	95, 96	sstrd 3480	. . . . . . . . . 10 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
98	97	3adant2 1024	. . . . . . . . 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 22187	. . . . . . . . . 10 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ ( m + 1 ) e. NN ) -> x e. ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
100	19, 99	syl3an3 1299	. . . . . . . . 9 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> x e. ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
101	98, 100	sseldd 3471	. . . . . . . 8 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> x e. ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
102	101	eldifbd 3455	. . . . . . 7 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> -. x e. ( M ` m ) )
103	102	3expa 1205	. . . . . 6 |- ( ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) /\ m e. NN ) -> -. x e. ( M ` m ) )
104	103	ralrimiva 2846	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> A. m e. NN -. x e. ( M ` m ) )
105		eluni2 4226	. . . . . . . . 9 |- ( x e. U. ran M <-> E. y e. ran M x e. y )
106		ffn 5746	. . . . . . . . . . 11 |- ( M : NN --> ( Clsd ` J ) -> M Fn NN )
107	9, 106	syl 17	. . . . . . . . . 10 |- ( ph -> M Fn NN )
108		eleq2 2502	. . . . . . . . . . 11 |- ( y = ( M ` m ) -> ( x e. y <-> x e. ( M ` m ) ) )
109	108	rexrn 6039	. . . . . . . . . 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 260	. . . . . . . 8 |- ( ph -> ( x e. U. ran M <-> E. m e. NN x e. ( M ` m ) ) )
112	111	notbid 295	. . . . . . 7 |- ( ph -> ( -. x e. U. ran M <-> -. E. m e. NN x e. ( M ` m ) ) )
113		ralnex 2878	. . . . . . 7 |- ( A. m e. NN -. x e. ( M ` m ) <-> -. E. m e. NN x e. ( M ` m ) )
114	112, 113	syl6bbr 266	. . . . . 6 |- ( ph -> ( -. x e. U. ran M <-> A. m e. NN -. x e. ( M ` m ) ) )
115	114	biimpar 487	. . . . 5 |- ( ( ph /\ A. m e. NN -. x e. ( M ` m ) ) -> -. x e. U. ran M )
116	104, 115	syldan 472	. . . 4 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> -. x e. U. ran M )
117	74, 116	eldifd 3453	. . 3 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( ( C ( ball ` D ) R ) \ U. ran M ) )
118		ne0i 3773	. . 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 1773	1 |- ( ph -> ( ( C ( ball ` D ) R ) \ U. ran M ) =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   E.wex 1659   e. wcel 1870   =/= wne 2625   A.wral 2782   E.wrex 2783   _Vcvv 3087   \ cdif 3439   C_ wss 3442   (/)c0 3767   <.cop 4008   U.cuni 4222   class class class wbr 4426   {copab 4483   X. cxp 4852   dom cdm 4854   ran crn 4855   o. ccom 4858   Fun wfun 5595   Fn wfn 5596   -->wf 5597   -onto->wfo 5599   ` cfv 5601  (class class class)co 6305   |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806   RRcr 9537   0cc0 9538   1c1 9539   + caddc 9541   RR*cxr 9673   < clt 9674   / cdiv 10268   NNcn 10609   RR+crp 11302   *Metcxmt 18890   Metcme 18891   ballcbl 18892   MetOpencmopn 18895   Topctop 19848   Clsdccld 19962   clsccl 19964   ~~> tclm 20173   Caucca 22116   CMetcms 22117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ico 11641  df-rest 15280  df-topgen 15301  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-top 19852  df-bases 19853  df-topon 19854  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lm 20176  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-cfil 22118  df-cau 22119  df-cmet 22120

This theorem is referenced by:  bcthlem5  22189