Theorem bcthlem4 21529

Description: Lemma for bcth 21531. 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 21488	. . . . . . 7 |- ( D e. ( CMet ` X ) -> D e. ( Met ` X ) )
3	1, 2	syl 16	. . . . . 6 |- ( ph -> D e. ( Met ` X ) )
4		metxmet 20600	. . . . . 6 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
5	3, 4	syl 16	. . . . 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 21527	. . . . 5 |- ( ph -> A. n e. NN ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) )
15		elrp 11222	. . . . . . . . 9 |- ( r e. RR+ <-> ( r e. RR /\ 0 < r ) )
16		nnrecl 10793	. . . . . . . . 9 |- ( ( r e. RR /\ 0 < r ) -> E. m e. NN ( 1 / m ) < r )
17	15, 16	sylbi 195	. . . . . . . 8 |- ( r e. RR+ -> E. m e. NN ( 1 / m ) < r )
18	17	adantl 466	. . . . . . 7 |- ( ( ph /\ r e. RR+ ) -> E. m e. NN ( 1 / m ) < r )
19		peano2nn 10548	. . . . . . . . . 10 |- ( m e. NN -> ( m + 1 ) e. NN )
20	19	adantl 466	. . . . . . . . 9 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( m + 1 ) e. NN )
21		oveq1 6291	. . . . . . . . . . . . . . . . 17 |- ( k = m -> ( k + 1 ) = ( m + 1 ) )
22	21	fveq2d 5870	. . . . . . . . . . . . . . . 16 |- ( k = m -> ( g ` ( k + 1 ) ) = ( g ` ( m + 1 ) ) )
23		id 22	. . . . . . . . . . . . . . . . 17 |- ( k = m -> k = m )
24		fveq2 5866	. . . . . . . . . . . . . . . . 17 |- ( k = m -> ( g ` k ) = ( g ` m ) )
25	23, 24	oveq12d 6302	. . . . . . . . . . . . . . . 16 |- ( k = m -> ( k F ( g ` k ) ) = ( m F ( g ` m ) ) )
26	22, 25	eleq12d 2549	. . . . . . . . . . . . . . 15 |- ( k = m -> ( ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) <-> ( g ` ( m + 1 ) ) e. ( m F ( g ` m ) ) ) )
27	26	rspccva 3213	. . . . . . . . . . . . . 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 471	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( g ` ( m + 1 ) ) e. ( m F ( g ` m ) ) )
29	6	ffvelrnda 6021	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( g ` m ) e. ( X X. RR+ ) )
30	7, 1, 8	bcthlem1 21526	. . . . . . . . . . . . . . 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 615	. . . . . . . . . . . . . 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 210	. . . . . . . . . . . 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 1009	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) )
35	34	adantlr 714	. . . . . . . . . 10 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) )
36	33	simp1d 1008	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( g ` ( m + 1 ) ) e. ( X X. RR+ ) )
37		xp2nd 6815	. . . . . . . . . . . . . 14 |- ( ( g ` ( m + 1 ) ) e. ( X X. RR+ ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR+ )
38	36, 37	syl 16	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR+ )
39	38	rpred 11256	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR )
40	39	adantlr 714	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR )
41		nnrecre 10572	. . . . . . . . . . . 12 |- ( m e. NN -> ( 1 / m ) e. RR )
42	41	adantl 466	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 1 / m ) e. RR )
43		rpre 11226	. . . . . . . . . . . 12 |- ( r e. RR+ -> r e. RR )
44	43	ad2antlr 726	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> r e. RR )
45		lttr 9661	. . . . . . . . . . 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 1228	. . . . . . . . . 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 675	. . . . . . . . 9 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( ( 1 / m ) < r -> ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) )
48		fveq2 5866	. . . . . . . . . . . 12 |- ( n = ( m + 1 ) -> ( g ` n ) = ( g ` ( m + 1 ) ) )
49	48	fveq2d 5870	. . . . . . . . . . 11 |- ( n = ( m + 1 ) -> ( 2nd ` ( g ` n ) ) = ( 2nd ` ( g ` ( m + 1 ) ) ) )
50	49	breq1d 4457	. . . . . . . . . 10 |- ( n = ( m + 1 ) -> ( ( 2nd ` ( g ` n ) ) < r <-> ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) )
51	50	rspcev 3214	. . . . . . . . 9 |- ( ( ( m + 1 ) e. NN /\ ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) -> E. n e. NN ( 2nd ` ( g ` n ) ) < r )
52	20, 47, 51	syl6an 545	. . . . . . . 8 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( ( 1 / m ) < r -> E. n e. NN ( 2nd ` ( g ` n ) ) < r ) )
53	52	rexlimdva 2955	. . . . . . 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 2878	. . . . 5 |- ( ph -> A. r e. RR+ E. n e. NN ( 2nd ` ( g ` n ) ) < r )
56	5, 6, 14, 55	caubl 21509	. . . 4 |- ( ph -> ( 1st o. g ) e. ( Cau ` D ) )
57	7	cmetcau 21491	. . . 4 |- ( ( D e. ( CMet ` X ) /\ ( 1st o. g ) e. ( Cau ` D ) ) -> ( 1st o. g ) e. dom ( ~~> t ` J ) )
58	1, 56, 57	syl2anc 661	. . 3 |- ( ph -> ( 1st o. g ) e. dom ( ~~> t ` J ) )
59		fo1st 6804	. . . . . 6 |- 1st : _V -onto-> _V
60		fofun 5796	. . . . . 6 |- ( 1st : _V -onto-> _V -> Fun 1st )
61	59, 60	ax-mp 5	. . . . 5 |- Fun 1st
62		vex 3116	. . . . 5 |- g e. _V
63		cofunexg 6748	. . . . 5 |- ( ( Fun 1st /\ g e. _V ) -> ( 1st o. g ) e. _V )
64	61, 62, 63	mp2an 672	. . . 4 |- ( 1st o. g ) e. _V
65	64	eldm 5200	. . 3 |- ( ( 1st o. g ) e. dom ( ~~> t ` J ) <-> E. x ( 1st o. g ) ( ~~> t ` J ) x )
66	58, 65	sylib 196	. 2 |- ( ph -> E. x ( 1st o. g ) ( ~~> t ` J ) x )
67		1nn 10547	. . . . . 6 |- 1 e. NN
68	7, 1, 8, 9, 10, 11, 6, 12, 13	bcthlem3 21528	. . . . . 6 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ 1 e. NN ) -> x e. ( ( ball ` D ) ` ( g ` 1 ) ) )
69	67, 68	mp3an3 1313	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( ( ball ` D ) ` ( g ` 1 ) ) )
70	12	fveq2d 5870	. . . . . . 7 |- ( ph -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( ( ball ` D ) ` <. C , R >. ) )
71		df-ov 6287	. . . . . . 7 |- ( C ( ball ` D ) R ) = ( ( ball ` D ) ` <. C , R >. )
72	70, 71	syl6eqr 2526	. . . . . 6 |- ( ph -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( C ( ball ` D ) R ) )
73	72	adantr 465	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( C ( ball ` D ) R ) )
74	69, 73	eleqtrd 2557	. . . 4 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( C ( ball ` D ) R ) )
75	7	mopntop 20706	. . . . . . . . . . . . . 14 |- ( D e. ( *Met ` X ) -> J e. Top )
76	5, 75	syl 16	. . . . . . . . . . . . 13 |- ( ph -> J e. Top )
77	76	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> J e. Top )
78	5	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> D e. ( *Met ` X ) )
79		xp1st 6814	. . . . . . . . . . . . . . 15 |- ( ( g ` ( m + 1 ) ) e. ( X X. RR+ ) -> ( 1st ` ( g ` ( m + 1 ) ) ) e. X )
80	36, 79	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( 1st ` ( g ` ( m + 1 ) ) ) e. X )
81	38	rpxrd 11257	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR* )
82		blssm 20684	. . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( ( 1st ` ( g ` ( m + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( m + 1 ) ) ) ) C_ X )
84		1st2nd2 6821	. . . . . . . . . . . . . . . 16 |- ( ( g ` ( m + 1 ) ) e. ( X X. RR+ ) -> ( g ` ( m + 1 ) ) = <. ( 1st ` ( g ` ( m + 1 ) ) ) , ( 2nd ` ( g ` ( m + 1 ) ) ) >. )
85	36, 84	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ph /\ m e. NN ) -> ( g ` ( m + 1 ) ) = <. ( 1st ` ( g ` ( m + 1 ) ) ) , ( 2nd ` ( g ` ( m + 1 ) ) ) >. )
86	85	fveq2d 5870	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) = ( ( ball ` D ) ` <. ( 1st ` ( g ` ( m + 1 ) ) ) , ( 2nd ` ( g ` ( m + 1 ) ) ) >. ) )
87		df-ov 6287	. . . . . . . . . . . . . 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 2527	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( ( 1st ` ( g ` ( m + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( m + 1 ) ) ) ) = ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
89	7	mopnuni 20707	. . . . . . . . . . . . . . 15 |- ( D e. ( *Met ` X ) -> X = U. J )
90	5, 89	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> X = U. J )
91	90	adantr 465	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> X = U. J )
92	83, 88, 91	3sstr3d 3546	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ U. J )
93		eqid 2467	. . . . . . . . . . . . 13 |- U. J = U. J
94	93	sscls 19351	. . . . . . . . . . . 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 661	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) )
96	33	simp3d 1010	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
97	95, 96	sstrd 3514	. . . . . . . . . 10 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
98	97	3adant2 1015	. . . . . . . . 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 21528	. . . . . . . . . 10 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ ( m + 1 ) e. NN ) -> x e. ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
100	19, 99	syl3an3 1263	. . . . . . . . 9 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> x e. ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
101	98, 100	sseldd 3505	. . . . . . . 8 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> x e. ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
102	101	eldifbd 3489	. . . . . . 7 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> -. x e. ( M ` m ) )
103	102	3expa 1196	. . . . . 6 |- ( ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) /\ m e. NN ) -> -. x e. ( M ` m ) )
104	103	ralrimiva 2878	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> A. m e. NN -. x e. ( M ` m ) )
105		eluni2 4249	. . . . . . . . 9 |- ( x e. U. ran M <-> E. y e. ran M x e. y )
106		ffn 5731	. . . . . . . . . . 11 |- ( M : NN --> ( Clsd ` J ) -> M Fn NN )
107	9, 106	syl 16	. . . . . . . . . 10 |- ( ph -> M Fn NN )
108		eleq2 2540	. . . . . . . . . . 11 |- ( y = ( M ` m ) -> ( x e. y <-> x e. ( M ` m ) ) )
109	108	rexrn 6023	. . . . . . . . . 10 |- ( M Fn NN -> ( E. y e. ran M x e. y <-> E. m e. NN x e. ( M ` m ) ) )
110	107, 109	syl 16	. . . . . . . . 9 |- ( ph -> ( E. y e. ran M x e. y <-> E. m e. NN x e. ( M ` m ) ) )
111	105, 110	syl5bb 257	. . . . . . . 8 |- ( ph -> ( x e. U. ran M <-> E. m e. NN x e. ( M ` m ) ) )
112	111	notbid 294	. . . . . . 7 |- ( ph -> ( -. x e. U. ran M <-> -. E. m e. NN x e. ( M ` m ) ) )
113		ralnex 2910	. . . . . . 7 |- ( A. m e. NN -. x e. ( M ` m ) <-> -. E. m e. NN x e. ( M ` m ) )
114	112, 113	syl6bbr 263	. . . . . 6 |- ( ph -> ( -. x e. U. ran M <-> A. m e. NN -. x e. ( M ` m ) ) )
115	114	biimpar 485	. . . . 5 |- ( ( ph /\ A. m e. NN -. x e. ( M ` m ) ) -> -. x e. U. ran M )
116	104, 115	syldan 470	. . . 4 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> -. x e. U. ran M )
117	74, 116	eldifd 3487	. . 3 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( ( C ( ball ` D ) R ) \ U. ran M ) )
118		ne0i 3791	. . 3 |- ( x e. ( ( C ( ball ` D ) R ) \ U. ran M ) -> ( ( C ( ball ` D ) R ) \ U. ran M ) =/= (/) )
119	117, 118	syl 16	. 2 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> ( ( C ( ball ` D ) R ) \ U. ran M ) =/= (/) )
120	66, 119	exlimddv 1702	1 |- ( ph -> ( ( C ( ball ` D ) R ) \ U. ran M ) =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   E.wex 1596   e. wcel 1767   =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113   \ cdif 3473   C_ wss 3476   (/)c0 3785   <.cop 4033   U.cuni 4245   class class class wbr 4447   {copab 4504   X. cxp 4997   dom cdm 4999   ran crn 5000   o. ccom 5003   Fun wfun 5582   Fn wfn 5583   -->wf 5584   -onto->wfo 5586   ` cfv 5588  (class class class)co 6284   |-> cmpt2 6286   1stc1st 6782   2ndc2nd 6783   RRcr 9491   0cc0 9492   1c1 9493   + caddc 9495   RR*cxr 9627   < clt 9628   / cdiv 10206   NNcn 10536   RR+crp 11220   *Metcxmt 18202   Metcme 18203   ballcbl 18204   MetOpencmopn 18207   Topctop 19189   Clsdccld 19311   clsccl 19313   ~~> tclm 19521   Caucca 21455   CMetcms 21456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ico 11535  df-rest 14678  df-topgen 14699  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-top 19194  df-bases 19196  df-topon 19197  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lm 19524  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-cfil 21457  df-cau 21458  df-cmet 21459

This theorem is referenced by:  bcthlem5  21530