Theorem bcthlem4 22373

Description: Lemma for bcth 22375. 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 22334	. . . . . . 7 |- ( D e. ( CMet ` X ) -> D e. ( Met ` X ) )
3	1, 2	syl 17	. . . . . 6 |- ( ph -> D e. ( Met ` X ) )
4		metxmet 21427	. . . . . 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 22371	. . . . 5 |- ( ph -> A. n e. NN ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) )
15		elrp 11327	. . . . . . . . 9 |- ( r e. RR+ <-> ( r e. RR /\ 0 < r ) )
16		nnrecl 10891	. . . . . . . . 9 |- ( ( r e. RR /\ 0 < r ) -> E. m e. NN ( 1 / m ) < r )
17	15, 16	sylbi 200	. . . . . . . 8 |- ( r e. RR+ -> E. m e. NN ( 1 / m ) < r )
18	17	adantl 473	. . . . . . 7 |- ( ( ph /\ r e. RR+ ) -> E. m e. NN ( 1 / m ) < r )
19		peano2nn 10643	. . . . . . . . . 10 |- ( m e. NN -> ( m + 1 ) e. NN )
20	19	adantl 473	. . . . . . . . 9 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( m + 1 ) e. NN )
21		oveq1 6315	. . . . . . . . . . . . . . . . 17 |- ( k = m -> ( k + 1 ) = ( m + 1 ) )
22	21	fveq2d 5883	. . . . . . . . . . . . . . . 16 |- ( k = m -> ( g ` ( k + 1 ) ) = ( g ` ( m + 1 ) ) )
23		id 22	. . . . . . . . . . . . . . . . 17 |- ( k = m -> k = m )
24		fveq2 5879	. . . . . . . . . . . . . . . . 17 |- ( k = m -> ( g ` k ) = ( g ` m ) )
25	23, 24	oveq12d 6326	. . . . . . . . . . . . . . . 16 |- ( k = m -> ( k F ( g ` k ) ) = ( m F ( g ` m ) ) )
26	22, 25	eleq12d 2543	. . . . . . . . . . . . . . 15 |- ( k = m -> ( ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) <-> ( g ` ( m + 1 ) ) e. ( m F ( g ` m ) ) ) )
27	26	rspccva 3135	. . . . . . . . . . . . . 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 479	. . . . . . . . . . . . 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 22370	. . . . . . . . . . . . . . 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 626	. . . . . . . . . . . . . 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 215	. . . . . . . . . . . 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 1043	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) )
35	34	adantlr 729	. . . . . . . . . 10 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) < ( 1 / m ) )
36	33	simp1d 1042	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( g ` ( m + 1 ) ) e. ( X X. RR+ ) )
37		xp2nd 6843	. . . . . . . . . . . . . 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 11364	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR )
40	39	adantlr 729	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR )
41		nnrecre 10668	. . . . . . . . . . . 12 |- ( m e. NN -> ( 1 / m ) e. RR )
42	41	adantl 473	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( 1 / m ) e. RR )
43		rpre 11331	. . . . . . . . . . . 12 |- ( r e. RR+ -> r e. RR )
44	43	ad2antlr 741	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> r e. RR )
45		lttr 9728	. . . . . . . . . . 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 1292	. . . . . . . . . 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 689	. . . . . . . . 9 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( ( 1 / m ) < r -> ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) )
48		fveq2 5879	. . . . . . . . . . . 12 |- ( n = ( m + 1 ) -> ( g ` n ) = ( g ` ( m + 1 ) ) )
49	48	fveq2d 5883	. . . . . . . . . . 11 |- ( n = ( m + 1 ) -> ( 2nd ` ( g ` n ) ) = ( 2nd ` ( g ` ( m + 1 ) ) ) )
50	49	breq1d 4405	. . . . . . . . . 10 |- ( n = ( m + 1 ) -> ( ( 2nd ` ( g ` n ) ) < r <-> ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) )
51	50	rspcev 3136	. . . . . . . . 9 |- ( ( ( m + 1 ) e. NN /\ ( 2nd ` ( g ` ( m + 1 ) ) ) < r ) -> E. n e. NN ( 2nd ` ( g ` n ) ) < r )
52	20, 47, 51	syl6an 554	. . . . . . . 8 |- ( ( ( ph /\ r e. RR+ ) /\ m e. NN ) -> ( ( 1 / m ) < r -> E. n e. NN ( 2nd ` ( g ` n ) ) < r ) )
53	52	rexlimdva 2871	. . . . . . 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 2809	. . . . 5 |- ( ph -> A. r e. RR+ E. n e. NN ( 2nd ` ( g ` n ) ) < r )
56	5, 6, 14, 55	caubl 22355	. . . 4 |- ( ph -> ( 1st o. g ) e. ( Cau ` D ) )
57	7	cmetcau 22337	. . . 4 |- ( ( D e. ( CMet ` X ) /\ ( 1st o. g ) e. ( Cau ` D ) ) -> ( 1st o. g ) e. dom ( ~~> t ` J ) )
58	1, 56, 57	syl2anc 673	. . 3 |- ( ph -> ( 1st o. g ) e. dom ( ~~> t ` J ) )
59		fo1st 6832	. . . . . 6 |- 1st : _V -onto-> _V
60		fofun 5807	. . . . . 6 |- ( 1st : _V -onto-> _V -> Fun 1st )
61	59, 60	ax-mp 5	. . . . 5 |- Fun 1st
62		vex 3034	. . . . 5 |- g e. _V
63		cofunexg 6776	. . . . 5 |- ( ( Fun 1st /\ g e. _V ) -> ( 1st o. g ) e. _V )
64	61, 62, 63	mp2an 686	. . . 4 |- ( 1st o. g ) e. _V
65	64	eldm 5037	. . 3 |- ( ( 1st o. g ) e. dom ( ~~> t ` J ) <-> E. x ( 1st o. g ) ( ~~> t ` J ) x )
66	58, 65	sylib 201	. 2 |- ( ph -> E. x ( 1st o. g ) ( ~~> t ` J ) x )
67		1nn 10642	. . . . . 6 |- 1 e. NN
68	7, 1, 8, 9, 10, 11, 6, 12, 13	bcthlem3 22372	. . . . . 6 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ 1 e. NN ) -> x e. ( ( ball ` D ) ` ( g ` 1 ) ) )
69	67, 68	mp3an3 1379	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( ( ball ` D ) ` ( g ` 1 ) ) )
70	12	fveq2d 5883	. . . . . . 7 |- ( ph -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( ( ball ` D ) ` <. C , R >. ) )
71		df-ov 6311	. . . . . . 7 |- ( C ( ball ` D ) R ) = ( ( ball ` D ) ` <. C , R >. )
72	70, 71	syl6eqr 2523	. . . . . 6 |- ( ph -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( C ( ball ` D ) R ) )
73	72	adantr 472	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> ( ( ball ` D ) ` ( g ` 1 ) ) = ( C ( ball ` D ) R ) )
74	69, 73	eleqtrd 2551	. . . 4 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( C ( ball ` D ) R ) )
75	7	mopntop 21533	. . . . . . . . . . . . . 14 |- ( D e. ( *Met ` X ) -> J e. Top )
76	5, 75	syl 17	. . . . . . . . . . . . 13 |- ( ph -> J e. Top )
77	76	adantr 472	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> J e. Top )
78	5	adantr 472	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> D e. ( *Met ` X ) )
79		xp1st 6842	. . . . . . . . . . . . . . 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 11365	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( 2nd ` ( g ` ( m + 1 ) ) ) e. RR* )
82		blssm 21511	. . . . . . . . . . . . . 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 1292	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( ( 1st ` ( g ` ( m + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( m + 1 ) ) ) ) C_ X )
84		1st2nd2 6849	. . . . . . . . . . . . . . . 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 5883	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) = ( ( ball ` D ) ` <. ( 1st ` ( g ` ( m + 1 ) ) ) , ( 2nd ` ( g ` ( m + 1 ) ) ) >. ) )
87		df-ov 6311	. . . . . . . . . . . . . 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 2524	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> ( ( 1st ` ( g ` ( m + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( m + 1 ) ) ) ) = ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
89	7	mopnuni 21534	. . . . . . . . . . . . . . 15 |- ( D e. ( *Met ` X ) -> X = U. J )
90	5, 89	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> X = U. J )
91	90	adantr 472	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. NN ) -> X = U. J )
92	83, 88, 91	3sstr3d 3460	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ U. J )
93		eqid 2471	. . . . . . . . . . . . 13 |- U. J = U. J
94	93	sscls 20148	. . . . . . . . . . . 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 673	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) )
96	33	simp3d 1044	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
97	95, 96	sstrd 3428	. . . . . . . . . 10 |- ( ( ph /\ m e. NN ) -> ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) C_ ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
98	97	3adant2 1049	. . . . . . . . 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 22372	. . . . . . . . . 10 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ ( m + 1 ) e. NN ) -> x e. ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
100	19, 99	syl3an3 1327	. . . . . . . . 9 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> x e. ( ( ball ` D ) ` ( g ` ( m + 1 ) ) ) )
101	98, 100	sseldd 3419	. . . . . . . 8 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> x e. ( ( ( ball ` D ) ` ( g ` m ) ) \ ( M ` m ) ) )
102	101	eldifbd 3403	. . . . . . 7 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ m e. NN ) -> -. x e. ( M ` m ) )
103	102	3expa 1231	. . . . . 6 |- ( ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) /\ m e. NN ) -> -. x e. ( M ` m ) )
104	103	ralrimiva 2809	. . . . 5 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> A. m e. NN -. x e. ( M ` m ) )
105		eluni2 4194	. . . . . . . . 9 |- ( x e. U. ran M <-> E. y e. ran M x e. y )
106		ffn 5739	. . . . . . . . . . 11 |- ( M : NN --> ( Clsd ` J ) -> M Fn NN )
107	9, 106	syl 17	. . . . . . . . . 10 |- ( ph -> M Fn NN )
108		eleq2 2538	. . . . . . . . . . 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 265	. . . . . . . 8 |- ( ph -> ( x e. U. ran M <-> E. m e. NN x e. ( M ` m ) ) )
112	111	notbid 301	. . . . . . 7 |- ( ph -> ( -. x e. U. ran M <-> -. E. m e. NN x e. ( M ` m ) ) )
113		ralnex 2834	. . . . . . 7 |- ( A. m e. NN -. x e. ( M ` m ) <-> -. E. m e. NN x e. ( M ` m ) )
114	112, 113	syl6bbr 271	. . . . . 6 |- ( ph -> ( -. x e. U. ran M <-> A. m e. NN -. x e. ( M ` m ) ) )
115	114	biimpar 493	. . . . 5 |- ( ( ph /\ A. m e. NN -. x e. ( M ` m ) ) -> -. x e. U. ran M )
116	104, 115	syldan 478	. . . 4 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> -. x e. U. ran M )
117	74, 116	eldifd 3401	. . 3 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x ) -> x e. ( ( C ( ball ` D ) R ) \ U. ran M ) )
118		ne0i 3728	. . 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 1789	1 |- ( ph -> ( ( C ( ball ` D ) R ) \ U. ran M ) =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   E.wex 1671   e. wcel 1904   =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031   \ cdif 3387   C_ wss 3390   (/)c0 3722   <.cop 3965   U.cuni 4190   class class class wbr 4395   {copab 4453   X. cxp 4837   dom cdm 4839   ran crn 4840   o. ccom 4843   Fun wfun 5583   Fn wfn 5584   -->wf 5585   -onto->wfo 5587   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   RR*cxr 9692   < clt 9693   / cdiv 10291   NNcn 10631   RR+crp 11325   *Metcxmt 19032   Metcme 19033   ballcbl 19034   MetOpencmopn 19037   Topctop 19994   Clsdccld 20108   clsccl 20110   ~~> tclm 20319   Caucca 22301   CMetcms 22302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ico 11666  df-rest 15399  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-top 19998  df-bases 19999  df-topon 20000  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lm 20322  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-cfil 22303  df-cau 22304  df-cmet 22305

This theorem is referenced by:  bcthlem5  22374