Theorem heiborlem8 32143

Description: Lemma for heibor 32146. The previous lemmas establish that the sequence M is Cauchy, so using completeness we now consider the convergent point Y. By assumption, U is an open cover, so Y is an element of some Z e. U, and some ball centered at Y is contained in Z. But the sequence contains arbitrarily small balls close to Y, so some element ball ( M ` n ) of the sequence is contained in Z. And finally we arrive at a contradiction, because { Z } is a finite subcover of U that covers ball ( M ` n ), yet ball ( M ` n ) e. K. For convenience, we write this contradiction as ph -> ps where ph is all the accumulated hypotheses and ps is anything at all. (Contributed by Jeff Madsen, 22-Jan-2014.)

Hypotheses

Ref	Expression
heibor.1	|- J = ( MetOpen ` D )
heibor.3	|- K = { u | -. E. v e. ( ~P U i^i Fin ) u C_ U. v }
heibor.4	|- G = { <. y , n >. | ( n e. NN0 /\ y e. ( F ` n ) /\ ( y B n ) e. K ) }
heibor.5	|- B = ( z e. X , m e. NN0 |-> ( z ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) )
heibor.6	|- ( ph -> D e. ( CMet ` X ) )
heibor.7	|- ( ph -> F : NN0 --> ( ~P X i^i Fin ) )
heibor.8	|- ( ph -> A. n e. NN0 X = U_ y e. ( F ` n ) ( y B n ) )
heibor.9	|- ( ph -> A. x e. G ( ( T ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( T ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) )
heibor.10	|- ( ph -> C G 0 )
heibor.11	|- S = seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) )
heibor.12	|- M = ( n e. NN |-> <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. )
heibor.13	|- ( ph -> U C_ J )
heibor.14	|- Y e. _V
heibor.15	|- ( ph -> Y e. Z )
heibor.16	|- ( ph -> Z e. U )
heibor.17	|- ( ph -> ( 1st o. M ) ( ~~> t ` J ) Y )

Assertion

Ref	Expression
heiborlem8	|- ( ph -> ps )

Distinct variable groups:   x, n, y, u, F   x, G   ph, x   m, n, u, v, x, y, z, D   m, M, u, x, y, z   T, m, n, x, y, z   B, n, u, v, y   m, J, n, u, v, x, y, z   U, n, u, v, x, y, z   ps, y, z   S, m, n, u, v, x, y, z   m, X, n, u, v, x, y, z   C, m, n, u, v, y   n, K, x, y, z   x, Y   v, Z, x   x, B

Allowed substitution hints:   ph( y, z, v, u, m, n)   ps( x, v, u, m, n)   B( z, m)   C( x, z)   T( v, u)   U( m)   F( z, v, m)   G( y, z, v, u, m, n)   K( v, u, m)   M( v, n)   Y( y, z, v, u, m, n)   Z( y, z, u, m, n)

Proof of Theorem heiborlem8

Dummy variables t k r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		heibor.6	. . . 4 |- ( ph -> D e. ( CMet ` X ) )
2		cmetmet 22249	. . . 4 |- ( D e. ( CMet ` X ) -> D e. ( Met ` X ) )
3		metxmet 21342	. . . 4 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
4	1, 2, 3	3syl 18	. . 3 |- ( ph -> D e. ( *Met ` X ) )
5		heibor.13	. . . 4 |- ( ph -> U C_ J )
6		heibor.16	. . . 4 |- ( ph -> Z e. U )
7	5, 6	sseldd 3432	. . 3 |- ( ph -> Z e. J )
8		heibor.15	. . 3 |- ( ph -> Y e. Z )
9		heibor.1	. . . 4 |- J = ( MetOpen ` D )
10	9	mopni2 21501	. . 3 |- ( ( D e. ( *Met ` X ) /\ Z e. J /\ Y e. Z ) -> E. x e. RR+ ( Y ( ball ` D ) x ) C_ Z )
11	4, 7, 8, 10	syl3anc 1267	. 2 |- ( ph -> E. x e. RR+ ( Y ( ball ` D ) x ) C_ Z )
12		rphalfcl 11324	. . . . . 6 |- ( x e. RR+ -> ( x / 2 ) e. RR+ )
13		breq2 4405	. . . . . . . 8 |- ( r = ( x / 2 ) -> ( ( 2nd ` ( M ` k ) ) < r <-> ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) )
14	13	rexbidv 2900	. . . . . . 7 |- ( r = ( x / 2 ) -> ( E. k e. NN ( 2nd ` ( M ` k ) ) < r <-> E. k e. NN ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) )
15		heibor.3	. . . . . . . 8 |- K = { u | -. E. v e. ( ~P U i^i Fin ) u C_ U. v }
16		heibor.4	. . . . . . . 8 |- G = { <. y , n >. | ( n e. NN0 /\ y e. ( F ` n ) /\ ( y B n ) e. K ) }
17		heibor.5	. . . . . . . 8 |- B = ( z e. X , m e. NN0 |-> ( z ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) )
18		heibor.7	. . . . . . . 8 |- ( ph -> F : NN0 --> ( ~P X i^i Fin ) )
19		heibor.8	. . . . . . . 8 |- ( ph -> A. n e. NN0 X = U_ y e. ( F ` n ) ( y B n ) )
20		heibor.9	. . . . . . . 8 |- ( ph -> A. x e. G ( ( T ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( T ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) )
21		heibor.10	. . . . . . . 8 |- ( ph -> C G 0 )
22		heibor.11	. . . . . . . 8 |- S = seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) )
23		heibor.12	. . . . . . . 8 |- M = ( n e. NN |-> <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. )
24	9, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23	heiborlem7 32142	. . . . . . 7 |- A. r e. RR+ E. k e. NN ( 2nd ` ( M ` k ) ) < r
25	14, 24	vtoclri 3123	. . . . . 6 |- ( ( x / 2 ) e. RR+ -> E. k e. NN ( 2nd ` ( M ` k ) ) < ( x / 2 ) )
26	12, 25	syl 17	. . . . 5 |- ( x e. RR+ -> E. k e. NN ( 2nd ` ( M ` k ) ) < ( x / 2 ) )
27	26	adantl 468	. . . 4 |- ( ( ph /\ x e. RR+ ) -> E. k e. NN ( 2nd ` ( M ` k ) ) < ( x / 2 ) )
28		nnnn0 10873	. . . . . . 7 |- ( k e. NN -> k e. NN0 )
29	9, 15, 16, 17, 1, 18, 19, 20, 21, 22	heiborlem4 32139	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> ( S ` k ) G k )
30		fvex 5873	. . . . . . . . . 10 |- ( S ` k ) e. _V
31		vex 3047	. . . . . . . . . 10 |- k e. _V
32	9, 15, 16, 30, 31	heiborlem2 32137	. . . . . . . . 9 |- ( ( S ` k ) G k <-> ( k e. NN0 /\ ( S ` k ) e. ( F ` k ) /\ ( ( S ` k ) B k ) e. K ) )
33	32	simp3bi 1024	. . . . . . . 8 |- ( ( S ` k ) G k -> ( ( S ` k ) B k ) e. K )
34	29, 33	syl 17	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( ( S ` k ) B k ) e. K )
35	28, 34	sylan2 477	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( ( S ` k ) B k ) e. K )
36	35	ad2ant2r 752	. . . . 5 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( S ` k ) B k ) e. K )
37	4	ad2antrr 731	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> D e. ( *Met ` X ) )
38	9, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23	heiborlem5 32140	. . . . . . . . . . . . 13 |- ( ph -> M : NN --> ( X X. RR+ ) )
39	38	ffvelrnda 6020	. . . . . . . . . . . 12 |- ( ( ph /\ k e. NN ) -> ( M ` k ) e. ( X X. RR+ ) )
40	39	ad2ant2r 752	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( M ` k ) e. ( X X. RR+ ) )
41		xp1st 6820	. . . . . . . . . . 11 |- ( ( M ` k ) e. ( X X. RR+ ) -> ( 1st ` ( M ` k ) ) e. X )
42	40, 41	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 1st ` ( M ` k ) ) e. X )
43		2nn 10764	. . . . . . . . . . . . . . 15 |- 2 e. NN
44		nnexpcl 12282	. . . . . . . . . . . . . . 15 |- ( ( 2 e. NN /\ k e. NN0 ) -> ( 2 ^ k ) e. NN )
45	43, 28, 44	sylancr 668	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( 2 ^ k ) e. NN )
46	45	nnrpd 11336	. . . . . . . . . . . . 13 |- ( k e. NN -> ( 2 ^ k ) e. RR+ )
47	46	rpreccld 11348	. . . . . . . . . . . 12 |- ( k e. NN -> ( 1 / ( 2 ^ k ) ) e. RR+ )
48	47	ad2antrl 733	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 1 / ( 2 ^ k ) ) e. RR+ )
49	48	rpxrd 11339	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 1 / ( 2 ^ k ) ) e. RR* )
50		xp2nd 6821	. . . . . . . . . . . 12 |- ( ( M ` k ) e. ( X X. RR+ ) -> ( 2nd ` ( M ` k ) ) e. RR+ )
51	40, 50	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 2nd ` ( M ` k ) ) e. RR+ )
52	51	rpxrd 11339	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 2nd ` ( M ` k ) ) e. RR* )
53		1le3 10823	. . . . . . . . . . . . . 14 |- 1 <_ 3
54		elrp 11301	. . . . . . . . . . . . . . 15 |- ( ( 2 ^ k ) e. RR+ <-> ( ( 2 ^ k ) e. RR /\ 0 < ( 2 ^ k ) ) )
55		1re 9639	. . . . . . . . . . . . . . . 16 |- 1 e. RR
56		3re 10680	. . . . . . . . . . . . . . . 16 |- 3 e. RR
57		lediv1 10467	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. RR /\ 3 e. RR /\ ( ( 2 ^ k ) e. RR /\ 0 < ( 2 ^ k ) ) ) -> ( 1 <_ 3 <-> ( 1 / ( 2 ^ k ) ) <_ ( 3 / ( 2 ^ k ) ) ) )
58	55, 56, 57	mp3an12 1353	. . . . . . . . . . . . . . 15 |- ( ( ( 2 ^ k ) e. RR /\ 0 < ( 2 ^ k ) ) -> ( 1 <_ 3 <-> ( 1 / ( 2 ^ k ) ) <_ ( 3 / ( 2 ^ k ) ) ) )
59	54, 58	sylbi 199	. . . . . . . . . . . . . 14 |- ( ( 2 ^ k ) e. RR+ -> ( 1 <_ 3 <-> ( 1 / ( 2 ^ k ) ) <_ ( 3 / ( 2 ^ k ) ) ) )
60	53, 59	mpbii 215	. . . . . . . . . . . . 13 |- ( ( 2 ^ k ) e. RR+ -> ( 1 / ( 2 ^ k ) ) <_ ( 3 / ( 2 ^ k ) ) )
61	46, 60	syl 17	. . . . . . . . . . . 12 |- ( k e. NN -> ( 1 / ( 2 ^ k ) ) <_ ( 3 / ( 2 ^ k ) ) )
62	61	ad2antrl 733	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 1 / ( 2 ^ k ) ) <_ ( 3 / ( 2 ^ k ) ) )
63		fveq2 5863	. . . . . . . . . . . . . . . 16 |- ( n = k -> ( S ` n ) = ( S ` k ) )
64		oveq2 6296	. . . . . . . . . . . . . . . . 17 |- ( n = k -> ( 2 ^ n ) = ( 2 ^ k ) )
65	64	oveq2d 6304	. . . . . . . . . . . . . . . 16 |- ( n = k -> ( 3 / ( 2 ^ n ) ) = ( 3 / ( 2 ^ k ) ) )
66	63, 65	opeq12d 4173	. . . . . . . . . . . . . . 15 |- ( n = k -> <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. = <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. )
67		opex 4663	. . . . . . . . . . . . . . 15 |- <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. e. _V
68	66, 23, 67	fvmpt 5946	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( M ` k ) = <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. )
69	68	fveq2d 5867	. . . . . . . . . . . . 13 |- ( k e. NN -> ( 2nd ` ( M ` k ) ) = ( 2nd ` <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. ) )
70		ovex 6316	. . . . . . . . . . . . . 14 |- ( 3 / ( 2 ^ k ) ) e. _V
71	30, 70	op2nd 6799	. . . . . . . . . . . . 13 |- ( 2nd ` <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. ) = ( 3 / ( 2 ^ k ) )
72	69, 71	syl6eq 2500	. . . . . . . . . . . 12 |- ( k e. NN -> ( 2nd ` ( M ` k ) ) = ( 3 / ( 2 ^ k ) ) )
73	72	ad2antrl 733	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 2nd ` ( M ` k ) ) = ( 3 / ( 2 ^ k ) ) )
74	62, 73	breqtrrd 4428	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 1 / ( 2 ^ k ) ) <_ ( 2nd ` ( M ` k ) ) )
75		ssbl 21431	. . . . . . . . . 10 |- ( ( ( D e. ( *Met ` X ) /\ ( 1st ` ( M ` k ) ) e. X ) /\ ( ( 1 / ( 2 ^ k ) ) e. RR* /\ ( 2nd ` ( M ` k ) ) e. RR* ) /\ ( 1 / ( 2 ^ k ) ) <_ ( 2nd ` ( M ` k ) ) ) -> ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 1 / ( 2 ^ k ) ) ) C_ ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 2nd ` ( M ` k ) ) ) )
76	37, 42, 49, 52, 74, 75	syl221anc 1278	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 1 / ( 2 ^ k ) ) ) C_ ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 2nd ` ( M ` k ) ) ) )
77	28	ad2antrl 733	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> k e. NN0 )
78		oveq1 6295	. . . . . . . . . . . 12 |- ( z = ( 1st ` ( M ` k ) ) -> ( z ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) = ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) )
79		oveq2 6296	. . . . . . . . . . . . . 14 |- ( m = k -> ( 2 ^ m ) = ( 2 ^ k ) )
80	79	oveq2d 6304	. . . . . . . . . . . . 13 |- ( m = k -> ( 1 / ( 2 ^ m ) ) = ( 1 / ( 2 ^ k ) ) )
81	80	oveq2d 6304	. . . . . . . . . . . 12 |- ( m = k -> ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) = ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 1 / ( 2 ^ k ) ) ) )
82		ovex 6316	. . . . . . . . . . . 12 |- ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 1 / ( 2 ^ k ) ) ) e. _V
83	78, 81, 17, 82	ovmpt2 6429	. . . . . . . . . . 11 |- ( ( ( 1st ` ( M ` k ) ) e. X /\ k e. NN0 ) -> ( ( 1st ` ( M ` k ) ) B k ) = ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 1 / ( 2 ^ k ) ) ) )
84	42, 77, 83	syl2anc 666	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( 1st ` ( M ` k ) ) B k ) = ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 1 / ( 2 ^ k ) ) ) )
85	68	fveq2d 5867	. . . . . . . . . . . . 13 |- ( k e. NN -> ( 1st ` ( M ` k ) ) = ( 1st ` <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. ) )
86	30, 70	op1st 6798	. . . . . . . . . . . . 13 |- ( 1st ` <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. ) = ( S ` k )
87	85, 86	syl6eq 2500	. . . . . . . . . . . 12 |- ( k e. NN -> ( 1st ` ( M ` k ) ) = ( S ` k ) )
88	87	ad2antrl 733	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 1st ` ( M ` k ) ) = ( S ` k ) )
89	88	oveq1d 6303	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( 1st ` ( M ` k ) ) B k ) = ( ( S ` k ) B k ) )
90	84, 89	eqtr3d 2486	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 1 / ( 2 ^ k ) ) ) = ( ( S ` k ) B k ) )
91		1st2nd2 6827	. . . . . . . . . . . 12 |- ( ( M ` k ) e. ( X X. RR+ ) -> ( M ` k ) = <. ( 1st ` ( M ` k ) ) , ( 2nd ` ( M ` k ) ) >. )
92	40, 91	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( M ` k ) = <. ( 1st ` ( M ` k ) ) , ( 2nd ` ( M ` k ) ) >. )
93	92	fveq2d 5867	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( ball ` D ) ` ( M ` k ) ) = ( ( ball ` D ) ` <. ( 1st ` ( M ` k ) ) , ( 2nd ` ( M ` k ) ) >. ) )
94		df-ov 6291	. . . . . . . . . 10 |- ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 2nd ` ( M ` k ) ) ) = ( ( ball ` D ) ` <. ( 1st ` ( M ` k ) ) , ( 2nd ` ( M ` k ) ) >. )
95	93, 94	syl6reqr 2503	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 2nd ` ( M ` k ) ) ) = ( ( ball ` D ) ` ( M ` k ) ) )
96	76, 90, 95	3sstr3d 3473	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( S ` k ) B k ) C_ ( ( ball ` D ) ` ( M ` k ) ) )
97	9	mopntop 21448	. . . . . . . . . . 11 |- ( D e. ( *Met ` X ) -> J e. Top )
98	37, 97	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> J e. Top )
99		blssm 21426	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` X ) /\ ( 1st ` ( M ` k ) ) e. X /\ ( 2nd ` ( M ` k ) ) e. RR* ) -> ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 2nd ` ( M ` k ) ) ) C_ X )
100	37, 42, 52, 99	syl3anc 1267	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 2nd ` ( M ` k ) ) ) C_ X )
101	9	mopnuni 21449	. . . . . . . . . . . 12 |- ( D e. ( *Met ` X ) -> X = U. J )
102	37, 101	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> X = U. J )
103	100, 95, 102	3sstr3d 3473	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( ball ` D ) ` ( M ` k ) ) C_ U. J )
104		eqid 2450	. . . . . . . . . . 11 |- U. J = U. J
105	104	sscls 20064	. . . . . . . . . 10 |- ( ( J e. Top /\ ( ( ball ` D ) ` ( M ` k ) ) C_ U. J ) -> ( ( ball ` D ) ` ( M ` k ) ) C_ ( ( cls ` J ) ` ( ( ball ` D ) ` ( M ` k ) ) ) )
106	98, 103, 105	syl2anc 666	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( ball ` D ) ` ( M ` k ) ) C_ ( ( cls ` J ) ` ( ( ball ` D ) ` ( M ` k ) ) ) )
107	95	fveq2d 5867	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( cls ` J ) ` ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 2nd ` ( M ` k ) ) ) ) = ( ( cls ` J ) ` ( ( ball ` D ) ` ( M ` k ) ) ) )
108	12	ad2antlr 732	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( x / 2 ) e. RR+ )
109	108	rpxrd 11339	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( x / 2 ) e. RR* )
110		simprr 765	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 2nd ` ( M ` k ) ) < ( x / 2 ) )
111	9	blsscls 21515	. . . . . . . . . . . 12 |- ( ( ( D e. ( *Met ` X ) /\ ( 1st ` ( M ` k ) ) e. X ) /\ ( ( 2nd ` ( M ` k ) ) e. RR* /\ ( x / 2 ) e. RR* /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( cls ` J ) ` ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 2nd ` ( M ` k ) ) ) ) C_ ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( x / 2 ) ) )
112	37, 42, 52, 109, 110, 111	syl23anc 1274	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( cls ` J ) ` ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 2nd ` ( M ` k ) ) ) ) C_ ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( x / 2 ) ) )
113	107, 112	eqsstr3d 3466	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( cls ` J ) ` ( ( ball ` D ) ` ( M ` k ) ) ) C_ ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( x / 2 ) ) )
114		rpre 11305	. . . . . . . . . . . 12 |- ( x e. RR+ -> x e. RR )
115	114	ad2antlr 732	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> x e. RR )
116		heibor.17	. . . . . . . . . . . . . . 15 |- ( ph -> ( 1st o. M ) ( ~~> t ` J ) Y )
117	9, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23	heiborlem6 32141	. . . . . . . . . . . . . . . . 17 |- ( ph -> A. t e. NN ( ( ball ` D ) ` ( M ` ( t + 1 ) ) ) C_ ( ( ball ` D ) ` ( M ` t ) ) )
118	4, 38, 117, 9	caublcls 22271	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( 1st o. M ) ( ~~> t ` J ) Y /\ k e. NN ) -> Y e. ( ( cls ` J ) ` ( ( ball ` D ) ` ( M ` k ) ) ) )
119	118	3expia 1209	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( 1st o. M ) ( ~~> t ` J ) Y ) -> ( k e. NN -> Y e. ( ( cls ` J ) ` ( ( ball ` D ) ` ( M ` k ) ) ) ) )
120	116, 119	mpdan 673	. . . . . . . . . . . . . 14 |- ( ph -> ( k e. NN -> Y e. ( ( cls ` J ) ` ( ( ball ` D ) ` ( M ` k ) ) ) ) )
121	120	imp 431	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. NN ) -> Y e. ( ( cls ` J ) ` ( ( ball ` D ) ` ( M ` k ) ) ) )
122	121	ad2ant2r 752	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> Y e. ( ( cls ` J ) ` ( ( ball ` D ) ` ( M ` k ) ) ) )
123	113, 122	sseldd 3432	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> Y e. ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( x / 2 ) ) )
124		blhalf 21413	. . . . . . . . . . 11 |- ( ( ( D e. ( *Met ` X ) /\ ( 1st ` ( M ` k ) ) e. X ) /\ ( x e. RR /\ Y e. ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( x / 2 ) ) ) ) -> ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( x / 2 ) ) C_ ( Y ( ball ` D ) x ) )
125	37, 42, 115, 123, 124	syl22anc 1268	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( x / 2 ) ) C_ ( Y ( ball ` D ) x ) )
126	113, 125	sstrd 3441	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( cls ` J ) ` ( ( ball ` D ) ` ( M ` k ) ) ) C_ ( Y ( ball ` D ) x ) )
127	106, 126	sstrd 3441	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( ball ` D ) ` ( M ` k ) ) C_ ( Y ( ball ` D ) x ) )
128	96, 127	sstrd 3441	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( S ` k ) B k ) C_ ( Y ( ball ` D ) x ) )
129		sstr2 3438	. . . . . . 7 |- ( ( ( S ` k ) B k ) C_ ( Y ( ball ` D ) x ) -> ( ( Y ( ball ` D ) x ) C_ Z -> ( ( S ` k ) B k ) C_ Z ) )
130	128, 129	syl 17	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( Y ( ball ` D ) x ) C_ Z -> ( ( S ` k ) B k ) C_ Z ) )
131		unisng 4213	. . . . . . . . . . . . 13 |- ( Z e. U -> U. { Z } = Z )
132	6, 131	syl 17	. . . . . . . . . . . 12 |- ( ph -> U. { Z } = Z )
133	132	sseq2d 3459	. . . . . . . . . . 11 |- ( ph -> ( ( ( S ` k ) B k ) C_ U. { Z } <-> ( ( S ` k ) B k ) C_ Z ) )
134	133	biimpar 488	. . . . . . . . . 10 |- ( ( ph /\ ( ( S ` k ) B k ) C_ Z ) -> ( ( S ` k ) B k ) C_ U. { Z } )
135	6	snssd 4116	. . . . . . . . . . . . 13 |- ( ph -> { Z } C_ U )
136		snex 4640	. . . . . . . . . . . . . 14 |- { Z } e. _V
137	136	elpw 3956	. . . . . . . . . . . . 13 |- ( { Z } e. ~P U <-> { Z } C_ U )
138	135, 137	sylibr 216	. . . . . . . . . . . 12 |- ( ph -> { Z } e. ~P U )
139		snfi 7647	. . . . . . . . . . . . 13 |- { Z } e. Fin
140	139	a1i 11	. . . . . . . . . . . 12 |- ( ph -> { Z } e. Fin )
141	138, 140	elind 3617	. . . . . . . . . . 11 |- ( ph -> { Z } e. ( ~P U i^i Fin ) )
142		unieq 4205	. . . . . . . . . . . . 13 |- ( v = { Z } -> U. v = U. { Z } )
143	142	sseq2d 3459	. . . . . . . . . . . 12 |- ( v = { Z } -> ( ( ( S ` k ) B k ) C_ U. v <-> ( ( S ` k ) B k ) C_ U. { Z } ) )
144	143	rspcev 3149	. . . . . . . . . . 11 |- ( ( { Z } e. ( ~P U i^i Fin ) /\ ( ( S ` k ) B k ) C_ U. { Z } ) -> E. v e. ( ~P U i^i Fin ) ( ( S ` k ) B k ) C_ U. v )
145	141, 144	sylan 474	. . . . . . . . . 10 |- ( ( ph /\ ( ( S ` k ) B k ) C_ U. { Z } ) -> E. v e. ( ~P U i^i Fin ) ( ( S ` k ) B k ) C_ U. v )
146	134, 145	syldan 473	. . . . . . . . 9 |- ( ( ph /\ ( ( S ` k ) B k ) C_ Z ) -> E. v e. ( ~P U i^i Fin ) ( ( S ` k ) B k ) C_ U. v )
147		ovex 6316	. . . . . . . . . . 11 |- ( ( S ` k ) B k ) e. _V
148		sseq1 3452	. . . . . . . . . . . . 13 |- ( u = ( ( S ` k ) B k ) -> ( u C_ U. v <-> ( ( S ` k ) B k ) C_ U. v ) )
149	148	rexbidv 2900	. . . . . . . . . . . 12 |- ( u = ( ( S ` k ) B k ) -> ( E. v e. ( ~P U i^i Fin ) u C_ U. v <-> E. v e. ( ~P U i^i Fin ) ( ( S ` k ) B k ) C_ U. v ) )
150	149	notbid 296	. . . . . . . . . . 11 |- ( u = ( ( S ` k ) B k ) -> ( -. E. v e. ( ~P U i^i Fin ) u C_ U. v <-> -. E. v e. ( ~P U i^i Fin ) ( ( S ` k ) B k ) C_ U. v ) )
151	147, 150, 15	elab2 3187	. . . . . . . . . 10 |- ( ( ( S ` k ) B k ) e. K <-> -. E. v e. ( ~P U i^i Fin ) ( ( S ` k ) B k ) C_ U. v )
152	151	con2bii 334	. . . . . . . . 9 |- ( E. v e. ( ~P U i^i Fin ) ( ( S ` k ) B k ) C_ U. v <-> -. ( ( S ` k ) B k ) e. K )
153	146, 152	sylib 200	. . . . . . . 8 |- ( ( ph /\ ( ( S ` k ) B k ) C_ Z ) -> -. ( ( S ` k ) B k ) e. K )
154	153	ex 436	. . . . . . 7 |- ( ph -> ( ( ( S ` k ) B k ) C_ Z -> -. ( ( S ` k ) B k ) e. K ) )
155	154	ad2antrr 731	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( ( S ` k ) B k ) C_ Z -> -. ( ( S ` k ) B k ) e. K ) )
156	130, 155	syld 45	. . . . 5 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( Y ( ball ` D ) x ) C_ Z -> -. ( ( S ` k ) B k ) e. K ) )
157	36, 156	mt2d 121	. . . 4 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> -. ( Y ( ball ` D ) x ) C_ Z )
158	27, 157	rexlimddv 2882	. . 3 |- ( ( ph /\ x e. RR+ ) -> -. ( Y ( ball ` D ) x ) C_ Z )
159	158	nrexdv 2842	. 2 |- ( ph -> -. E. x e. RR+ ( Y ( ball ` D ) x ) C_ Z )
160	11, 159	pm2.21dd 178	1 |- ( ph -> ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   {cab 2436   A.wral 2736   E.wrex 2737   _Vcvv 3044   i^i cin 3402   C_ wss 3403   ifcif 3880   ~Pcpw 3950   {csn 3967   <.cop 3973   U.cuni 4197   U_ciun 4277   class class class wbr 4401   {copab 4459   |-> cmpt 4460   X. cxp 4831   o. ccom 4837   -->wf 5577   ` cfv 5581  (class class class)co 6288   |-> cmpt2 6290   1stc1st 6788   2ndc2nd 6789   Fincfn 7566   RRcr 9535   0cc0 9536   1c1 9537   + caddc 9539   RR*cxr 9671   < clt 9672   <_ cle 9673   - cmin 9857   / cdiv 10266   NNcn 10606   2c2 10656   3c3 10657   NN0cn0 10866   RR+crp 11299   seqcseq 12210   ^cexp 12269   *Metcxmt 18948   Metcme 18949   ballcbl 18950   MetOpencmopn 18953   Topctop 19910   clsccl 20026   ~~> tclm 20235   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-1o 7179  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  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-3 10666  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-fl 12025  df-seq 12211  df-exp 12270  df-topgen 15335  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-top 19914  df-bases 19915  df-topon 19916  df-cld 20027  df-ntr 20028  df-cls 20029  df-lm 20238  df-cmet 22220

This theorem is referenced by:  heiborlem9  32144