Theorem heiborlem8 32214

Description: Lemma for heibor 32217. 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 22334	. . . 4 |- ( D e. ( CMet ` X ) -> D e. ( Met ` X ) )
3		metxmet 21427	. . . 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 3419	. . 3 |- ( ph -> Z e. J )
8		heibor.15	. . 3 |- ( ph -> Y e. Z )
9		heibor.1	. . . 4 |- J = ( MetOpen ` D )
10	9	mopni2 21586	. . 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 1292	. 2 |- ( ph -> E. x e. RR+ ( Y ( ball ` D ) x ) C_ Z )
12		rphalfcl 11350	. . . . . 6 |- ( x e. RR+ -> ( x / 2 ) e. RR+ )
13		breq2 4399	. . . . . . . 8 |- ( r = ( x / 2 ) -> ( ( 2nd ` ( M ` k ) ) < r <-> ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) )
14	13	rexbidv 2892	. . . . . . 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 32213	. . . . . . 7 |- A. r e. RR+ E. k e. NN ( 2nd ` ( M ` k ) ) < r
25	14, 24	vtoclri 3110	. . . . . 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 473	. . . 4 |- ( ( ph /\ x e. RR+ ) -> E. k e. NN ( 2nd ` ( M ` k ) ) < ( x / 2 ) )
28		nnnn0 10900	. . . . . . 7 |- ( k e. NN -> k e. NN0 )
29	9, 15, 16, 17, 1, 18, 19, 20, 21, 22	heiborlem4 32210	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> ( S ` k ) G k )
30		fvex 5889	. . . . . . . . . 10 |- ( S ` k ) e. _V
31		vex 3034	. . . . . . . . . 10 |- k e. _V
32	9, 15, 16, 30, 31	heiborlem2 32208	. . . . . . . . 9 |- ( ( S ` k ) G k <-> ( k e. NN0 /\ ( S ` k ) e. ( F ` k ) /\ ( ( S ` k ) B k ) e. K ) )
33	32	simp3bi 1047	. . . . . . . 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 482	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( ( S ` k ) B k ) e. K )
36	35	ad2ant2r 761	. . . . 5 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( S ` k ) B k ) e. K )
37	4	ad2antrr 740	. . . . . . . . . 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 32211	. . . . . . . . . . . . 13 |- ( ph -> M : NN --> ( X X. RR+ ) )
39	38	ffvelrnda 6037	. . . . . . . . . . . 12 |- ( ( ph /\ k e. NN ) -> ( M ` k ) e. ( X X. RR+ ) )
40	39	ad2ant2r 761	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( M ` k ) e. ( X X. RR+ ) )
41		xp1st 6842	. . . . . . . . . . 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 10790	. . . . . . . . . . . . . . 15 |- 2 e. NN
44		nnexpcl 12323	. . . . . . . . . . . . . . 15 |- ( ( 2 e. NN /\ k e. NN0 ) -> ( 2 ^ k ) e. NN )
45	43, 28, 44	sylancr 676	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( 2 ^ k ) e. NN )
46	45	nnrpd 11362	. . . . . . . . . . . . 13 |- ( k e. NN -> ( 2 ^ k ) e. RR+ )
47	46	rpreccld 11374	. . . . . . . . . . . 12 |- ( k e. NN -> ( 1 / ( 2 ^ k ) ) e. RR+ )
48	47	ad2antrl 742	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 1 / ( 2 ^ k ) ) e. RR+ )
49	48	rpxrd 11365	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 1 / ( 2 ^ k ) ) e. RR* )
50		xp2nd 6843	. . . . . . . . . . . 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 11365	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 2nd ` ( M ` k ) ) e. RR* )
53		1le3 10849	. . . . . . . . . . . . . 14 |- 1 <_ 3
54		elrp 11327	. . . . . . . . . . . . . . 15 |- ( ( 2 ^ k ) e. RR+ <-> ( ( 2 ^ k ) e. RR /\ 0 < ( 2 ^ k ) ) )
55		1re 9660	. . . . . . . . . . . . . . . 16 |- 1 e. RR
56		3re 10705	. . . . . . . . . . . . . . . 16 |- 3 e. RR
57		lediv1 10492	. . . . . . . . . . . . . . . 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 1380	. . . . . . . . . . . . . . 15 |- ( ( ( 2 ^ k ) e. RR /\ 0 < ( 2 ^ k ) ) -> ( 1 <_ 3 <-> ( 1 / ( 2 ^ k ) ) <_ ( 3 / ( 2 ^ k ) ) ) )
59	54, 58	sylbi 200	. . . . . . . . . . . . . 14 |- ( ( 2 ^ k ) e. RR+ -> ( 1 <_ 3 <-> ( 1 / ( 2 ^ k ) ) <_ ( 3 / ( 2 ^ k ) ) ) )
60	53, 59	mpbii 216	. . . . . . . . . . . . 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 742	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 1 / ( 2 ^ k ) ) <_ ( 3 / ( 2 ^ k ) ) )
63		fveq2 5879	. . . . . . . . . . . . . . . 16 |- ( n = k -> ( S ` n ) = ( S ` k ) )
64		oveq2 6316	. . . . . . . . . . . . . . . . 17 |- ( n = k -> ( 2 ^ n ) = ( 2 ^ k ) )
65	64	oveq2d 6324	. . . . . . . . . . . . . . . 16 |- ( n = k -> ( 3 / ( 2 ^ n ) ) = ( 3 / ( 2 ^ k ) ) )
66	63, 65	opeq12d 4166	. . . . . . . . . . . . . . 15 |- ( n = k -> <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. = <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. )
67		opex 4664	. . . . . . . . . . . . . . 15 |- <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. e. _V
68	66, 23, 67	fvmpt 5963	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( M ` k ) = <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. )
69	68	fveq2d 5883	. . . . . . . . . . . . 13 |- ( k e. NN -> ( 2nd ` ( M ` k ) ) = ( 2nd ` <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. ) )
70		ovex 6336	. . . . . . . . . . . . . 14 |- ( 3 / ( 2 ^ k ) ) e. _V
71	30, 70	op2nd 6821	. . . . . . . . . . . . 13 |- ( 2nd ` <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. ) = ( 3 / ( 2 ^ k ) )
72	69, 71	syl6eq 2521	. . . . . . . . . . . 12 |- ( k e. NN -> ( 2nd ` ( M ` k ) ) = ( 3 / ( 2 ^ k ) ) )
73	72	ad2antrl 742	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 2nd ` ( M ` k ) ) = ( 3 / ( 2 ^ k ) ) )
74	62, 73	breqtrrd 4422	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 1 / ( 2 ^ k ) ) <_ ( 2nd ` ( M ` k ) ) )
75		ssbl 21516	. . . . . . . . . 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 1303	. . . . . . . . 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 742	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> k e. NN0 )
78		oveq1 6315	. . . . . . . . . . . 12 |- ( z = ( 1st ` ( M ` k ) ) -> ( z ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) = ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) )
79		oveq2 6316	. . . . . . . . . . . . . 14 |- ( m = k -> ( 2 ^ m ) = ( 2 ^ k ) )
80	79	oveq2d 6324	. . . . . . . . . . . . 13 |- ( m = k -> ( 1 / ( 2 ^ m ) ) = ( 1 / ( 2 ^ k ) ) )
81	80	oveq2d 6324	. . . . . . . . . . . 12 |- ( m = k -> ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) = ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 1 / ( 2 ^ k ) ) ) )
82		ovex 6336	. . . . . . . . . . . 12 |- ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 1 / ( 2 ^ k ) ) ) e. _V
83	78, 81, 17, 82	ovmpt2 6451	. . . . . . . . . . 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 673	. . . . . . . . . 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 5883	. . . . . . . . . . . . 13 |- ( k e. NN -> ( 1st ` ( M ` k ) ) = ( 1st ` <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. ) )
86	30, 70	op1st 6820	. . . . . . . . . . . . 13 |- ( 1st ` <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. ) = ( S ` k )
87	85, 86	syl6eq 2521	. . . . . . . . . . . 12 |- ( k e. NN -> ( 1st ` ( M ` k ) ) = ( S ` k ) )
88	87	ad2antrl 742	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 1st ` ( M ` k ) ) = ( S ` k ) )
89	88	oveq1d 6323	. . . . . . . . . 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 2507	. . . . . . . . 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 6849	. . . . . . . . . . . 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 5883	. . . . . . . . . 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 6311	. . . . . . . . . 10 |- ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( 2nd ` ( M ` k ) ) ) = ( ( ball ` D ) ` <. ( 1st ` ( M ` k ) ) , ( 2nd ` ( M ` k ) ) >. )
95	93, 94	syl6reqr 2524	. . . . . . . . 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 3460	. . . . . . . 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 21533	. . . . . . . . . . 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 21511	. . . . . . . . . . . 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 1292	. . . . . . . . . . 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 21534	. . . . . . . . . . . 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 3460	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( ball ` D ) ` ( M ` k ) ) C_ U. J )
104		eqid 2471	. . . . . . . . . . 11 |- U. J = U. J
105	104	sscls 20148	. . . . . . . . . 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 673	. . . . . . . . 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 5883	. . . . . . . . . . 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 741	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( x / 2 ) e. RR+ )
109	108	rpxrd 11365	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( x / 2 ) e. RR* )
110		simprr 774	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( 2nd ` ( M ` k ) ) < ( x / 2 ) )
111	9	blsscls 21600	. . . . . . . . . . . 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 1299	. . . . . . . . . . 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 3453	. . . . . . . . . 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 11331	. . . . . . . . . . . 12 |- ( x e. RR+ -> x e. RR )
115	114	ad2antlr 741	. . . . . . . . . . 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 32212	. . . . . . . . . . . . . . . . 17 |- ( ph -> A. t e. NN ( ( ball ` D ) ` ( M ` ( t + 1 ) ) ) C_ ( ( ball ` D ) ` ( M ` t ) ) )
118	4, 38, 117, 9	caublcls 22356	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( 1st o. M ) ( ~~> t ` J ) Y /\ k e. NN ) -> Y e. ( ( cls ` J ) ` ( ( ball ` D ) ` ( M ` k ) ) ) )
119	118	3expia 1233	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( 1st o. M ) ( ~~> t ` J ) Y ) -> ( k e. NN -> Y e. ( ( cls ` J ) ` ( ( ball ` D ) ` ( M ` k ) ) ) ) )
120	116, 119	mpdan 681	. . . . . . . . . . . . . 14 |- ( ph -> ( k e. NN -> Y e. ( ( cls ` J ) ` ( ( ball ` D ) ` ( M ` k ) ) ) ) )
121	120	imp 436	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. NN ) -> Y e. ( ( cls ` J ) ` ( ( ball ` D ) ` ( M ` k ) ) ) )
122	121	ad2ant2r 761	. . . . . . . . . . . 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 3419	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> Y e. ( ( 1st ` ( M ` k ) ) ( ball ` D ) ( x / 2 ) ) )
124		blhalf 21498	. . . . . . . . . . 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 1293	. . . . . . . . . 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 3428	. . . . . . . . 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 3428	. . . . . . . 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 3428	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ ( k e. NN /\ ( 2nd ` ( M ` k ) ) < ( x / 2 ) ) ) -> ( ( S ` k ) B k ) C_ ( Y ( ball ` D ) x ) )
129		sstr2 3425	. . . . . . 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 4206	. . . . . . . . . . . . 13 |- ( Z e. U -> U. { Z } = Z )
132	6, 131	syl 17	. . . . . . . . . . . 12 |- ( ph -> U. { Z } = Z )
133	132	sseq2d 3446	. . . . . . . . . . 11 |- ( ph -> ( ( ( S ` k ) B k ) C_ U. { Z } <-> ( ( S ` k ) B k ) C_ Z ) )
134	133	biimpar 493	. . . . . . . . . 10 |- ( ( ph /\ ( ( S ` k ) B k ) C_ Z ) -> ( ( S ` k ) B k ) C_ U. { Z } )
135	6	snssd 4108	. . . . . . . . . . . . 13 |- ( ph -> { Z } C_ U )
136		snex 4641	. . . . . . . . . . . . . 14 |- { Z } e. _V
137	136	elpw 3948	. . . . . . . . . . . . 13 |- ( { Z } e. ~P U <-> { Z } C_ U )
138	135, 137	sylibr 217	. . . . . . . . . . . 12 |- ( ph -> { Z } e. ~P U )
139		snfi 7668	. . . . . . . . . . . . 13 |- { Z } e. Fin
140	139	a1i 11	. . . . . . . . . . . 12 |- ( ph -> { Z } e. Fin )
141	138, 140	elind 3609	. . . . . . . . . . 11 |- ( ph -> { Z } e. ( ~P U i^i Fin ) )
142		unieq 4198	. . . . . . . . . . . . 13 |- ( v = { Z } -> U. v = U. { Z } )
143	142	sseq2d 3446	. . . . . . . . . . . 12 |- ( v = { Z } -> ( ( ( S ` k ) B k ) C_ U. v <-> ( ( S ` k ) B k ) C_ U. { Z } ) )
144	143	rspcev 3136	. . . . . . . . . . 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 479	. . . . . . . . . 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 478	. . . . . . . . 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 6336	. . . . . . . . . . 11 |- ( ( S ` k ) B k ) e. _V
148		sseq1 3439	. . . . . . . . . . . . 13 |- ( u = ( ( S ` k ) B k ) -> ( u C_ U. v <-> ( ( S ` k ) B k ) C_ U. v ) )
149	148	rexbidv 2892	. . . . . . . . . . . 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 301	. . . . . . . . . . 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 3176	. . . . . . . . . 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 339	. . . . . . . . 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 201	. . . . . . . 8 |- ( ( ph /\ ( ( S ` k ) B k ) C_ Z ) -> -. ( ( S ` k ) B k ) e. K )
154	153	ex 441	. . . . . . 7 |- ( ph -> ( ( ( S ` k ) B k ) C_ Z -> -. ( ( S ` k ) B k ) e. K ) )
155	154	ad2antrr 740	. . . . . 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 44	. . . . 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 2875	. . 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 179	1 |- ( ph -> ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   {cab 2457   A.wral 2756   E.wrex 2757   _Vcvv 3031   i^i cin 3389   C_ wss 3390   ifcif 3872   ~Pcpw 3942   {csn 3959   <.cop 3965   U.cuni 4190   U_ciun 4269   class class class wbr 4395   {copab 4453   |-> cmpt 4454   X. cxp 4837   o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811   Fincfn 7587   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   RR*cxr 9692   < clt 9693   <_ cle 9694   - cmin 9880   / cdiv 10291   NNcn 10631   2c2 10681   3c3 10682   NN0cn0 10893   RR+crp 11325   seqcseq 12251   ^cexp 12310   *Metcxmt 19032   Metcme 19033   ballcbl 19034   MetOpencmopn 19037   Topctop 19994   clsccl 20110   ~~> tclm 20319   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-1o 7200  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  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-3 10691  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-fl 12061  df-seq 12252  df-exp 12311  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-top 19998  df-bases 19999  df-topon 20000  df-cld 20111  df-ntr 20112  df-cls 20113  df-lm 20322  df-cmet 22305

This theorem is referenced by:  heiborlem9  32215