Step |
Hyp |
Ref |
Expression |
1 |
|
heibor.1 |
|- J = ( MetOpen ` D ) |
2 |
|
heibor1.3 |
|- ( ph -> D e. ( Met ` X ) ) |
3 |
|
heibor1.4 |
|- ( ph -> J e. Comp ) |
4 |
|
heibor1.5 |
|- ( ph -> F e. ( Cau ` D ) ) |
5 |
|
heibor1.6 |
|- ( ph -> F : NN --> X ) |
6 |
|
metxmet |
|- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) ) |
7 |
2 6
|
syl |
|- ( ph -> D e. ( *Met ` X ) ) |
8 |
1
|
mopntop |
|- ( D e. ( *Met ` X ) -> J e. Top ) |
9 |
7 8
|
syl |
|- ( ph -> J e. Top ) |
10 |
|
imassrn |
|- ( F " u ) C_ ran F |
11 |
5
|
frnd |
|- ( ph -> ran F C_ X ) |
12 |
1
|
mopnuni |
|- ( D e. ( *Met ` X ) -> X = U. J ) |
13 |
7 12
|
syl |
|- ( ph -> X = U. J ) |
14 |
11 13
|
sseqtrd |
|- ( ph -> ran F C_ U. J ) |
15 |
10 14
|
sstrid |
|- ( ph -> ( F " u ) C_ U. J ) |
16 |
|
eqid |
|- U. J = U. J |
17 |
16
|
clscld |
|- ( ( J e. Top /\ ( F " u ) C_ U. J ) -> ( ( cls ` J ) ` ( F " u ) ) e. ( Clsd ` J ) ) |
18 |
9 15 17
|
syl2anc |
|- ( ph -> ( ( cls ` J ) ` ( F " u ) ) e. ( Clsd ` J ) ) |
19 |
|
eleq1a |
|- ( ( ( cls ` J ) ` ( F " u ) ) e. ( Clsd ` J ) -> ( k = ( ( cls ` J ) ` ( F " u ) ) -> k e. ( Clsd ` J ) ) ) |
20 |
18 19
|
syl |
|- ( ph -> ( k = ( ( cls ` J ) ` ( F " u ) ) -> k e. ( Clsd ` J ) ) ) |
21 |
20
|
rexlimdvw |
|- ( ph -> ( E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) -> k e. ( Clsd ` J ) ) ) |
22 |
21
|
abssdv |
|- ( ph -> { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } C_ ( Clsd ` J ) ) |
23 |
|
fvex |
|- ( Clsd ` J ) e. _V |
24 |
23
|
elpw2 |
|- ( { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } e. ~P ( Clsd ` J ) <-> { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } C_ ( Clsd ` J ) ) |
25 |
22 24
|
sylibr |
|- ( ph -> { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } e. ~P ( Clsd ` J ) ) |
26 |
|
elin |
|- ( r e. ( ~P { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } i^i Fin ) <-> ( r e. ~P { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } /\ r e. Fin ) ) |
27 |
|
velpw |
|- ( r e. ~P { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } <-> r C_ { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) |
28 |
|
ssabral |
|- ( r C_ { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } <-> A. k e. r E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) |
29 |
27 28
|
bitri |
|- ( r e. ~P { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } <-> A. k e. r E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) |
30 |
29
|
anbi1i |
|- ( ( r e. ~P { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } /\ r e. Fin ) <-> ( A. k e. r E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) /\ r e. Fin ) ) |
31 |
26 30
|
bitri |
|- ( r e. ( ~P { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } i^i Fin ) <-> ( A. k e. r E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) /\ r e. Fin ) ) |
32 |
|
raleq |
|- ( m = (/) -> ( A. k e. m E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) <-> A. k e. (/) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) ) |
33 |
32
|
anbi2d |
|- ( m = (/) -> ( ( ph /\ A. k e. m E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) <-> ( ph /\ A. k e. (/) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) ) ) |
34 |
|
inteq |
|- ( m = (/) -> |^| m = |^| (/) ) |
35 |
34
|
sseq2d |
|- ( m = (/) -> ( ( F " k ) C_ |^| m <-> ( F " k ) C_ |^| (/) ) ) |
36 |
35
|
rexbidv |
|- ( m = (/) -> ( E. k e. ran ZZ>= ( F " k ) C_ |^| m <-> E. k e. ran ZZ>= ( F " k ) C_ |^| (/) ) ) |
37 |
33 36
|
imbi12d |
|- ( m = (/) -> ( ( ( ph /\ A. k e. m E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| m ) <-> ( ( ph /\ A. k e. (/) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| (/) ) ) ) |
38 |
|
raleq |
|- ( m = y -> ( A. k e. m E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) <-> A. k e. y E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) ) |
39 |
38
|
anbi2d |
|- ( m = y -> ( ( ph /\ A. k e. m E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) <-> ( ph /\ A. k e. y E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) ) ) |
40 |
|
inteq |
|- ( m = y -> |^| m = |^| y ) |
41 |
40
|
sseq2d |
|- ( m = y -> ( ( F " k ) C_ |^| m <-> ( F " k ) C_ |^| y ) ) |
42 |
41
|
rexbidv |
|- ( m = y -> ( E. k e. ran ZZ>= ( F " k ) C_ |^| m <-> E. k e. ran ZZ>= ( F " k ) C_ |^| y ) ) |
43 |
39 42
|
imbi12d |
|- ( m = y -> ( ( ( ph /\ A. k e. m E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| m ) <-> ( ( ph /\ A. k e. y E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| y ) ) ) |
44 |
|
raleq |
|- ( m = ( y u. { n } ) -> ( A. k e. m E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) <-> A. k e. ( y u. { n } ) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) ) |
45 |
44
|
anbi2d |
|- ( m = ( y u. { n } ) -> ( ( ph /\ A. k e. m E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) <-> ( ph /\ A. k e. ( y u. { n } ) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) ) ) |
46 |
|
inteq |
|- ( m = ( y u. { n } ) -> |^| m = |^| ( y u. { n } ) ) |
47 |
46
|
sseq2d |
|- ( m = ( y u. { n } ) -> ( ( F " k ) C_ |^| m <-> ( F " k ) C_ |^| ( y u. { n } ) ) ) |
48 |
47
|
rexbidv |
|- ( m = ( y u. { n } ) -> ( E. k e. ran ZZ>= ( F " k ) C_ |^| m <-> E. k e. ran ZZ>= ( F " k ) C_ |^| ( y u. { n } ) ) ) |
49 |
45 48
|
imbi12d |
|- ( m = ( y u. { n } ) -> ( ( ( ph /\ A. k e. m E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| m ) <-> ( ( ph /\ A. k e. ( y u. { n } ) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| ( y u. { n } ) ) ) ) |
50 |
|
raleq |
|- ( m = r -> ( A. k e. m E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) <-> A. k e. r E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) ) |
51 |
50
|
anbi2d |
|- ( m = r -> ( ( ph /\ A. k e. m E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) <-> ( ph /\ A. k e. r E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) ) ) |
52 |
|
inteq |
|- ( m = r -> |^| m = |^| r ) |
53 |
52
|
sseq2d |
|- ( m = r -> ( ( F " k ) C_ |^| m <-> ( F " k ) C_ |^| r ) ) |
54 |
53
|
rexbidv |
|- ( m = r -> ( E. k e. ran ZZ>= ( F " k ) C_ |^| m <-> E. k e. ran ZZ>= ( F " k ) C_ |^| r ) ) |
55 |
51 54
|
imbi12d |
|- ( m = r -> ( ( ( ph /\ A. k e. m E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| m ) <-> ( ( ph /\ A. k e. r E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| r ) ) ) |
56 |
|
uzf |
|- ZZ>= : ZZ --> ~P ZZ |
57 |
|
ffn |
|- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ ) |
58 |
56 57
|
ax-mp |
|- ZZ>= Fn ZZ |
59 |
|
0z |
|- 0 e. ZZ |
60 |
|
fnfvelrn |
|- ( ( ZZ>= Fn ZZ /\ 0 e. ZZ ) -> ( ZZ>= ` 0 ) e. ran ZZ>= ) |
61 |
58 59 60
|
mp2an |
|- ( ZZ>= ` 0 ) e. ran ZZ>= |
62 |
|
ssv |
|- ( F " ( ZZ>= ` 0 ) ) C_ _V |
63 |
|
int0 |
|- |^| (/) = _V |
64 |
62 63
|
sseqtrri |
|- ( F " ( ZZ>= ` 0 ) ) C_ |^| (/) |
65 |
|
imaeq2 |
|- ( k = ( ZZ>= ` 0 ) -> ( F " k ) = ( F " ( ZZ>= ` 0 ) ) ) |
66 |
65
|
sseq1d |
|- ( k = ( ZZ>= ` 0 ) -> ( ( F " k ) C_ |^| (/) <-> ( F " ( ZZ>= ` 0 ) ) C_ |^| (/) ) ) |
67 |
66
|
rspcev |
|- ( ( ( ZZ>= ` 0 ) e. ran ZZ>= /\ ( F " ( ZZ>= ` 0 ) ) C_ |^| (/) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| (/) ) |
68 |
61 64 67
|
mp2an |
|- E. k e. ran ZZ>= ( F " k ) C_ |^| (/) |
69 |
68
|
a1i |
|- ( ( ph /\ A. k e. (/) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| (/) ) |
70 |
|
ssun1 |
|- y C_ ( y u. { n } ) |
71 |
|
ssralv |
|- ( y C_ ( y u. { n } ) -> ( A. k e. ( y u. { n } ) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) -> A. k e. y E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) ) |
72 |
70 71
|
ax-mp |
|- ( A. k e. ( y u. { n } ) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) -> A. k e. y E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) |
73 |
72
|
anim2i |
|- ( ( ph /\ A. k e. ( y u. { n } ) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> ( ph /\ A. k e. y E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) ) |
74 |
73
|
imim1i |
|- ( ( ( ph /\ A. k e. y E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| y ) -> ( ( ph /\ A. k e. ( y u. { n } ) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| y ) ) |
75 |
|
ssun2 |
|- { n } C_ ( y u. { n } ) |
76 |
|
ssralv |
|- ( { n } C_ ( y u. { n } ) -> ( A. k e. ( y u. { n } ) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) -> A. k e. { n } E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) ) |
77 |
75 76
|
ax-mp |
|- ( A. k e. ( y u. { n } ) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) -> A. k e. { n } E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) |
78 |
|
vex |
|- n e. _V |
79 |
|
eqeq1 |
|- ( k = n -> ( k = ( ( cls ` J ) ` ( F " u ) ) <-> n = ( ( cls ` J ) ` ( F " u ) ) ) ) |
80 |
79
|
rexbidv |
|- ( k = n -> ( E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) <-> E. u e. ran ZZ>= n = ( ( cls ` J ) ` ( F " u ) ) ) ) |
81 |
78 80
|
ralsn |
|- ( A. k e. { n } E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) <-> E. u e. ran ZZ>= n = ( ( cls ` J ) ` ( F " u ) ) ) |
82 |
77 81
|
sylib |
|- ( A. k e. ( y u. { n } ) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) -> E. u e. ran ZZ>= n = ( ( cls ` J ) ` ( F " u ) ) ) |
83 |
|
uzin2 |
|- ( ( u e. ran ZZ>= /\ k e. ran ZZ>= ) -> ( u i^i k ) e. ran ZZ>= ) |
84 |
10 11
|
sstrid |
|- ( ph -> ( F " u ) C_ X ) |
85 |
84 13
|
sseqtrd |
|- ( ph -> ( F " u ) C_ U. J ) |
86 |
16
|
sscls |
|- ( ( J e. Top /\ ( F " u ) C_ U. J ) -> ( F " u ) C_ ( ( cls ` J ) ` ( F " u ) ) ) |
87 |
9 85 86
|
syl2anc |
|- ( ph -> ( F " u ) C_ ( ( cls ` J ) ` ( F " u ) ) ) |
88 |
|
sseq2 |
|- ( n = ( ( cls ` J ) ` ( F " u ) ) -> ( ( F " u ) C_ n <-> ( F " u ) C_ ( ( cls ` J ) ` ( F " u ) ) ) ) |
89 |
87 88
|
syl5ibrcom |
|- ( ph -> ( n = ( ( cls ` J ) ` ( F " u ) ) -> ( F " u ) C_ n ) ) |
90 |
|
inss2 |
|- ( u i^i k ) C_ k |
91 |
|
inss1 |
|- ( u i^i k ) C_ u |
92 |
|
imass2 |
|- ( ( u i^i k ) C_ k -> ( F " ( u i^i k ) ) C_ ( F " k ) ) |
93 |
|
imass2 |
|- ( ( u i^i k ) C_ u -> ( F " ( u i^i k ) ) C_ ( F " u ) ) |
94 |
92 93
|
anim12i |
|- ( ( ( u i^i k ) C_ k /\ ( u i^i k ) C_ u ) -> ( ( F " ( u i^i k ) ) C_ ( F " k ) /\ ( F " ( u i^i k ) ) C_ ( F " u ) ) ) |
95 |
|
ssin |
|- ( ( ( F " ( u i^i k ) ) C_ ( F " k ) /\ ( F " ( u i^i k ) ) C_ ( F " u ) ) <-> ( F " ( u i^i k ) ) C_ ( ( F " k ) i^i ( F " u ) ) ) |
96 |
94 95
|
sylib |
|- ( ( ( u i^i k ) C_ k /\ ( u i^i k ) C_ u ) -> ( F " ( u i^i k ) ) C_ ( ( F " k ) i^i ( F " u ) ) ) |
97 |
90 91 96
|
mp2an |
|- ( F " ( u i^i k ) ) C_ ( ( F " k ) i^i ( F " u ) ) |
98 |
|
ss2in |
|- ( ( ( F " k ) C_ |^| y /\ ( F " u ) C_ n ) -> ( ( F " k ) i^i ( F " u ) ) C_ ( |^| y i^i n ) ) |
99 |
97 98
|
sstrid |
|- ( ( ( F " k ) C_ |^| y /\ ( F " u ) C_ n ) -> ( F " ( u i^i k ) ) C_ ( |^| y i^i n ) ) |
100 |
78
|
intunsn |
|- |^| ( y u. { n } ) = ( |^| y i^i n ) |
101 |
99 100
|
sseqtrrdi |
|- ( ( ( F " k ) C_ |^| y /\ ( F " u ) C_ n ) -> ( F " ( u i^i k ) ) C_ |^| ( y u. { n } ) ) |
102 |
101
|
expcom |
|- ( ( F " u ) C_ n -> ( ( F " k ) C_ |^| y -> ( F " ( u i^i k ) ) C_ |^| ( y u. { n } ) ) ) |
103 |
89 102
|
syl6 |
|- ( ph -> ( n = ( ( cls ` J ) ` ( F " u ) ) -> ( ( F " k ) C_ |^| y -> ( F " ( u i^i k ) ) C_ |^| ( y u. { n } ) ) ) ) |
104 |
103
|
impd |
|- ( ph -> ( ( n = ( ( cls ` J ) ` ( F " u ) ) /\ ( F " k ) C_ |^| y ) -> ( F " ( u i^i k ) ) C_ |^| ( y u. { n } ) ) ) |
105 |
|
imaeq2 |
|- ( m = ( u i^i k ) -> ( F " m ) = ( F " ( u i^i k ) ) ) |
106 |
105
|
sseq1d |
|- ( m = ( u i^i k ) -> ( ( F " m ) C_ |^| ( y u. { n } ) <-> ( F " ( u i^i k ) ) C_ |^| ( y u. { n } ) ) ) |
107 |
106
|
rspcev |
|- ( ( ( u i^i k ) e. ran ZZ>= /\ ( F " ( u i^i k ) ) C_ |^| ( y u. { n } ) ) -> E. m e. ran ZZ>= ( F " m ) C_ |^| ( y u. { n } ) ) |
108 |
107
|
expcom |
|- ( ( F " ( u i^i k ) ) C_ |^| ( y u. { n } ) -> ( ( u i^i k ) e. ran ZZ>= -> E. m e. ran ZZ>= ( F " m ) C_ |^| ( y u. { n } ) ) ) |
109 |
104 108
|
syl6 |
|- ( ph -> ( ( n = ( ( cls ` J ) ` ( F " u ) ) /\ ( F " k ) C_ |^| y ) -> ( ( u i^i k ) e. ran ZZ>= -> E. m e. ran ZZ>= ( F " m ) C_ |^| ( y u. { n } ) ) ) ) |
110 |
109
|
com23 |
|- ( ph -> ( ( u i^i k ) e. ran ZZ>= -> ( ( n = ( ( cls ` J ) ` ( F " u ) ) /\ ( F " k ) C_ |^| y ) -> E. m e. ran ZZ>= ( F " m ) C_ |^| ( y u. { n } ) ) ) ) |
111 |
83 110
|
syl5 |
|- ( ph -> ( ( u e. ran ZZ>= /\ k e. ran ZZ>= ) -> ( ( n = ( ( cls ` J ) ` ( F " u ) ) /\ ( F " k ) C_ |^| y ) -> E. m e. ran ZZ>= ( F " m ) C_ |^| ( y u. { n } ) ) ) ) |
112 |
111
|
rexlimdvv |
|- ( ph -> ( E. u e. ran ZZ>= E. k e. ran ZZ>= ( n = ( ( cls ` J ) ` ( F " u ) ) /\ ( F " k ) C_ |^| y ) -> E. m e. ran ZZ>= ( F " m ) C_ |^| ( y u. { n } ) ) ) |
113 |
|
reeanv |
|- ( E. u e. ran ZZ>= E. k e. ran ZZ>= ( n = ( ( cls ` J ) ` ( F " u ) ) /\ ( F " k ) C_ |^| y ) <-> ( E. u e. ran ZZ>= n = ( ( cls ` J ) ` ( F " u ) ) /\ E. k e. ran ZZ>= ( F " k ) C_ |^| y ) ) |
114 |
|
imaeq2 |
|- ( m = k -> ( F " m ) = ( F " k ) ) |
115 |
114
|
sseq1d |
|- ( m = k -> ( ( F " m ) C_ |^| ( y u. { n } ) <-> ( F " k ) C_ |^| ( y u. { n } ) ) ) |
116 |
115
|
cbvrexvw |
|- ( E. m e. ran ZZ>= ( F " m ) C_ |^| ( y u. { n } ) <-> E. k e. ran ZZ>= ( F " k ) C_ |^| ( y u. { n } ) ) |
117 |
112 113 116
|
3imtr3g |
|- ( ph -> ( ( E. u e. ran ZZ>= n = ( ( cls ` J ) ` ( F " u ) ) /\ E. k e. ran ZZ>= ( F " k ) C_ |^| y ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| ( y u. { n } ) ) ) |
118 |
117
|
expd |
|- ( ph -> ( E. u e. ran ZZ>= n = ( ( cls ` J ) ` ( F " u ) ) -> ( E. k e. ran ZZ>= ( F " k ) C_ |^| y -> E. k e. ran ZZ>= ( F " k ) C_ |^| ( y u. { n } ) ) ) ) |
119 |
82 118
|
syl5 |
|- ( ph -> ( A. k e. ( y u. { n } ) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) -> ( E. k e. ran ZZ>= ( F " k ) C_ |^| y -> E. k e. ran ZZ>= ( F " k ) C_ |^| ( y u. { n } ) ) ) ) |
120 |
119
|
imp |
|- ( ( ph /\ A. k e. ( y u. { n } ) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> ( E. k e. ran ZZ>= ( F " k ) C_ |^| y -> E. k e. ran ZZ>= ( F " k ) C_ |^| ( y u. { n } ) ) ) |
121 |
74 120
|
sylcom |
|- ( ( ( ph /\ A. k e. y E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| y ) -> ( ( ph /\ A. k e. ( y u. { n } ) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| ( y u. { n } ) ) ) |
122 |
121
|
a1i |
|- ( y e. Fin -> ( ( ( ph /\ A. k e. y E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| y ) -> ( ( ph /\ A. k e. ( y u. { n } ) E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| ( y u. { n } ) ) ) ) |
123 |
37 43 49 55 69 122
|
findcard2 |
|- ( r e. Fin -> ( ( ph /\ A. k e. r E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| r ) ) |
124 |
123
|
com12 |
|- ( ( ph /\ A. k e. r E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) ) -> ( r e. Fin -> E. k e. ran ZZ>= ( F " k ) C_ |^| r ) ) |
125 |
124
|
impr |
|- ( ( ph /\ ( A. k e. r E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) /\ r e. Fin ) ) -> E. k e. ran ZZ>= ( F " k ) C_ |^| r ) |
126 |
5
|
ffnd |
|- ( ph -> F Fn NN ) |
127 |
|
inss1 |
|- ( k i^i NN ) C_ k |
128 |
|
imass2 |
|- ( ( k i^i NN ) C_ k -> ( F " ( k i^i NN ) ) C_ ( F " k ) ) |
129 |
127 128
|
ax-mp |
|- ( F " ( k i^i NN ) ) C_ ( F " k ) |
130 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
131 |
|
1z |
|- 1 e. ZZ |
132 |
|
fnfvelrn |
|- ( ( ZZ>= Fn ZZ /\ 1 e. ZZ ) -> ( ZZ>= ` 1 ) e. ran ZZ>= ) |
133 |
58 131 132
|
mp2an |
|- ( ZZ>= ` 1 ) e. ran ZZ>= |
134 |
130 133
|
eqeltri |
|- NN e. ran ZZ>= |
135 |
|
uzin2 |
|- ( ( k e. ran ZZ>= /\ NN e. ran ZZ>= ) -> ( k i^i NN ) e. ran ZZ>= ) |
136 |
134 135
|
mpan2 |
|- ( k e. ran ZZ>= -> ( k i^i NN ) e. ran ZZ>= ) |
137 |
|
uzn0 |
|- ( ( k i^i NN ) e. ran ZZ>= -> ( k i^i NN ) =/= (/) ) |
138 |
136 137
|
syl |
|- ( k e. ran ZZ>= -> ( k i^i NN ) =/= (/) ) |
139 |
|
n0 |
|- ( ( k i^i NN ) =/= (/) <-> E. y y e. ( k i^i NN ) ) |
140 |
138 139
|
sylib |
|- ( k e. ran ZZ>= -> E. y y e. ( k i^i NN ) ) |
141 |
|
fnfun |
|- ( F Fn NN -> Fun F ) |
142 |
|
inss2 |
|- ( k i^i NN ) C_ NN |
143 |
|
fndm |
|- ( F Fn NN -> dom F = NN ) |
144 |
142 143
|
sseqtrrid |
|- ( F Fn NN -> ( k i^i NN ) C_ dom F ) |
145 |
|
funfvima2 |
|- ( ( Fun F /\ ( k i^i NN ) C_ dom F ) -> ( y e. ( k i^i NN ) -> ( F ` y ) e. ( F " ( k i^i NN ) ) ) ) |
146 |
141 144 145
|
syl2anc |
|- ( F Fn NN -> ( y e. ( k i^i NN ) -> ( F ` y ) e. ( F " ( k i^i NN ) ) ) ) |
147 |
|
ne0i |
|- ( ( F ` y ) e. ( F " ( k i^i NN ) ) -> ( F " ( k i^i NN ) ) =/= (/) ) |
148 |
146 147
|
syl6 |
|- ( F Fn NN -> ( y e. ( k i^i NN ) -> ( F " ( k i^i NN ) ) =/= (/) ) ) |
149 |
148
|
exlimdv |
|- ( F Fn NN -> ( E. y y e. ( k i^i NN ) -> ( F " ( k i^i NN ) ) =/= (/) ) ) |
150 |
140 149
|
syl5 |
|- ( F Fn NN -> ( k e. ran ZZ>= -> ( F " ( k i^i NN ) ) =/= (/) ) ) |
151 |
150
|
imp |
|- ( ( F Fn NN /\ k e. ran ZZ>= ) -> ( F " ( k i^i NN ) ) =/= (/) ) |
152 |
|
ssn0 |
|- ( ( ( F " ( k i^i NN ) ) C_ ( F " k ) /\ ( F " ( k i^i NN ) ) =/= (/) ) -> ( F " k ) =/= (/) ) |
153 |
129 151 152
|
sylancr |
|- ( ( F Fn NN /\ k e. ran ZZ>= ) -> ( F " k ) =/= (/) ) |
154 |
|
ssn0 |
|- ( ( ( F " k ) C_ |^| r /\ ( F " k ) =/= (/) ) -> |^| r =/= (/) ) |
155 |
154
|
expcom |
|- ( ( F " k ) =/= (/) -> ( ( F " k ) C_ |^| r -> |^| r =/= (/) ) ) |
156 |
153 155
|
syl |
|- ( ( F Fn NN /\ k e. ran ZZ>= ) -> ( ( F " k ) C_ |^| r -> |^| r =/= (/) ) ) |
157 |
156
|
rexlimdva |
|- ( F Fn NN -> ( E. k e. ran ZZ>= ( F " k ) C_ |^| r -> |^| r =/= (/) ) ) |
158 |
126 157
|
syl |
|- ( ph -> ( E. k e. ran ZZ>= ( F " k ) C_ |^| r -> |^| r =/= (/) ) ) |
159 |
158
|
adantr |
|- ( ( ph /\ ( A. k e. r E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) /\ r e. Fin ) ) -> ( E. k e. ran ZZ>= ( F " k ) C_ |^| r -> |^| r =/= (/) ) ) |
160 |
125 159
|
mpd |
|- ( ( ph /\ ( A. k e. r E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) /\ r e. Fin ) ) -> |^| r =/= (/) ) |
161 |
160
|
necomd |
|- ( ( ph /\ ( A. k e. r E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) /\ r e. Fin ) ) -> (/) =/= |^| r ) |
162 |
161
|
neneqd |
|- ( ( ph /\ ( A. k e. r E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) /\ r e. Fin ) ) -> -. (/) = |^| r ) |
163 |
31 162
|
sylan2b |
|- ( ( ph /\ r e. ( ~P { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } i^i Fin ) ) -> -. (/) = |^| r ) |
164 |
163
|
nrexdv |
|- ( ph -> -. E. r e. ( ~P { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } i^i Fin ) (/) = |^| r ) |
165 |
|
0ex |
|- (/) e. _V |
166 |
|
zex |
|- ZZ e. _V |
167 |
166
|
pwex |
|- ~P ZZ e. _V |
168 |
|
frn |
|- ( ZZ>= : ZZ --> ~P ZZ -> ran ZZ>= C_ ~P ZZ ) |
169 |
56 168
|
ax-mp |
|- ran ZZ>= C_ ~P ZZ |
170 |
167 169
|
ssexi |
|- ran ZZ>= e. _V |
171 |
170
|
abrexex |
|- { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } e. _V |
172 |
|
elfi |
|- ( ( (/) e. _V /\ { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } e. _V ) -> ( (/) e. ( fi ` { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) <-> E. r e. ( ~P { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } i^i Fin ) (/) = |^| r ) ) |
173 |
165 171 172
|
mp2an |
|- ( (/) e. ( fi ` { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) <-> E. r e. ( ~P { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } i^i Fin ) (/) = |^| r ) |
174 |
164 173
|
sylnibr |
|- ( ph -> -. (/) e. ( fi ` { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) ) |
175 |
|
cmptop |
|- ( J e. Comp -> J e. Top ) |
176 |
|
cmpfi |
|- ( J e. Top -> ( J e. Comp <-> A. m e. ~P ( Clsd ` J ) ( -. (/) e. ( fi ` m ) -> |^| m =/= (/) ) ) ) |
177 |
175 176
|
syl |
|- ( J e. Comp -> ( J e. Comp <-> A. m e. ~P ( Clsd ` J ) ( -. (/) e. ( fi ` m ) -> |^| m =/= (/) ) ) ) |
178 |
177
|
ibi |
|- ( J e. Comp -> A. m e. ~P ( Clsd ` J ) ( -. (/) e. ( fi ` m ) -> |^| m =/= (/) ) ) |
179 |
|
fveq2 |
|- ( m = { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } -> ( fi ` m ) = ( fi ` { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) ) |
180 |
179
|
eleq2d |
|- ( m = { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } -> ( (/) e. ( fi ` m ) <-> (/) e. ( fi ` { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) ) ) |
181 |
180
|
notbid |
|- ( m = { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } -> ( -. (/) e. ( fi ` m ) <-> -. (/) e. ( fi ` { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) ) ) |
182 |
|
inteq |
|- ( m = { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } -> |^| m = |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) |
183 |
182
|
neeq1d |
|- ( m = { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } -> ( |^| m =/= (/) <-> |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } =/= (/) ) ) |
184 |
|
n0 |
|- ( |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } =/= (/) <-> E. y y e. |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) |
185 |
183 184
|
bitrdi |
|- ( m = { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } -> ( |^| m =/= (/) <-> E. y y e. |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) ) |
186 |
181 185
|
imbi12d |
|- ( m = { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } -> ( ( -. (/) e. ( fi ` m ) -> |^| m =/= (/) ) <-> ( -. (/) e. ( fi ` { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) -> E. y y e. |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) ) ) |
187 |
186
|
rspccv |
|- ( A. m e. ~P ( Clsd ` J ) ( -. (/) e. ( fi ` m ) -> |^| m =/= (/) ) -> ( { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } e. ~P ( Clsd ` J ) -> ( -. (/) e. ( fi ` { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) -> E. y y e. |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) ) ) |
188 |
178 187
|
syl |
|- ( J e. Comp -> ( { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } e. ~P ( Clsd ` J ) -> ( -. (/) e. ( fi ` { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) -> E. y y e. |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) ) ) |
189 |
3 25 174 188
|
syl3c |
|- ( ph -> E. y y e. |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) |
190 |
|
lmrel |
|- Rel ( ~~>t ` J ) |
191 |
|
r19.23v |
|- ( A. u e. ran ZZ>= ( k = ( ( cls ` J ) ` ( F " u ) ) -> y e. k ) <-> ( E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) -> y e. k ) ) |
192 |
191
|
albii |
|- ( A. k A. u e. ran ZZ>= ( k = ( ( cls ` J ) ` ( F " u ) ) -> y e. k ) <-> A. k ( E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) -> y e. k ) ) |
193 |
|
fvex |
|- ( ( cls ` J ) ` ( F " u ) ) e. _V |
194 |
|
eleq2 |
|- ( k = ( ( cls ` J ) ` ( F " u ) ) -> ( y e. k <-> y e. ( ( cls ` J ) ` ( F " u ) ) ) ) |
195 |
193 194
|
ceqsalv |
|- ( A. k ( k = ( ( cls ` J ) ` ( F " u ) ) -> y e. k ) <-> y e. ( ( cls ` J ) ` ( F " u ) ) ) |
196 |
195
|
ralbii |
|- ( A. u e. ran ZZ>= A. k ( k = ( ( cls ` J ) ` ( F " u ) ) -> y e. k ) <-> A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) |
197 |
|
ralcom4 |
|- ( A. u e. ran ZZ>= A. k ( k = ( ( cls ` J ) ` ( F " u ) ) -> y e. k ) <-> A. k A. u e. ran ZZ>= ( k = ( ( cls ` J ) ` ( F " u ) ) -> y e. k ) ) |
198 |
196 197
|
bitr3i |
|- ( A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) <-> A. k A. u e. ran ZZ>= ( k = ( ( cls ` J ) ` ( F " u ) ) -> y e. k ) ) |
199 |
|
vex |
|- y e. _V |
200 |
199
|
elintab |
|- ( y e. |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } <-> A. k ( E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) -> y e. k ) ) |
201 |
192 198 200
|
3bitr4i |
|- ( A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) <-> y e. |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) |
202 |
|
eqid |
|- ( ( cls ` J ) ` ( F " NN ) ) = ( ( cls ` J ) ` ( F " NN ) ) |
203 |
|
imaeq2 |
|- ( u = NN -> ( F " u ) = ( F " NN ) ) |
204 |
203
|
fveq2d |
|- ( u = NN -> ( ( cls ` J ) ` ( F " u ) ) = ( ( cls ` J ) ` ( F " NN ) ) ) |
205 |
204
|
rspceeqv |
|- ( ( NN e. ran ZZ>= /\ ( ( cls ` J ) ` ( F " NN ) ) = ( ( cls ` J ) ` ( F " NN ) ) ) -> E. u e. ran ZZ>= ( ( cls ` J ) ` ( F " NN ) ) = ( ( cls ` J ) ` ( F " u ) ) ) |
206 |
134 202 205
|
mp2an |
|- E. u e. ran ZZ>= ( ( cls ` J ) ` ( F " NN ) ) = ( ( cls ` J ) ` ( F " u ) ) |
207 |
|
fvex |
|- ( ( cls ` J ) ` ( F " NN ) ) e. _V |
208 |
|
eqeq1 |
|- ( k = ( ( cls ` J ) ` ( F " NN ) ) -> ( k = ( ( cls ` J ) ` ( F " u ) ) <-> ( ( cls ` J ) ` ( F " NN ) ) = ( ( cls ` J ) ` ( F " u ) ) ) ) |
209 |
208
|
rexbidv |
|- ( k = ( ( cls ` J ) ` ( F " NN ) ) -> ( E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) <-> E. u e. ran ZZ>= ( ( cls ` J ) ` ( F " NN ) ) = ( ( cls ` J ) ` ( F " u ) ) ) ) |
210 |
207 209
|
elab |
|- ( ( ( cls ` J ) ` ( F " NN ) ) e. { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } <-> E. u e. ran ZZ>= ( ( cls ` J ) ` ( F " NN ) ) = ( ( cls ` J ) ` ( F " u ) ) ) |
211 |
206 210
|
mpbir |
|- ( ( cls ` J ) ` ( F " NN ) ) e. { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } |
212 |
|
intss1 |
|- ( ( ( cls ` J ) ` ( F " NN ) ) e. { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } -> |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } C_ ( ( cls ` J ) ` ( F " NN ) ) ) |
213 |
211 212
|
ax-mp |
|- |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } C_ ( ( cls ` J ) ` ( F " NN ) ) |
214 |
|
imassrn |
|- ( F " NN ) C_ ran F |
215 |
214 14
|
sstrid |
|- ( ph -> ( F " NN ) C_ U. J ) |
216 |
16
|
clsss3 |
|- ( ( J e. Top /\ ( F " NN ) C_ U. J ) -> ( ( cls ` J ) ` ( F " NN ) ) C_ U. J ) |
217 |
9 215 216
|
syl2anc |
|- ( ph -> ( ( cls ` J ) ` ( F " NN ) ) C_ U. J ) |
218 |
217 13
|
sseqtrrd |
|- ( ph -> ( ( cls ` J ) ` ( F " NN ) ) C_ X ) |
219 |
213 218
|
sstrid |
|- ( ph -> |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } C_ X ) |
220 |
219
|
sselda |
|- ( ( ph /\ y e. |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) -> y e. X ) |
221 |
201 220
|
sylan2b |
|- ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) -> y e. X ) |
222 |
|
1zzd |
|- ( ph -> 1 e. ZZ ) |
223 |
130 7 222
|
iscau3 |
|- ( ph -> ( F e. ( Cau ` D ) <-> ( F e. ( X ^pm CC ) /\ A. y e. RR+ E. m e. NN A. k e. ( ZZ>= ` m ) ( k e. dom F /\ ( F ` k ) e. X /\ A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y ) ) ) ) |
224 |
4 223
|
mpbid |
|- ( ph -> ( F e. ( X ^pm CC ) /\ A. y e. RR+ E. m e. NN A. k e. ( ZZ>= ` m ) ( k e. dom F /\ ( F ` k ) e. X /\ A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y ) ) ) |
225 |
224
|
simprd |
|- ( ph -> A. y e. RR+ E. m e. NN A. k e. ( ZZ>= ` m ) ( k e. dom F /\ ( F ` k ) e. X /\ A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y ) ) |
226 |
|
simp3 |
|- ( ( k e. dom F /\ ( F ` k ) e. X /\ A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y ) -> A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y ) |
227 |
226
|
ralimi |
|- ( A. k e. ( ZZ>= ` m ) ( k e. dom F /\ ( F ` k ) e. X /\ A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y ) -> A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y ) |
228 |
227
|
reximi |
|- ( E. m e. NN A. k e. ( ZZ>= ` m ) ( k e. dom F /\ ( F ` k ) e. X /\ A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y ) -> E. m e. NN A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y ) |
229 |
228
|
ralimi |
|- ( A. y e. RR+ E. m e. NN A. k e. ( ZZ>= ` m ) ( k e. dom F /\ ( F ` k ) e. X /\ A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y ) -> A. y e. RR+ E. m e. NN A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y ) |
230 |
225 229
|
syl |
|- ( ph -> A. y e. RR+ E. m e. NN A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y ) |
231 |
230
|
adantr |
|- ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) -> A. y e. RR+ E. m e. NN A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y ) |
232 |
|
rphalfcl |
|- ( r e. RR+ -> ( r / 2 ) e. RR+ ) |
233 |
|
breq2 |
|- ( y = ( r / 2 ) -> ( ( ( F ` k ) D ( F ` n ) ) < y <-> ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) ) ) |
234 |
233
|
2ralbidv |
|- ( y = ( r / 2 ) -> ( A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y <-> A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) ) ) |
235 |
234
|
rexbidv |
|- ( y = ( r / 2 ) -> ( E. m e. NN A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y <-> E. m e. NN A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) ) ) |
236 |
235
|
rspccva |
|- ( ( A. y e. RR+ E. m e. NN A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < y /\ ( r / 2 ) e. RR+ ) -> E. m e. NN A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) ) |
237 |
231 232 236
|
syl2an |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ r e. RR+ ) -> E. m e. NN A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) ) |
238 |
5
|
ffund |
|- ( ph -> Fun F ) |
239 |
238
|
ad2antrr |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> Fun F ) |
240 |
9
|
ad2antrr |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> J e. Top ) |
241 |
|
imassrn |
|- ( F " ( ZZ>= ` m ) ) C_ ran F |
242 |
241 14
|
sstrid |
|- ( ph -> ( F " ( ZZ>= ` m ) ) C_ U. J ) |
243 |
242
|
ad2antrr |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> ( F " ( ZZ>= ` m ) ) C_ U. J ) |
244 |
|
nnz |
|- ( m e. NN -> m e. ZZ ) |
245 |
|
fnfvelrn |
|- ( ( ZZ>= Fn ZZ /\ m e. ZZ ) -> ( ZZ>= ` m ) e. ran ZZ>= ) |
246 |
58 244 245
|
sylancr |
|- ( m e. NN -> ( ZZ>= ` m ) e. ran ZZ>= ) |
247 |
246
|
ad2antll |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> ( ZZ>= ` m ) e. ran ZZ>= ) |
248 |
|
simplr |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) |
249 |
|
imaeq2 |
|- ( u = ( ZZ>= ` m ) -> ( F " u ) = ( F " ( ZZ>= ` m ) ) ) |
250 |
249
|
fveq2d |
|- ( u = ( ZZ>= ` m ) -> ( ( cls ` J ) ` ( F " u ) ) = ( ( cls ` J ) ` ( F " ( ZZ>= ` m ) ) ) ) |
251 |
250
|
eleq2d |
|- ( u = ( ZZ>= ` m ) -> ( y e. ( ( cls ` J ) ` ( F " u ) ) <-> y e. ( ( cls ` J ) ` ( F " ( ZZ>= ` m ) ) ) ) ) |
252 |
251
|
rspcv |
|- ( ( ZZ>= ` m ) e. ran ZZ>= -> ( A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) -> y e. ( ( cls ` J ) ` ( F " ( ZZ>= ` m ) ) ) ) ) |
253 |
247 248 252
|
sylc |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> y e. ( ( cls ` J ) ` ( F " ( ZZ>= ` m ) ) ) ) |
254 |
7
|
ad2antrr |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> D e. ( *Met ` X ) ) |
255 |
221
|
adantr |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> y e. X ) |
256 |
232
|
ad2antrl |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> ( r / 2 ) e. RR+ ) |
257 |
256
|
rpxrd |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> ( r / 2 ) e. RR* ) |
258 |
1
|
blopn |
|- ( ( D e. ( *Met ` X ) /\ y e. X /\ ( r / 2 ) e. RR* ) -> ( y ( ball ` D ) ( r / 2 ) ) e. J ) |
259 |
254 255 257 258
|
syl3anc |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> ( y ( ball ` D ) ( r / 2 ) ) e. J ) |
260 |
|
blcntr |
|- ( ( D e. ( *Met ` X ) /\ y e. X /\ ( r / 2 ) e. RR+ ) -> y e. ( y ( ball ` D ) ( r / 2 ) ) ) |
261 |
254 255 256 260
|
syl3anc |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> y e. ( y ( ball ` D ) ( r / 2 ) ) ) |
262 |
16
|
clsndisj |
|- ( ( ( J e. Top /\ ( F " ( ZZ>= ` m ) ) C_ U. J /\ y e. ( ( cls ` J ) ` ( F " ( ZZ>= ` m ) ) ) ) /\ ( ( y ( ball ` D ) ( r / 2 ) ) e. J /\ y e. ( y ( ball ` D ) ( r / 2 ) ) ) ) -> ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) =/= (/) ) |
263 |
240 243 253 259 261 262
|
syl32anc |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) =/= (/) ) |
264 |
|
n0 |
|- ( ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) =/= (/) <-> E. n n e. ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) ) |
265 |
|
inss2 |
|- ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) C_ ( F " ( ZZ>= ` m ) ) |
266 |
265
|
sseli |
|- ( n e. ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) -> n e. ( F " ( ZZ>= ` m ) ) ) |
267 |
|
fvelima |
|- ( ( Fun F /\ n e. ( F " ( ZZ>= ` m ) ) ) -> E. k e. ( ZZ>= ` m ) ( F ` k ) = n ) |
268 |
266 267
|
sylan2 |
|- ( ( Fun F /\ n e. ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) ) -> E. k e. ( ZZ>= ` m ) ( F ` k ) = n ) |
269 |
|
inss1 |
|- ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) C_ ( y ( ball ` D ) ( r / 2 ) ) |
270 |
269
|
sseli |
|- ( n e. ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) -> n e. ( y ( ball ` D ) ( r / 2 ) ) ) |
271 |
270
|
adantl |
|- ( ( Fun F /\ n e. ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) ) -> n e. ( y ( ball ` D ) ( r / 2 ) ) ) |
272 |
|
eleq1a |
|- ( n e. ( y ( ball ` D ) ( r / 2 ) ) -> ( ( F ` k ) = n -> ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) ) |
273 |
271 272
|
syl |
|- ( ( Fun F /\ n e. ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) ) -> ( ( F ` k ) = n -> ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) ) |
274 |
273
|
reximdv |
|- ( ( Fun F /\ n e. ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) ) -> ( E. k e. ( ZZ>= ` m ) ( F ` k ) = n -> E. k e. ( ZZ>= ` m ) ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) ) |
275 |
268 274
|
mpd |
|- ( ( Fun F /\ n e. ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) ) -> E. k e. ( ZZ>= ` m ) ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) |
276 |
275
|
ex |
|- ( Fun F -> ( n e. ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) -> E. k e. ( ZZ>= ` m ) ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) ) |
277 |
276
|
exlimdv |
|- ( Fun F -> ( E. n n e. ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) -> E. k e. ( ZZ>= ` m ) ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) ) |
278 |
264 277
|
syl5bi |
|- ( Fun F -> ( ( ( y ( ball ` D ) ( r / 2 ) ) i^i ( F " ( ZZ>= ` m ) ) ) =/= (/) -> E. k e. ( ZZ>= ` m ) ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) ) |
279 |
239 263 278
|
sylc |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> E. k e. ( ZZ>= ` m ) ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) |
280 |
|
r19.29 |
|- ( ( A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) /\ E. k e. ( ZZ>= ` m ) ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) -> E. k e. ( ZZ>= ` m ) ( A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) ) |
281 |
|
uznnssnn |
|- ( m e. NN -> ( ZZ>= ` m ) C_ NN ) |
282 |
281
|
ad2antll |
|- ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) -> ( ZZ>= ` m ) C_ NN ) |
283 |
|
simprlr |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) |
284 |
7
|
ad3antrrr |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> D e. ( *Met ` X ) ) |
285 |
|
simplrl |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> r e. RR+ ) |
286 |
285 232
|
syl |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( r / 2 ) e. RR+ ) |
287 |
286
|
rpxrd |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( r / 2 ) e. RR* ) |
288 |
|
simpllr |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> y e. X ) |
289 |
5
|
ad3antrrr |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> F : NN --> X ) |
290 |
|
eluznn |
|- ( ( m e. NN /\ k e. ( ZZ>= ` m ) ) -> k e. NN ) |
291 |
290
|
ad2ant2lr |
|- ( ( ( r e. RR+ /\ m e. NN ) /\ ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) ) -> k e. NN ) |
292 |
291
|
ad2ant2lr |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> k e. NN ) |
293 |
289 292
|
ffvelrnd |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( F ` k ) e. X ) |
294 |
|
elbl3 |
|- ( ( ( D e. ( *Met ` X ) /\ ( r / 2 ) e. RR* ) /\ ( y e. X /\ ( F ` k ) e. X ) ) -> ( ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) <-> ( ( F ` k ) D y ) < ( r / 2 ) ) ) |
295 |
284 287 288 293 294
|
syl22anc |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) <-> ( ( F ` k ) D y ) < ( r / 2 ) ) ) |
296 |
283 295
|
mpbid |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( F ` k ) D y ) < ( r / 2 ) ) |
297 |
2
|
ad3antrrr |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> D e. ( Met ` X ) ) |
298 |
|
simprr |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> n e. ( ZZ>= ` k ) ) |
299 |
|
eluznn |
|- ( ( k e. NN /\ n e. ( ZZ>= ` k ) ) -> n e. NN ) |
300 |
292 298 299
|
syl2anc |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> n e. NN ) |
301 |
289 300
|
ffvelrnd |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( F ` n ) e. X ) |
302 |
|
metcl |
|- ( ( D e. ( Met ` X ) /\ ( F ` k ) e. X /\ ( F ` n ) e. X ) -> ( ( F ` k ) D ( F ` n ) ) e. RR ) |
303 |
297 293 301 302
|
syl3anc |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( F ` k ) D ( F ` n ) ) e. RR ) |
304 |
|
metcl |
|- ( ( D e. ( Met ` X ) /\ ( F ` k ) e. X /\ y e. X ) -> ( ( F ` k ) D y ) e. RR ) |
305 |
297 293 288 304
|
syl3anc |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( F ` k ) D y ) e. RR ) |
306 |
286
|
rpred |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( r / 2 ) e. RR ) |
307 |
|
lt2add |
|- ( ( ( ( ( F ` k ) D ( F ` n ) ) e. RR /\ ( ( F ` k ) D y ) e. RR ) /\ ( ( r / 2 ) e. RR /\ ( r / 2 ) e. RR ) ) -> ( ( ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) /\ ( ( F ` k ) D y ) < ( r / 2 ) ) -> ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) < ( ( r / 2 ) + ( r / 2 ) ) ) ) |
308 |
303 305 306 306 307
|
syl22anc |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) /\ ( ( F ` k ) D y ) < ( r / 2 ) ) -> ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) < ( ( r / 2 ) + ( r / 2 ) ) ) ) |
309 |
296 308
|
mpan2d |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) -> ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) < ( ( r / 2 ) + ( r / 2 ) ) ) ) |
310 |
285
|
rpcnd |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> r e. CC ) |
311 |
310
|
2halvesd |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( r / 2 ) + ( r / 2 ) ) = r ) |
312 |
311
|
breq2d |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) < ( ( r / 2 ) + ( r / 2 ) ) <-> ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) < r ) ) |
313 |
309 312
|
sylibd |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) -> ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) < r ) ) |
314 |
|
mettri2 |
|- ( ( D e. ( Met ` X ) /\ ( ( F ` k ) e. X /\ ( F ` n ) e. X /\ y e. X ) ) -> ( ( F ` n ) D y ) <_ ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) ) |
315 |
297 293 301 288 314
|
syl13anc |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( F ` n ) D y ) <_ ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) ) |
316 |
|
metcl |
|- ( ( D e. ( Met ` X ) /\ ( F ` n ) e. X /\ y e. X ) -> ( ( F ` n ) D y ) e. RR ) |
317 |
297 301 288 316
|
syl3anc |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( F ` n ) D y ) e. RR ) |
318 |
303 305
|
readdcld |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) e. RR ) |
319 |
285
|
rpred |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> r e. RR ) |
320 |
|
lelttr |
|- ( ( ( ( F ` n ) D y ) e. RR /\ ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) e. RR /\ r e. RR ) -> ( ( ( ( F ` n ) D y ) <_ ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) /\ ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) < r ) -> ( ( F ` n ) D y ) < r ) ) |
321 |
317 318 319 320
|
syl3anc |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( ( ( F ` n ) D y ) <_ ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) /\ ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) < r ) -> ( ( F ` n ) D y ) < r ) ) |
322 |
315 321
|
mpand |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( ( ( F ` k ) D ( F ` n ) ) + ( ( F ` k ) D y ) ) < r -> ( ( F ` n ) D y ) < r ) ) |
323 |
313 322
|
syld |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) /\ n e. ( ZZ>= ` k ) ) ) -> ( ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) -> ( ( F ` n ) D y ) < r ) ) |
324 |
323
|
anassrs |
|- ( ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) ) /\ n e. ( ZZ>= ` k ) ) -> ( ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) -> ( ( F ` n ) D y ) < r ) ) |
325 |
324
|
ralimdva |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ ( k e. ( ZZ>= ` m ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) ) -> ( A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) -> A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) ) |
326 |
325
|
expr |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ k e. ( ZZ>= ` m ) ) -> ( ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) -> ( A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) -> A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) ) ) |
327 |
326
|
com23 |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ k e. ( ZZ>= ` m ) ) -> ( A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) -> ( ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) -> A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) ) ) |
328 |
327
|
impd |
|- ( ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) /\ k e. ( ZZ>= ` m ) ) -> ( ( A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) -> A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) ) |
329 |
328
|
reximdva |
|- ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) -> ( E. k e. ( ZZ>= ` m ) ( A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) -> E. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) ) |
330 |
|
ssrexv |
|- ( ( ZZ>= ` m ) C_ NN -> ( E. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r -> E. k e. NN A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) ) |
331 |
282 329 330
|
sylsyld |
|- ( ( ( ph /\ y e. X ) /\ ( r e. RR+ /\ m e. NN ) ) -> ( E. k e. ( ZZ>= ` m ) ( A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) -> E. k e. NN A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) ) |
332 |
221 331
|
syldanl |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> ( E. k e. ( ZZ>= ` m ) ( A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) /\ ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) -> E. k e. NN A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) ) |
333 |
280 332
|
syl5 |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> ( ( A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) /\ E. k e. ( ZZ>= ` m ) ( F ` k ) e. ( y ( ball ` D ) ( r / 2 ) ) ) -> E. k e. NN A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) ) |
334 |
279 333
|
mpan2d |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ ( r e. RR+ /\ m e. NN ) ) -> ( A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) -> E. k e. NN A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) ) |
335 |
334
|
anassrs |
|- ( ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ r e. RR+ ) /\ m e. NN ) -> ( A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) -> E. k e. NN A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) ) |
336 |
335
|
rexlimdva |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ r e. RR+ ) -> ( E. m e. NN A. k e. ( ZZ>= ` m ) A. n e. ( ZZ>= ` k ) ( ( F ` k ) D ( F ` n ) ) < ( r / 2 ) -> E. k e. NN A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) ) |
337 |
237 336
|
mpd |
|- ( ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) /\ r e. RR+ ) -> E. k e. NN A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) |
338 |
337
|
ralrimiva |
|- ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) -> A. r e. RR+ E. k e. NN A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) |
339 |
|
eqidd |
|- ( ( ph /\ n e. NN ) -> ( F ` n ) = ( F ` n ) ) |
340 |
1 7 130 222 339 5
|
lmmbrf |
|- ( ph -> ( F ( ~~>t ` J ) y <-> ( y e. X /\ A. r e. RR+ E. k e. NN A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) ) ) |
341 |
340
|
adantr |
|- ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) -> ( F ( ~~>t ` J ) y <-> ( y e. X /\ A. r e. RR+ E. k e. NN A. n e. ( ZZ>= ` k ) ( ( F ` n ) D y ) < r ) ) ) |
342 |
221 338 341
|
mpbir2and |
|- ( ( ph /\ A. u e. ran ZZ>= y e. ( ( cls ` J ) ` ( F " u ) ) ) -> F ( ~~>t ` J ) y ) |
343 |
201 342
|
sylan2br |
|- ( ( ph /\ y e. |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) -> F ( ~~>t ` J ) y ) |
344 |
|
releldm |
|- ( ( Rel ( ~~>t ` J ) /\ F ( ~~>t ` J ) y ) -> F e. dom ( ~~>t ` J ) ) |
345 |
190 343 344
|
sylancr |
|- ( ( ph /\ y e. |^| { k | E. u e. ran ZZ>= k = ( ( cls ` J ) ` ( F " u ) ) } ) -> F e. dom ( ~~>t ` J ) ) |
346 |
189 345
|
exlimddv |
|- ( ph -> F e. dom ( ~~>t ` J ) ) |