Step |
Hyp |
Ref |
Expression |
1 |
|
nghmcn.j |
|- J = ( TopOpen ` S ) |
2 |
|
nghmcn.k |
|- K = ( TopOpen ` T ) |
3 |
|
nghmghm |
|- ( F e. ( S NGHom T ) -> F e. ( S GrpHom T ) ) |
4 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
5 |
|
eqid |
|- ( Base ` T ) = ( Base ` T ) |
6 |
4 5
|
ghmf |
|- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) ) |
7 |
3 6
|
syl |
|- ( F e. ( S NGHom T ) -> F : ( Base ` S ) --> ( Base ` T ) ) |
8 |
|
simprr |
|- ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) -> r e. RR+ ) |
9 |
|
eqid |
|- ( S normOp T ) = ( S normOp T ) |
10 |
9
|
nghmcl |
|- ( F e. ( S NGHom T ) -> ( ( S normOp T ) ` F ) e. RR ) |
11 |
|
nghmrcl1 |
|- ( F e. ( S NGHom T ) -> S e. NrmGrp ) |
12 |
|
nghmrcl2 |
|- ( F e. ( S NGHom T ) -> T e. NrmGrp ) |
13 |
9
|
nmoge0 |
|- ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) -> 0 <_ ( ( S normOp T ) ` F ) ) |
14 |
11 12 3 13
|
syl3anc |
|- ( F e. ( S NGHom T ) -> 0 <_ ( ( S normOp T ) ` F ) ) |
15 |
10 14
|
ge0p1rpd |
|- ( F e. ( S NGHom T ) -> ( ( ( S normOp T ) ` F ) + 1 ) e. RR+ ) |
16 |
15
|
adantr |
|- ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) -> ( ( ( S normOp T ) ` F ) + 1 ) e. RR+ ) |
17 |
8 16
|
rpdivcld |
|- ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) -> ( r / ( ( ( S normOp T ) ` F ) + 1 ) ) e. RR+ ) |
18 |
|
ngpms |
|- ( S e. NrmGrp -> S e. MetSp ) |
19 |
11 18
|
syl |
|- ( F e. ( S NGHom T ) -> S e. MetSp ) |
20 |
19
|
ad2antrr |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> S e. MetSp ) |
21 |
|
simplrl |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> x e. ( Base ` S ) ) |
22 |
|
simpr |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> y e. ( Base ` S ) ) |
23 |
|
eqid |
|- ( dist ` S ) = ( dist ` S ) |
24 |
4 23
|
mscl |
|- ( ( S e. MetSp /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( x ( dist ` S ) y ) e. RR ) |
25 |
20 21 22 24
|
syl3anc |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( x ( dist ` S ) y ) e. RR ) |
26 |
8
|
adantr |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> r e. RR+ ) |
27 |
26
|
rpred |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> r e. RR ) |
28 |
15
|
ad2antrr |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( ( S normOp T ) ` F ) + 1 ) e. RR+ ) |
29 |
25 27 28
|
ltmuldiv2d |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( ( ( ( S normOp T ) ` F ) + 1 ) x. ( x ( dist ` S ) y ) ) < r <-> ( x ( dist ` S ) y ) < ( r / ( ( ( S normOp T ) ` F ) + 1 ) ) ) ) |
30 |
|
ngpms |
|- ( T e. NrmGrp -> T e. MetSp ) |
31 |
12 30
|
syl |
|- ( F e. ( S NGHom T ) -> T e. MetSp ) |
32 |
31
|
ad2antrr |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> T e. MetSp ) |
33 |
7
|
ad2antrr |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> F : ( Base ` S ) --> ( Base ` T ) ) |
34 |
33 21
|
ffvelrnd |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( F ` x ) e. ( Base ` T ) ) |
35 |
33 22
|
ffvelrnd |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( F ` y ) e. ( Base ` T ) ) |
36 |
|
eqid |
|- ( dist ` T ) = ( dist ` T ) |
37 |
5 36
|
mscl |
|- ( ( T e. MetSp /\ ( F ` x ) e. ( Base ` T ) /\ ( F ` y ) e. ( Base ` T ) ) -> ( ( F ` x ) ( dist ` T ) ( F ` y ) ) e. RR ) |
38 |
32 34 35 37
|
syl3anc |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( F ` x ) ( dist ` T ) ( F ` y ) ) e. RR ) |
39 |
10
|
ad2antrr |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( S normOp T ) ` F ) e. RR ) |
40 |
39 25
|
remulcld |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( ( S normOp T ) ` F ) x. ( x ( dist ` S ) y ) ) e. RR ) |
41 |
28
|
rpred |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( ( S normOp T ) ` F ) + 1 ) e. RR ) |
42 |
41 25
|
remulcld |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( ( ( S normOp T ) ` F ) + 1 ) x. ( x ( dist ` S ) y ) ) e. RR ) |
43 |
9 4 23 36
|
nmods |
|- ( ( F e. ( S NGHom T ) /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( ( F ` x ) ( dist ` T ) ( F ` y ) ) <_ ( ( ( S normOp T ) ` F ) x. ( x ( dist ` S ) y ) ) ) |
44 |
43
|
3expa |
|- ( ( ( F e. ( S NGHom T ) /\ x e. ( Base ` S ) ) /\ y e. ( Base ` S ) ) -> ( ( F ` x ) ( dist ` T ) ( F ` y ) ) <_ ( ( ( S normOp T ) ` F ) x. ( x ( dist ` S ) y ) ) ) |
45 |
44
|
adantlrr |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( F ` x ) ( dist ` T ) ( F ` y ) ) <_ ( ( ( S normOp T ) ` F ) x. ( x ( dist ` S ) y ) ) ) |
46 |
|
msxms |
|- ( S e. MetSp -> S e. *MetSp ) |
47 |
20 46
|
syl |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> S e. *MetSp ) |
48 |
4 23
|
xmsge0 |
|- ( ( S e. *MetSp /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> 0 <_ ( x ( dist ` S ) y ) ) |
49 |
47 21 22 48
|
syl3anc |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> 0 <_ ( x ( dist ` S ) y ) ) |
50 |
39
|
lep1d |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( S normOp T ) ` F ) <_ ( ( ( S normOp T ) ` F ) + 1 ) ) |
51 |
39 41 25 49 50
|
lemul1ad |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( ( S normOp T ) ` F ) x. ( x ( dist ` S ) y ) ) <_ ( ( ( ( S normOp T ) ` F ) + 1 ) x. ( x ( dist ` S ) y ) ) ) |
52 |
38 40 42 45 51
|
letrd |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( F ` x ) ( dist ` T ) ( F ` y ) ) <_ ( ( ( ( S normOp T ) ` F ) + 1 ) x. ( x ( dist ` S ) y ) ) ) |
53 |
|
lelttr |
|- ( ( ( ( F ` x ) ( dist ` T ) ( F ` y ) ) e. RR /\ ( ( ( ( S normOp T ) ` F ) + 1 ) x. ( x ( dist ` S ) y ) ) e. RR /\ r e. RR ) -> ( ( ( ( F ` x ) ( dist ` T ) ( F ` y ) ) <_ ( ( ( ( S normOp T ) ` F ) + 1 ) x. ( x ( dist ` S ) y ) ) /\ ( ( ( ( S normOp T ) ` F ) + 1 ) x. ( x ( dist ` S ) y ) ) < r ) -> ( ( F ` x ) ( dist ` T ) ( F ` y ) ) < r ) ) |
54 |
38 42 27 53
|
syl3anc |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( ( ( F ` x ) ( dist ` T ) ( F ` y ) ) <_ ( ( ( ( S normOp T ) ` F ) + 1 ) x. ( x ( dist ` S ) y ) ) /\ ( ( ( ( S normOp T ) ` F ) + 1 ) x. ( x ( dist ` S ) y ) ) < r ) -> ( ( F ` x ) ( dist ` T ) ( F ` y ) ) < r ) ) |
55 |
52 54
|
mpand |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( ( ( ( S normOp T ) ` F ) + 1 ) x. ( x ( dist ` S ) y ) ) < r -> ( ( F ` x ) ( dist ` T ) ( F ` y ) ) < r ) ) |
56 |
29 55
|
sylbird |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( x ( dist ` S ) y ) < ( r / ( ( ( S normOp T ) ` F ) + 1 ) ) -> ( ( F ` x ) ( dist ` T ) ( F ` y ) ) < r ) ) |
57 |
21 22
|
ovresd |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( x ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) y ) = ( x ( dist ` S ) y ) ) |
58 |
57
|
breq1d |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( x ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) y ) < ( r / ( ( ( S normOp T ) ` F ) + 1 ) ) <-> ( x ( dist ` S ) y ) < ( r / ( ( ( S normOp T ) ` F ) + 1 ) ) ) ) |
59 |
34 35
|
ovresd |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( F ` x ) ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ( F ` y ) ) = ( ( F ` x ) ( dist ` T ) ( F ` y ) ) ) |
60 |
59
|
breq1d |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( ( F ` x ) ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ( F ` y ) ) < r <-> ( ( F ` x ) ( dist ` T ) ( F ` y ) ) < r ) ) |
61 |
56 58 60
|
3imtr4d |
|- ( ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) /\ y e. ( Base ` S ) ) -> ( ( x ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) y ) < ( r / ( ( ( S normOp T ) ` F ) + 1 ) ) -> ( ( F ` x ) ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ( F ` y ) ) < r ) ) |
62 |
61
|
ralrimiva |
|- ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) -> A. y e. ( Base ` S ) ( ( x ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) y ) < ( r / ( ( ( S normOp T ) ` F ) + 1 ) ) -> ( ( F ` x ) ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ( F ` y ) ) < r ) ) |
63 |
|
breq2 |
|- ( s = ( r / ( ( ( S normOp T ) ` F ) + 1 ) ) -> ( ( x ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) y ) < s <-> ( x ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) y ) < ( r / ( ( ( S normOp T ) ` F ) + 1 ) ) ) ) |
64 |
63
|
rspceaimv |
|- ( ( ( r / ( ( ( S normOp T ) ` F ) + 1 ) ) e. RR+ /\ A. y e. ( Base ` S ) ( ( x ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) y ) < ( r / ( ( ( S normOp T ) ` F ) + 1 ) ) -> ( ( F ` x ) ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ( F ` y ) ) < r ) ) -> E. s e. RR+ A. y e. ( Base ` S ) ( ( x ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) y ) < s -> ( ( F ` x ) ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ( F ` y ) ) < r ) ) |
65 |
17 62 64
|
syl2anc |
|- ( ( F e. ( S NGHom T ) /\ ( x e. ( Base ` S ) /\ r e. RR+ ) ) -> E. s e. RR+ A. y e. ( Base ` S ) ( ( x ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) y ) < s -> ( ( F ` x ) ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ( F ` y ) ) < r ) ) |
66 |
65
|
ralrimivva |
|- ( F e. ( S NGHom T ) -> A. x e. ( Base ` S ) A. r e. RR+ E. s e. RR+ A. y e. ( Base ` S ) ( ( x ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) y ) < s -> ( ( F ` x ) ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ( F ` y ) ) < r ) ) |
67 |
|
eqid |
|- ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) = ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) |
68 |
4 67
|
xmsxmet |
|- ( S e. *MetSp -> ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) e. ( *Met ` ( Base ` S ) ) ) |
69 |
19 46 68
|
3syl |
|- ( F e. ( S NGHom T ) -> ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) e. ( *Met ` ( Base ` S ) ) ) |
70 |
|
msxms |
|- ( T e. MetSp -> T e. *MetSp ) |
71 |
|
eqid |
|- ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) = ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) |
72 |
5 71
|
xmsxmet |
|- ( T e. *MetSp -> ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) e. ( *Met ` ( Base ` T ) ) ) |
73 |
31 70 72
|
3syl |
|- ( F e. ( S NGHom T ) -> ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) e. ( *Met ` ( Base ` T ) ) ) |
74 |
|
eqid |
|- ( MetOpen ` ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) ) = ( MetOpen ` ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) ) |
75 |
|
eqid |
|- ( MetOpen ` ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ) = ( MetOpen ` ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ) |
76 |
74 75
|
metcn |
|- ( ( ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) e. ( *Met ` ( Base ` S ) ) /\ ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) e. ( *Met ` ( Base ` T ) ) ) -> ( F e. ( ( MetOpen ` ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) ) Cn ( MetOpen ` ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ) ) <-> ( F : ( Base ` S ) --> ( Base ` T ) /\ A. x e. ( Base ` S ) A. r e. RR+ E. s e. RR+ A. y e. ( Base ` S ) ( ( x ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) y ) < s -> ( ( F ` x ) ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ( F ` y ) ) < r ) ) ) ) |
77 |
69 73 76
|
syl2anc |
|- ( F e. ( S NGHom T ) -> ( F e. ( ( MetOpen ` ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) ) Cn ( MetOpen ` ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ) ) <-> ( F : ( Base ` S ) --> ( Base ` T ) /\ A. x e. ( Base ` S ) A. r e. RR+ E. s e. RR+ A. y e. ( Base ` S ) ( ( x ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) y ) < s -> ( ( F ` x ) ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ( F ` y ) ) < r ) ) ) ) |
78 |
7 66 77
|
mpbir2and |
|- ( F e. ( S NGHom T ) -> F e. ( ( MetOpen ` ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) ) Cn ( MetOpen ` ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ) ) ) |
79 |
1 4 67
|
mstopn |
|- ( S e. MetSp -> J = ( MetOpen ` ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) ) ) |
80 |
19 79
|
syl |
|- ( F e. ( S NGHom T ) -> J = ( MetOpen ` ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) ) ) |
81 |
2 5 71
|
mstopn |
|- ( T e. MetSp -> K = ( MetOpen ` ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ) ) |
82 |
31 81
|
syl |
|- ( F e. ( S NGHom T ) -> K = ( MetOpen ` ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ) ) |
83 |
80 82
|
oveq12d |
|- ( F e. ( S NGHom T ) -> ( J Cn K ) = ( ( MetOpen ` ( ( dist ` S ) |` ( ( Base ` S ) X. ( Base ` S ) ) ) ) Cn ( MetOpen ` ( ( dist ` T ) |` ( ( Base ` T ) X. ( Base ` T ) ) ) ) ) ) |
84 |
78 83
|
eleqtrrd |
|- ( F e. ( S NGHom T ) -> F e. ( J Cn K ) ) |