Step |
Hyp |
Ref |
Expression |
1 |
|
nmofval.1 |
|- N = ( S normOp T ) |
2 |
|
nmoi.2 |
|- V = ( Base ` S ) |
3 |
|
nmoi.3 |
|- L = ( norm ` S ) |
4 |
|
nmoi.4 |
|- M = ( norm ` T ) |
5 |
|
nmoi2.z |
|- .0. = ( 0g ` S ) |
6 |
|
nmoleub.1 |
|- ( ph -> S e. NrmGrp ) |
7 |
|
nmoleub.2 |
|- ( ph -> T e. NrmGrp ) |
8 |
|
nmoleub.3 |
|- ( ph -> F e. ( S GrpHom T ) ) |
9 |
|
nmoleub.4 |
|- ( ph -> A e. RR* ) |
10 |
|
nmoleub.5 |
|- ( ph -> 0 <_ A ) |
11 |
7
|
ad2antrr |
|- ( ( ( ph /\ ( N ` F ) <_ A ) /\ ( x e. V /\ x =/= .0. ) ) -> T e. NrmGrp ) |
12 |
|
eqid |
|- ( Base ` T ) = ( Base ` T ) |
13 |
2 12
|
ghmf |
|- ( F e. ( S GrpHom T ) -> F : V --> ( Base ` T ) ) |
14 |
8 13
|
syl |
|- ( ph -> F : V --> ( Base ` T ) ) |
15 |
14
|
ad2antrr |
|- ( ( ( ph /\ ( N ` F ) <_ A ) /\ ( x e. V /\ x =/= .0. ) ) -> F : V --> ( Base ` T ) ) |
16 |
|
simprl |
|- ( ( ( ph /\ ( N ` F ) <_ A ) /\ ( x e. V /\ x =/= .0. ) ) -> x e. V ) |
17 |
|
ffvelrn |
|- ( ( F : V --> ( Base ` T ) /\ x e. V ) -> ( F ` x ) e. ( Base ` T ) ) |
18 |
15 16 17
|
syl2anc |
|- ( ( ( ph /\ ( N ` F ) <_ A ) /\ ( x e. V /\ x =/= .0. ) ) -> ( F ` x ) e. ( Base ` T ) ) |
19 |
12 4
|
nmcl |
|- ( ( T e. NrmGrp /\ ( F ` x ) e. ( Base ` T ) ) -> ( M ` ( F ` x ) ) e. RR ) |
20 |
11 18 19
|
syl2anc |
|- ( ( ( ph /\ ( N ` F ) <_ A ) /\ ( x e. V /\ x =/= .0. ) ) -> ( M ` ( F ` x ) ) e. RR ) |
21 |
6
|
adantr |
|- ( ( ph /\ ( N ` F ) <_ A ) -> S e. NrmGrp ) |
22 |
2 3 5
|
nmrpcl |
|- ( ( S e. NrmGrp /\ x e. V /\ x =/= .0. ) -> ( L ` x ) e. RR+ ) |
23 |
22
|
3expb |
|- ( ( S e. NrmGrp /\ ( x e. V /\ x =/= .0. ) ) -> ( L ` x ) e. RR+ ) |
24 |
21 23
|
sylan |
|- ( ( ( ph /\ ( N ` F ) <_ A ) /\ ( x e. V /\ x =/= .0. ) ) -> ( L ` x ) e. RR+ ) |
25 |
20 24
|
rerpdivcld |
|- ( ( ( ph /\ ( N ` F ) <_ A ) /\ ( x e. V /\ x =/= .0. ) ) -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) e. RR ) |
26 |
25
|
rexrd |
|- ( ( ( ph /\ ( N ` F ) <_ A ) /\ ( x e. V /\ x =/= .0. ) ) -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) e. RR* ) |
27 |
1
|
nmocl |
|- ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) -> ( N ` F ) e. RR* ) |
28 |
6 7 8 27
|
syl3anc |
|- ( ph -> ( N ` F ) e. RR* ) |
29 |
28
|
ad2antrr |
|- ( ( ( ph /\ ( N ` F ) <_ A ) /\ ( x e. V /\ x =/= .0. ) ) -> ( N ` F ) e. RR* ) |
30 |
9
|
ad2antrr |
|- ( ( ( ph /\ ( N ` F ) <_ A ) /\ ( x e. V /\ x =/= .0. ) ) -> A e. RR* ) |
31 |
6 7 8
|
3jca |
|- ( ph -> ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) ) |
32 |
31
|
adantr |
|- ( ( ph /\ ( N ` F ) <_ A ) -> ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) ) |
33 |
1 2 3 4 5
|
nmoi2 |
|- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( x e. V /\ x =/= .0. ) ) -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ ( N ` F ) ) |
34 |
32 33
|
sylan |
|- ( ( ( ph /\ ( N ` F ) <_ A ) /\ ( x e. V /\ x =/= .0. ) ) -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ ( N ` F ) ) |
35 |
|
simplr |
|- ( ( ( ph /\ ( N ` F ) <_ A ) /\ ( x e. V /\ x =/= .0. ) ) -> ( N ` F ) <_ A ) |
36 |
26 29 30 34 35
|
xrletrd |
|- ( ( ( ph /\ ( N ` F ) <_ A ) /\ ( x e. V /\ x =/= .0. ) ) -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) |
37 |
36
|
expr |
|- ( ( ( ph /\ ( N ` F ) <_ A ) /\ x e. V ) -> ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) ) |
38 |
37
|
ralrimiva |
|- ( ( ph /\ ( N ` F ) <_ A ) -> A. x e. V ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) ) |
39 |
|
0le0 |
|- 0 <_ 0 |
40 |
|
simpllr |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x = .0. ) -> A e. RR ) |
41 |
40
|
recnd |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x = .0. ) -> A e. CC ) |
42 |
41
|
mul01d |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x = .0. ) -> ( A x. 0 ) = 0 ) |
43 |
39 42
|
breqtrrid |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x = .0. ) -> 0 <_ ( A x. 0 ) ) |
44 |
|
fveq2 |
|- ( x = .0. -> ( F ` x ) = ( F ` .0. ) ) |
45 |
8
|
ad2antrr |
|- ( ( ( ph /\ A e. RR ) /\ x e. V ) -> F e. ( S GrpHom T ) ) |
46 |
|
eqid |
|- ( 0g ` T ) = ( 0g ` T ) |
47 |
5 46
|
ghmid |
|- ( F e. ( S GrpHom T ) -> ( F ` .0. ) = ( 0g ` T ) ) |
48 |
45 47
|
syl |
|- ( ( ( ph /\ A e. RR ) /\ x e. V ) -> ( F ` .0. ) = ( 0g ` T ) ) |
49 |
44 48
|
sylan9eqr |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x = .0. ) -> ( F ` x ) = ( 0g ` T ) ) |
50 |
49
|
fveq2d |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x = .0. ) -> ( M ` ( F ` x ) ) = ( M ` ( 0g ` T ) ) ) |
51 |
7
|
ad3antrrr |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x = .0. ) -> T e. NrmGrp ) |
52 |
4 46
|
nm0 |
|- ( T e. NrmGrp -> ( M ` ( 0g ` T ) ) = 0 ) |
53 |
51 52
|
syl |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x = .0. ) -> ( M ` ( 0g ` T ) ) = 0 ) |
54 |
50 53
|
eqtrd |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x = .0. ) -> ( M ` ( F ` x ) ) = 0 ) |
55 |
|
fveq2 |
|- ( x = .0. -> ( L ` x ) = ( L ` .0. ) ) |
56 |
6
|
ad2antrr |
|- ( ( ( ph /\ A e. RR ) /\ x e. V ) -> S e. NrmGrp ) |
57 |
3 5
|
nm0 |
|- ( S e. NrmGrp -> ( L ` .0. ) = 0 ) |
58 |
56 57
|
syl |
|- ( ( ( ph /\ A e. RR ) /\ x e. V ) -> ( L ` .0. ) = 0 ) |
59 |
55 58
|
sylan9eqr |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x = .0. ) -> ( L ` x ) = 0 ) |
60 |
59
|
oveq2d |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x = .0. ) -> ( A x. ( L ` x ) ) = ( A x. 0 ) ) |
61 |
43 54 60
|
3brtr4d |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x = .0. ) -> ( M ` ( F ` x ) ) <_ ( A x. ( L ` x ) ) ) |
62 |
61
|
a1d |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x = .0. ) -> ( ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) -> ( M ` ( F ` x ) ) <_ ( A x. ( L ` x ) ) ) ) |
63 |
|
simpr |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x =/= .0. ) -> x =/= .0. ) |
64 |
7
|
ad2antrr |
|- ( ( ( ph /\ A e. RR ) /\ x e. V ) -> T e. NrmGrp ) |
65 |
14
|
adantr |
|- ( ( ph /\ A e. RR ) -> F : V --> ( Base ` T ) ) |
66 |
65 17
|
sylan |
|- ( ( ( ph /\ A e. RR ) /\ x e. V ) -> ( F ` x ) e. ( Base ` T ) ) |
67 |
64 66 19
|
syl2anc |
|- ( ( ( ph /\ A e. RR ) /\ x e. V ) -> ( M ` ( F ` x ) ) e. RR ) |
68 |
67
|
adantr |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x =/= .0. ) -> ( M ` ( F ` x ) ) e. RR ) |
69 |
|
simpllr |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x =/= .0. ) -> A e. RR ) |
70 |
6
|
adantr |
|- ( ( ph /\ A e. RR ) -> S e. NrmGrp ) |
71 |
22
|
3expa |
|- ( ( ( S e. NrmGrp /\ x e. V ) /\ x =/= .0. ) -> ( L ` x ) e. RR+ ) |
72 |
70 71
|
sylanl1 |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x =/= .0. ) -> ( L ` x ) e. RR+ ) |
73 |
68 69 72
|
ledivmul2d |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x =/= .0. ) -> ( ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A <-> ( M ` ( F ` x ) ) <_ ( A x. ( L ` x ) ) ) ) |
74 |
73
|
biimpd |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x =/= .0. ) -> ( ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A -> ( M ` ( F ` x ) ) <_ ( A x. ( L ` x ) ) ) ) |
75 |
63 74
|
embantd |
|- ( ( ( ( ph /\ A e. RR ) /\ x e. V ) /\ x =/= .0. ) -> ( ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) -> ( M ` ( F ` x ) ) <_ ( A x. ( L ` x ) ) ) ) |
76 |
62 75
|
pm2.61dane |
|- ( ( ( ph /\ A e. RR ) /\ x e. V ) -> ( ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) -> ( M ` ( F ` x ) ) <_ ( A x. ( L ` x ) ) ) ) |
77 |
76
|
ralimdva |
|- ( ( ph /\ A e. RR ) -> ( A. x e. V ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) -> A. x e. V ( M ` ( F ` x ) ) <_ ( A x. ( L ` x ) ) ) ) |
78 |
7
|
adantr |
|- ( ( ph /\ A e. RR ) -> T e. NrmGrp ) |
79 |
8
|
adantr |
|- ( ( ph /\ A e. RR ) -> F e. ( S GrpHom T ) ) |
80 |
|
simpr |
|- ( ( ph /\ A e. RR ) -> A e. RR ) |
81 |
10
|
adantr |
|- ( ( ph /\ A e. RR ) -> 0 <_ A ) |
82 |
1 2 3 4
|
nmolb |
|- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ A e. RR /\ 0 <_ A ) -> ( A. x e. V ( M ` ( F ` x ) ) <_ ( A x. ( L ` x ) ) -> ( N ` F ) <_ A ) ) |
83 |
70 78 79 80 81 82
|
syl311anc |
|- ( ( ph /\ A e. RR ) -> ( A. x e. V ( M ` ( F ` x ) ) <_ ( A x. ( L ` x ) ) -> ( N ` F ) <_ A ) ) |
84 |
77 83
|
syld |
|- ( ( ph /\ A e. RR ) -> ( A. x e. V ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) -> ( N ` F ) <_ A ) ) |
85 |
84
|
imp |
|- ( ( ( ph /\ A e. RR ) /\ A. x e. V ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) ) -> ( N ` F ) <_ A ) |
86 |
85
|
an32s |
|- ( ( ( ph /\ A. x e. V ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) ) /\ A e. RR ) -> ( N ` F ) <_ A ) |
87 |
28
|
ad2antrr |
|- ( ( ( ph /\ A. x e. V ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) ) /\ A = +oo ) -> ( N ` F ) e. RR* ) |
88 |
|
pnfge |
|- ( ( N ` F ) e. RR* -> ( N ` F ) <_ +oo ) |
89 |
87 88
|
syl |
|- ( ( ( ph /\ A. x e. V ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) ) /\ A = +oo ) -> ( N ` F ) <_ +oo ) |
90 |
|
simpr |
|- ( ( ( ph /\ A. x e. V ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) ) /\ A = +oo ) -> A = +oo ) |
91 |
89 90
|
breqtrrd |
|- ( ( ( ph /\ A. x e. V ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) ) /\ A = +oo ) -> ( N ` F ) <_ A ) |
92 |
|
ge0nemnf |
|- ( ( A e. RR* /\ 0 <_ A ) -> A =/= -oo ) |
93 |
9 10 92
|
syl2anc |
|- ( ph -> A =/= -oo ) |
94 |
9 93
|
jca |
|- ( ph -> ( A e. RR* /\ A =/= -oo ) ) |
95 |
|
xrnemnf |
|- ( ( A e. RR* /\ A =/= -oo ) <-> ( A e. RR \/ A = +oo ) ) |
96 |
94 95
|
sylib |
|- ( ph -> ( A e. RR \/ A = +oo ) ) |
97 |
96
|
adantr |
|- ( ( ph /\ A. x e. V ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) ) -> ( A e. RR \/ A = +oo ) ) |
98 |
86 91 97
|
mpjaodan |
|- ( ( ph /\ A. x e. V ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) ) -> ( N ` F ) <_ A ) |
99 |
38 98
|
impbida |
|- ( ph -> ( ( N ` F ) <_ A <-> A. x e. V ( x =/= .0. -> ( ( M ` ( F ` x ) ) / ( L ` x ) ) <_ A ) ) ) |