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