Step | Hyp | Ref
| Expression |
1 | | nmoleub2.n |
. 2
⊢ 𝑁 = (𝑆 normOp 𝑇) |
2 | | nmoleub2.v |
. 2
⊢ 𝑉 = (Base‘𝑆) |
3 | | nmoleub2.l |
. 2
⊢ 𝐿 = (norm‘𝑆) |
4 | | nmoleub2.m |
. 2
⊢ 𝑀 = (norm‘𝑇) |
5 | | nmoleub2.g |
. 2
⊢ 𝐺 = (Scalar‘𝑆) |
6 | | nmoleub2.w |
. 2
⊢ 𝐾 = (Base‘𝐺) |
7 | | nmoleub2.s |
. 2
⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩
ℂMod)) |
8 | | nmoleub2.t |
. 2
⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩
ℂMod)) |
9 | | nmoleub2.f |
. 2
⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) |
10 | | nmoleub2.a |
. 2
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
11 | | nmoleub2.r |
. 2
⊢ (𝜑 → 𝑅 ∈
ℝ+) |
12 | | lmghm 19935 |
. . . . . . . . 9
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
13 | | eqid 2739 |
. . . . . . . . . 10
⊢
(0g‘𝑆) = (0g‘𝑆) |
14 | | eqid 2739 |
. . . . . . . . . 10
⊢
(0g‘𝑇) = (0g‘𝑇) |
15 | 13, 14 | ghmid 18495 |
. . . . . . . . 9
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇)) |
16 | 9, 12, 15 | 3syl 18 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘(0g‘𝑆)) = (0g‘𝑇)) |
17 | 16 | fveq2d 6691 |
. . . . . . 7
⊢ (𝜑 → (𝑀‘(𝐹‘(0g‘𝑆))) = (𝑀‘(0g‘𝑇))) |
18 | 8 | elin1d 4098 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 ∈ NrmMod) |
19 | | nlmngp 23443 |
. . . . . . . 8
⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp) |
20 | 4, 14 | nm0 23395 |
. . . . . . . 8
⊢ (𝑇 ∈ NrmGrp → (𝑀‘(0g‘𝑇)) = 0) |
21 | 18, 19, 20 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → (𝑀‘(0g‘𝑇)) = 0) |
22 | 17, 21 | eqtrd 2774 |
. . . . . 6
⊢ (𝜑 → (𝑀‘(𝐹‘(0g‘𝑆))) = 0) |
23 | 22 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (𝑀‘(𝐹‘(0g‘𝑆))) = 0) |
24 | 23 | oveq1d 7198 |
. . . 4
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) = (0 / 𝑅)) |
25 | 11 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 𝑅 ∈
ℝ+) |
26 | 25 | rpcnd 12529 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 𝑅 ∈ ℂ) |
27 | 25 | rpne0d 12532 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 𝑅 ≠ 0) |
28 | 26, 27 | div0d 11506 |
. . . 4
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (0 / 𝑅) = 0) |
29 | 24, 28 | eqtrd 2774 |
. . 3
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) = 0) |
30 | 7 | elin1d 4098 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ NrmMod) |
31 | | nlmngp 23443 |
. . . . . . 7
⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp) |
32 | 3, 13 | nm0 23395 |
. . . . . . 7
⊢ (𝑆 ∈ NrmGrp → (𝐿‘(0g‘𝑆)) = 0) |
33 | 30, 31, 32 | 3syl 18 |
. . . . . 6
⊢ (𝜑 → (𝐿‘(0g‘𝑆)) = 0) |
34 | 33 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (𝐿‘(0g‘𝑆)) = 0) |
35 | 25 | rpgt0d 12530 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 0 < 𝑅) |
36 | 34, 35 | eqbrtrd 5062 |
. . . 4
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (𝐿‘(0g‘𝑆)) < 𝑅) |
37 | | fveq2 6687 |
. . . . . . 7
⊢ (𝑥 = (0g‘𝑆) → (𝐿‘𝑥) = (𝐿‘(0g‘𝑆))) |
38 | 37 | breq1d 5050 |
. . . . . 6
⊢ (𝑥 = (0g‘𝑆) → ((𝐿‘𝑥) < 𝑅 ↔ (𝐿‘(0g‘𝑆)) < 𝑅)) |
39 | | 2fveq3 6692 |
. . . . . . . 8
⊢ (𝑥 = (0g‘𝑆) → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(𝐹‘(0g‘𝑆)))) |
40 | 39 | oveq1d 7198 |
. . . . . . 7
⊢ (𝑥 = (0g‘𝑆) → ((𝑀‘(𝐹‘𝑥)) / 𝑅) = ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅)) |
41 | 40 | breq1d 5050 |
. . . . . 6
⊢ (𝑥 = (0g‘𝑆) → (((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴 ↔ ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) ≤ 𝐴)) |
42 | 38, 41 | imbi12d 348 |
. . . . 5
⊢ (𝑥 = (0g‘𝑆) → (((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴) ↔ ((𝐿‘(0g‘𝑆)) < 𝑅 → ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) ≤ 𝐴))) |
43 | 30, 31 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 ∈ NrmGrp) |
44 | 2, 3 | nmcl 23382 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) → (𝐿‘𝑥) ∈ ℝ) |
45 | 43, 44 | sylan 583 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐿‘𝑥) ∈ ℝ) |
46 | 11 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈
ℝ+) |
47 | 46 | rpred 12527 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ ℝ) |
48 | | nmoleub2lem2.7 |
. . . . . . . . 9
⊢ (((𝐿‘𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿‘𝑥) < 𝑅 → (𝐿‘𝑥)𝑂𝑅)) |
49 | 45, 47, 48 | syl2anc 587 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝑥) < 𝑅 → (𝐿‘𝑥)𝑂𝑅)) |
50 | 49 | imim1d 82 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴) → ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))) |
51 | 50 | ralimdva 3092 |
. . . . . 6
⊢ (𝜑 → (∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴) → ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))) |
52 | 51 | imp 410 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) |
53 | | ngpgrp 23365 |
. . . . . . 7
⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ Grp) |
54 | 2, 13 | grpidcl 18262 |
. . . . . . 7
⊢ (𝑆 ∈ Grp →
(0g‘𝑆)
∈ 𝑉) |
55 | 43, 53, 54 | 3syl 18 |
. . . . . 6
⊢ (𝜑 → (0g‘𝑆) ∈ 𝑉) |
56 | 55 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (0g‘𝑆) ∈ 𝑉) |
57 | 42, 52, 56 | rspcdva 3531 |
. . . 4
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → ((𝐿‘(0g‘𝑆)) < 𝑅 → ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) ≤ 𝐴)) |
58 | 36, 57 | mpd 15 |
. . 3
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) ≤ 𝐴) |
59 | 29, 58 | eqbrtrrd 5064 |
. 2
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 0 ≤ 𝐴) |
60 | | simp-4l 783 |
. . . . 5
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝜑) |
61 | 60, 7 | syl 17 |
. . . 4
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝑆 ∈ (NrmMod ∩
ℂMod)) |
62 | 60, 8 | syl 17 |
. . . 4
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝑇 ∈ (NrmMod ∩
ℂMod)) |
63 | 60, 9 | syl 17 |
. . . 4
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝐹 ∈ (𝑆 LMHom 𝑇)) |
64 | 60, 10 | syl 17 |
. . . 4
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝐴 ∈
ℝ*) |
65 | 60, 11 | syl 17 |
. . . 4
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝑅 ∈
ℝ+) |
66 | | nmoleub2a.5 |
. . . . 5
⊢ (𝜑 → ℚ ⊆ 𝐾) |
67 | 60, 66 | syl 17 |
. . . 4
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → ℚ ⊆ 𝐾) |
68 | | eqid 2739 |
. . . 4
⊢ (
·𝑠 ‘𝑆) = ( ·𝑠
‘𝑆) |
69 | | simpllr 776 |
. . . 4
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝐴 ∈ ℝ) |
70 | 59 | ad3antrrr 730 |
. . . 4
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 0 ≤ 𝐴) |
71 | | simplrl 777 |
. . . 4
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝑦 ∈ 𝑉) |
72 | | simplrr 778 |
. . . 4
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝑦 ≠ (0g‘𝑆)) |
73 | 52 | ad3antrrr 730 |
. . . . 5
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) |
74 | | fveq2 6687 |
. . . . . . . 8
⊢ (𝑥 = (𝑧( ·𝑠
‘𝑆)𝑦) → (𝐿‘𝑥) = (𝐿‘(𝑧( ·𝑠
‘𝑆)𝑦))) |
75 | 74 | breq1d 5050 |
. . . . . . 7
⊢ (𝑥 = (𝑧( ·𝑠
‘𝑆)𝑦) → ((𝐿‘𝑥) < 𝑅 ↔ (𝐿‘(𝑧( ·𝑠
‘𝑆)𝑦)) < 𝑅)) |
76 | | 2fveq3 6692 |
. . . . . . . . 9
⊢ (𝑥 = (𝑧( ·𝑠
‘𝑆)𝑦) → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(𝐹‘(𝑧( ·𝑠
‘𝑆)𝑦)))) |
77 | 76 | oveq1d 7198 |
. . . . . . . 8
⊢ (𝑥 = (𝑧( ·𝑠
‘𝑆)𝑦) → ((𝑀‘(𝐹‘𝑥)) / 𝑅) = ((𝑀‘(𝐹‘(𝑧( ·𝑠
‘𝑆)𝑦))) / 𝑅)) |
78 | 77 | breq1d 5050 |
. . . . . . 7
⊢ (𝑥 = (𝑧( ·𝑠
‘𝑆)𝑦) → (((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴 ↔ ((𝑀‘(𝐹‘(𝑧( ·𝑠
‘𝑆)𝑦))) / 𝑅) ≤ 𝐴)) |
79 | 75, 78 | imbi12d 348 |
. . . . . 6
⊢ (𝑥 = (𝑧( ·𝑠
‘𝑆)𝑦) → (((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴) ↔ ((𝐿‘(𝑧( ·𝑠
‘𝑆)𝑦)) < 𝑅 → ((𝑀‘(𝐹‘(𝑧( ·𝑠
‘𝑆)𝑦))) / 𝑅) ≤ 𝐴))) |
80 | 79 | rspccv 3526 |
. . . . 5
⊢
(∀𝑥 ∈
𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴) → ((𝑧( ·𝑠
‘𝑆)𝑦) ∈ 𝑉 → ((𝐿‘(𝑧( ·𝑠
‘𝑆)𝑦)) < 𝑅 → ((𝑀‘(𝐹‘(𝑧( ·𝑠
‘𝑆)𝑦))) / 𝑅) ≤ 𝐴))) |
81 | 73, 80 | syl 17 |
. . . 4
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → ((𝑧( ·𝑠
‘𝑆)𝑦) ∈ 𝑉 → ((𝐿‘(𝑧( ·𝑠
‘𝑆)𝑦)) < 𝑅 → ((𝑀‘(𝐹‘(𝑧( ·𝑠
‘𝑆)𝑦))) / 𝑅) ≤ 𝐴))) |
82 | | simpr 488 |
. . . 4
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) |
83 | 1, 2, 3, 4, 5, 6, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 81, 82 | nmoleub2lem3 23880 |
. . 3
⊢ ¬
((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) |
84 | | iman 405 |
. . 3
⊢
(((((𝜑 ∧
∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) ↔ ¬ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦)))) |
85 | 83, 84 | mpbir 234 |
. 2
⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) |
86 | | nmoleub2lem2.6 |
. . 3
⊢ (((𝐿‘𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿‘𝑥)𝑂𝑅 → (𝐿‘𝑥) ≤ 𝑅)) |
87 | 45, 47, 86 | syl2anc 587 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝑥)𝑂𝑅 → (𝐿‘𝑥) ≤ 𝑅)) |
88 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 59, 85, 87 | nmoleub2lem 23879 |
1
⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))) |