| 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 21030 |
. . . . . . . . 9
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
| 13 | | eqid 2737 |
. . . . . . . . . 10
⊢
(0g‘𝑆) = (0g‘𝑆) |
| 14 | | eqid 2737 |
. . . . . . . . . 10
⊢
(0g‘𝑇) = (0g‘𝑇) |
| 15 | 13, 14 | ghmid 19240 |
. . . . . . . . 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 6910 |
. . . . . . 7
⊢ (𝜑 → (𝑀‘(𝐹‘(0g‘𝑆))) = (𝑀‘(0g‘𝑇))) |
| 18 | 8 | elin1d 4204 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 ∈ NrmMod) |
| 19 | | nlmngp 24698 |
. . . . . . . 8
⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp) |
| 20 | 4, 14 | nm0 24642 |
. . . . . . . 8
⊢ (𝑇 ∈ NrmGrp → (𝑀‘(0g‘𝑇)) = 0) |
| 21 | 18, 19, 20 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → (𝑀‘(0g‘𝑇)) = 0) |
| 22 | 17, 21 | eqtrd 2777 |
. . . . . 6
⊢ (𝜑 → (𝑀‘(𝐹‘(0g‘𝑆))) = 0) |
| 23 | 22 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (𝑀‘(𝐹‘(0g‘𝑆))) = 0) |
| 24 | 23 | oveq1d 7446 |
. . . 4
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) = (0 / 𝑅)) |
| 25 | 11 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 𝑅 ∈
ℝ+) |
| 26 | 25 | rpcnd 13079 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 𝑅 ∈ ℂ) |
| 27 | 25 | rpne0d 13082 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 𝑅 ≠ 0) |
| 28 | 26, 27 | div0d 12042 |
. . . 4
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (0 / 𝑅) = 0) |
| 29 | 24, 28 | eqtrd 2777 |
. . 3
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) = 0) |
| 30 | 7 | elin1d 4204 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ NrmMod) |
| 31 | | nlmngp 24698 |
. . . . . . 7
⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp) |
| 32 | 3, 13 | nm0 24642 |
. . . . . . 7
⊢ (𝑆 ∈ NrmGrp → (𝐿‘(0g‘𝑆)) = 0) |
| 33 | 30, 31, 32 | 3syl 18 |
. . . . . 6
⊢ (𝜑 → (𝐿‘(0g‘𝑆)) = 0) |
| 34 | 33 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (𝐿‘(0g‘𝑆)) = 0) |
| 35 | 25 | rpgt0d 13080 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 0 < 𝑅) |
| 36 | 34, 35 | eqbrtrd 5165 |
. . . 4
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (𝐿‘(0g‘𝑆)) < 𝑅) |
| 37 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑥 = (0g‘𝑆) → (𝐿‘𝑥) = (𝐿‘(0g‘𝑆))) |
| 38 | 37 | breq1d 5153 |
. . . . . 6
⊢ (𝑥 = (0g‘𝑆) → ((𝐿‘𝑥) < 𝑅 ↔ (𝐿‘(0g‘𝑆)) < 𝑅)) |
| 39 | | 2fveq3 6911 |
. . . . . . . 8
⊢ (𝑥 = (0g‘𝑆) → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(𝐹‘(0g‘𝑆)))) |
| 40 | 39 | oveq1d 7446 |
. . . . . . 7
⊢ (𝑥 = (0g‘𝑆) → ((𝑀‘(𝐹‘𝑥)) / 𝑅) = ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅)) |
| 41 | 40 | breq1d 5153 |
. . . . . 6
⊢ (𝑥 = (0g‘𝑆) → (((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴 ↔ ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) ≤ 𝐴)) |
| 42 | 38, 41 | imbi12d 344 |
. . . . 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 24629 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) → (𝐿‘𝑥) ∈ ℝ) |
| 45 | 43, 44 | sylan 580 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐿‘𝑥) ∈ ℝ) |
| 46 | 11 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈
ℝ+) |
| 47 | 46 | rpred 13077 |
. . . . . . . . 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 584 |
. . . . . . . 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 3167 |
. . . . . 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 406 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) |
| 53 | | ngpgrp 24612 |
. . . . . . 7
⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ Grp) |
| 54 | 2, 13 | grpidcl 18983 |
. . . . . . 7
⊢ (𝑆 ∈ Grp →
(0g‘𝑆)
∈ 𝑉) |
| 55 | 43, 53, 54 | 3syl 18 |
. . . . . 6
⊢ (𝜑 → (0g‘𝑆) ∈ 𝑉) |
| 56 | 55 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (0g‘𝑆) ∈ 𝑉) |
| 57 | 42, 52, 56 | rspcdva 3623 |
. . . 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 5167 |
. 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 2737 |
. . . 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 6906 |
. . . . . . . 8
⊢ (𝑥 = (𝑧( ·𝑠
‘𝑆)𝑦) → (𝐿‘𝑥) = (𝐿‘(𝑧( ·𝑠
‘𝑆)𝑦))) |
| 75 | 74 | breq1d 5153 |
. . . . . . 7
⊢ (𝑥 = (𝑧( ·𝑠
‘𝑆)𝑦) → ((𝐿‘𝑥) < 𝑅 ↔ (𝐿‘(𝑧( ·𝑠
‘𝑆)𝑦)) < 𝑅)) |
| 76 | | 2fveq3 6911 |
. . . . . . . . 9
⊢ (𝑥 = (𝑧( ·𝑠
‘𝑆)𝑦) → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(𝐹‘(𝑧( ·𝑠
‘𝑆)𝑦)))) |
| 77 | 76 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝑥 = (𝑧( ·𝑠
‘𝑆)𝑦) → ((𝑀‘(𝐹‘𝑥)) / 𝑅) = ((𝑀‘(𝐹‘(𝑧( ·𝑠
‘𝑆)𝑦))) / 𝑅)) |
| 78 | 77 | breq1d 5153 |
. . . . . . 7
⊢ (𝑥 = (𝑧( ·𝑠
‘𝑆)𝑦) → (((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴 ↔ ((𝑀‘(𝐹‘(𝑧( ·𝑠
‘𝑆)𝑦))) / 𝑅) ≤ 𝐴)) |
| 79 | 75, 78 | imbi12d 344 |
. . . . . 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 3619 |
. . . . 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 484 |
. . . 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 25148 |
. . 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 401 |
. . 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 231 |
. 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 584 |
. 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 25147 |
1
⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))) |