Proof of Theorem nmoleub
| Step | Hyp | Ref
| Expression |
| 1 | | nmoleub.2 |
. . . . . . . . 9
⊢ (𝜑 → 𝑇 ∈ NrmGrp) |
| 2 | 1 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → 𝑇 ∈ NrmGrp) |
| 3 | | nmoleub.3 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
| 4 | | nmoi.2 |
. . . . . . . . . . . 12
⊢ 𝑉 = (Base‘𝑆) |
| 5 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(Base‘𝑇) =
(Base‘𝑇) |
| 6 | 4, 5 | ghmf 19238 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑉⟶(Base‘𝑇)) |
| 7 | 3, 6 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:𝑉⟶(Base‘𝑇)) |
| 8 | 7 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → 𝐹:𝑉⟶(Base‘𝑇)) |
| 9 | | simprl 771 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → 𝑥 ∈ 𝑉) |
| 10 | | ffvelcdm 7101 |
. . . . . . . . 9
⊢ ((𝐹:𝑉⟶(Base‘𝑇) ∧ 𝑥 ∈ 𝑉) → (𝐹‘𝑥) ∈ (Base‘𝑇)) |
| 11 | 8, 9, 10 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝐹‘𝑥) ∈ (Base‘𝑇)) |
| 12 | | nmoi.4 |
. . . . . . . . 9
⊢ 𝑀 = (norm‘𝑇) |
| 13 | 5, 12 | nmcl 24629 |
. . . . . . . 8
⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇)) → (𝑀‘(𝐹‘𝑥)) ∈ ℝ) |
| 14 | 2, 11, 13 | syl2anc 584 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑀‘(𝐹‘𝑥)) ∈ ℝ) |
| 15 | | nmoleub.1 |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ NrmGrp) |
| 16 | 15 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) → 𝑆 ∈ NrmGrp) |
| 17 | | nmoi.3 |
. . . . . . . . . 10
⊢ 𝐿 = (norm‘𝑆) |
| 18 | | nmoi2.z |
. . . . . . . . . 10
⊢ 0 =
(0g‘𝑆) |
| 19 | 4, 17, 18 | nmrpcl 24633 |
. . . . . . . . 9
⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 ) → (𝐿‘𝑥) ∈
ℝ+) |
| 20 | 19 | 3expb 1121 |
. . . . . . . 8
⊢ ((𝑆 ∈ NrmGrp ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝐿‘𝑥) ∈
ℝ+) |
| 21 | 16, 20 | sylan 580 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝐿‘𝑥) ∈
ℝ+) |
| 22 | 14, 21 | rerpdivcld 13108 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ∈ ℝ) |
| 23 | 22 | rexrd 11311 |
. . . . 5
⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ∈
ℝ*) |
| 24 | | nmofval.1 |
. . . . . . . 8
⊢ 𝑁 = (𝑆 normOp 𝑇) |
| 25 | 24 | nmocl 24741 |
. . . . . . 7
⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘𝐹) ∈
ℝ*) |
| 26 | 15, 1, 3, 25 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝑁‘𝐹) ∈
ℝ*) |
| 27 | 26 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑁‘𝐹) ∈
ℝ*) |
| 28 | | nmoleub.4 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 29 | 28 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → 𝐴 ∈
ℝ*) |
| 30 | 15, 1, 3 | 3jca 1129 |
. . . . . . 7
⊢ (𝜑 → (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇))) |
| 31 | 30 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) → (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇))) |
| 32 | 24, 4, 17, 12, 18 | nmoi2 24751 |
. . . . . 6
⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ (𝑁‘𝐹)) |
| 33 | 31, 32 | sylan 580 |
. . . . 5
⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ (𝑁‘𝐹)) |
| 34 | | simplr 769 |
. . . . 5
⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑁‘𝐹) ≤ 𝐴) |
| 35 | 23, 27, 29, 33, 34 | xrletrd 13204 |
. . . 4
⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴) |
| 36 | 35 | expr 456 |
. . 3
⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ 𝑥 ∈ 𝑉) → (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) |
| 37 | 36 | ralrimiva 3146 |
. 2
⊢ ((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) → ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) |
| 38 | | 0le0 12367 |
. . . . . . . . . . 11
⊢ 0 ≤
0 |
| 39 | | simpllr 776 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → 𝐴 ∈ ℝ) |
| 40 | 39 | recnd 11289 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → 𝐴 ∈ ℂ) |
| 41 | 40 | mul01d 11460 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝐴 · 0) =
0) |
| 42 | 38, 41 | breqtrrid 5181 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → 0 ≤ (𝐴 · 0)) |
| 43 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 0 → (𝐹‘𝑥) = (𝐹‘ 0 )) |
| 44 | 3 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
| 45 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(0g‘𝑇) = (0g‘𝑇) |
| 46 | 18, 45 | ghmid 19240 |
. . . . . . . . . . . . . 14
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘ 0 ) =
(0g‘𝑇)) |
| 47 | 44, 46 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → (𝐹‘ 0 ) =
(0g‘𝑇)) |
| 48 | 43, 47 | sylan9eqr 2799 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝐹‘𝑥) = (0g‘𝑇)) |
| 49 | 48 | fveq2d 6910 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(0g‘𝑇))) |
| 50 | 1 | ad3antrrr 730 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → 𝑇 ∈ NrmGrp) |
| 51 | 12, 45 | nm0 24642 |
. . . . . . . . . . . 12
⊢ (𝑇 ∈ NrmGrp → (𝑀‘(0g‘𝑇)) = 0) |
| 52 | 50, 51 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝑀‘(0g‘𝑇)) = 0) |
| 53 | 49, 52 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝑀‘(𝐹‘𝑥)) = 0) |
| 54 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑥 = 0 → (𝐿‘𝑥) = (𝐿‘ 0 )) |
| 55 | 15 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → 𝑆 ∈ NrmGrp) |
| 56 | 17, 18 | nm0 24642 |
. . . . . . . . . . . . 13
⊢ (𝑆 ∈ NrmGrp → (𝐿‘ 0 ) = 0) |
| 57 | 55, 56 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → (𝐿‘ 0 ) = 0) |
| 58 | 54, 57 | sylan9eqr 2799 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝐿‘𝑥) = 0) |
| 59 | 58 | oveq2d 7447 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝐴 · (𝐿‘𝑥)) = (𝐴 · 0)) |
| 60 | 42, 53, 59 | 3brtr4d 5175 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) |
| 61 | 60 | a1d 25 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → ((𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)))) |
| 62 | | simpr 484 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → 𝑥 ≠ 0 ) |
| 63 | 1 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → 𝑇 ∈ NrmGrp) |
| 64 | 7 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹:𝑉⟶(Base‘𝑇)) |
| 65 | 64, 10 | sylan 580 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → (𝐹‘𝑥) ∈ (Base‘𝑇)) |
| 66 | 63, 65, 13 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → (𝑀‘(𝐹‘𝑥)) ∈ ℝ) |
| 67 | 66 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (𝑀‘(𝐹‘𝑥)) ∈ ℝ) |
| 68 | | simpllr 776 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → 𝐴 ∈ ℝ) |
| 69 | 15 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝑆 ∈ NrmGrp) |
| 70 | 19 | 3expa 1119 |
. . . . . . . . . . . 12
⊢ (((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (𝐿‘𝑥) ∈
ℝ+) |
| 71 | 69, 70 | sylanl1 680 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (𝐿‘𝑥) ∈
ℝ+) |
| 72 | 67, 68, 71 | ledivmul2d 13131 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴 ↔ (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)))) |
| 73 | 72 | biimpd 229 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴 → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)))) |
| 74 | 62, 73 | embantd 59 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → ((𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)))) |
| 75 | 61, 74 | pm2.61dane 3029 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → ((𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)))) |
| 76 | 75 | ralimdva 3167 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → (∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴) → ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)))) |
| 77 | 1 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝑇 ∈ NrmGrp) |
| 78 | 3 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
| 79 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ) |
| 80 | | nmoleub.5 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤ 𝐴) |
| 81 | 80 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 0 ≤ 𝐴) |
| 82 | 24, 4, 17, 12 | nmolb 24738 |
. . . . . . 7
⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) → (𝑁‘𝐹) ≤ 𝐴)) |
| 83 | 69, 77, 78, 79, 81, 82 | syl311anc 1386 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → (∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) → (𝑁‘𝐹) ≤ 𝐴)) |
| 84 | 76, 83 | syld 47 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → (∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴) → (𝑁‘𝐹) ≤ 𝐴)) |
| 85 | 84 | imp 406 |
. . . 4
⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) → (𝑁‘𝐹) ≤ 𝐴) |
| 86 | 85 | an32s 652 |
. . 3
⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) → (𝑁‘𝐹) ≤ 𝐴) |
| 87 | 26 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) ∧ 𝐴 = +∞) → (𝑁‘𝐹) ∈
ℝ*) |
| 88 | | pnfge 13172 |
. . . . 5
⊢ ((𝑁‘𝐹) ∈ ℝ* → (𝑁‘𝐹) ≤ +∞) |
| 89 | 87, 88 | syl 17 |
. . . 4
⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) ∧ 𝐴 = +∞) → (𝑁‘𝐹) ≤ +∞) |
| 90 | | simpr 484 |
. . . 4
⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) ∧ 𝐴 = +∞) → 𝐴 = +∞) |
| 91 | 89, 90 | breqtrrd 5171 |
. . 3
⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) ∧ 𝐴 = +∞) → (𝑁‘𝐹) ≤ 𝐴) |
| 92 | | ge0nemnf 13215 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 0 ≤ 𝐴) →
𝐴 ≠
-∞) |
| 93 | 28, 80, 92 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → 𝐴 ≠ -∞) |
| 94 | 28, 93 | jca 511 |
. . . . 5
⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠
-∞)) |
| 95 | | xrnemnf 13159 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐴 ≠ -∞)
↔ (𝐴 ∈ ℝ
∨ 𝐴 =
+∞)) |
| 96 | 94, 95 | sylib 218 |
. . . 4
⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
| 97 | 96 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
| 98 | 86, 91, 97 | mpjaodan 961 |
. 2
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) → (𝑁‘𝐹) ≤ 𝐴) |
| 99 | 37, 98 | impbida 801 |
1
⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴))) |