| Step | Hyp | Ref
| Expression |
| 1 | | mettrifi.3 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
| 2 | | eluzfz2 13572 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
| 3 | 1, 2 | syl 17 |
. 2
⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
| 4 | | eleq1 2829 |
. . . . . 6
⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁))) |
| 5 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) |
| 6 | 5 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑥 = 𝑀 → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) = ((𝐹‘𝑀)𝐷(𝐹‘𝑀))) |
| 7 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑥 = 𝑀 → (𝑥 − 1) = (𝑀 − 1)) |
| 8 | 7 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑥 = 𝑀 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑀 − 1))) |
| 9 | 8 | sumeq1d 15736 |
. . . . . . 7
⊢ (𝑥 = 𝑀 → Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) |
| 10 | 6, 9 | breq12d 5156 |
. . . . . 6
⊢ (𝑥 = 𝑀 → (((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) |
| 11 | 4, 10 | imbi12d 344 |
. . . . 5
⊢ (𝑥 = 𝑀 → ((𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) ↔ (𝑀 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))) |
| 12 | 11 | imbi2d 340 |
. . . 4
⊢ (𝑥 = 𝑀 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))) |
| 13 | | eleq1 2829 |
. . . . . 6
⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁))) |
| 14 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑥 = 𝑛 → (𝐹‘𝑥) = (𝐹‘𝑛)) |
| 15 | 14 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑥 = 𝑛 → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) = ((𝐹‘𝑀)𝐷(𝐹‘𝑛))) |
| 16 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑥 = 𝑛 → (𝑥 − 1) = (𝑛 − 1)) |
| 17 | 16 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑥 = 𝑛 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑛 − 1))) |
| 18 | 17 | sumeq1d 15736 |
. . . . . . 7
⊢ (𝑥 = 𝑛 → Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) |
| 19 | 15, 18 | breq12d 5156 |
. . . . . 6
⊢ (𝑥 = 𝑛 → (((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) |
| 20 | 13, 19 | imbi12d 344 |
. . . . 5
⊢ (𝑥 = 𝑛 → ((𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) ↔ (𝑛 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))) |
| 21 | 20 | imbi2d 340 |
. . . 4
⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))) |
| 22 | | eleq1 2829 |
. . . . . 6
⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁))) |
| 23 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1))) |
| 24 | 23 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) = ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1)))) |
| 25 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑥 = (𝑛 + 1) → (𝑥 − 1) = ((𝑛 + 1) − 1)) |
| 26 | 25 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑥 = (𝑛 + 1) → (𝑀...(𝑥 − 1)) = (𝑀...((𝑛 + 1) − 1))) |
| 27 | 26 | sumeq1d 15736 |
. . . . . . 7
⊢ (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) |
| 28 | 24, 27 | breq12d 5156 |
. . . . . 6
⊢ (𝑥 = (𝑛 + 1) → (((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) |
| 29 | 22, 28 | imbi12d 344 |
. . . . 5
⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))) |
| 30 | 29 | imbi2d 340 |
. . . 4
⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))) |
| 31 | | eleq1 2829 |
. . . . . 6
⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁))) |
| 32 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑥 = 𝑁 → (𝐹‘𝑥) = (𝐹‘𝑁)) |
| 33 | 32 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) = ((𝐹‘𝑀)𝐷(𝐹‘𝑁))) |
| 34 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑥 = 𝑁 → (𝑥 − 1) = (𝑁 − 1)) |
| 35 | 34 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑥 = 𝑁 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑁 − 1))) |
| 36 | 35 | sumeq1d 15736 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) |
| 37 | 33, 36 | breq12d 5156 |
. . . . . 6
⊢ (𝑥 = 𝑁 → (((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) |
| 38 | 31, 37 | imbi12d 344 |
. . . . 5
⊢ (𝑥 = 𝑁 → ((𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) ↔ (𝑁 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))) |
| 39 | 38 | imbi2d 340 |
. . . 4
⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))) |
| 40 | | 0le0 12367 |
. . . . . . . 8
⊢ 0 ≤
0 |
| 41 | 40 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ 0) |
| 42 | | mettrifi.2 |
. . . . . . . 8
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
| 43 | | eluzfz1 13571 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
| 44 | 1, 43 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
| 45 | | mettrifi.4 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑋) |
| 46 | 45 | ralrimiva 3146 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑋) |
| 47 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀)) |
| 48 | 47 | eleq1d 2826 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘𝑀) ∈ 𝑋)) |
| 49 | 48 | rspcv 3618 |
. . . . . . . . 9
⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑋 → (𝐹‘𝑀) ∈ 𝑋)) |
| 50 | 44, 46, 49 | sylc 65 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑋) |
| 51 | | met0 24353 |
. . . . . . . 8
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑀) ∈ 𝑋) → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) = 0) |
| 52 | 42, 50, 51 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) = 0) |
| 53 | | eluzel2 12883 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
| 54 | 1, 53 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 55 | 54 | zred 12722 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℝ) |
| 56 | 55 | ltm1d 12200 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀 − 1) < 𝑀) |
| 57 | | peano2zm 12660 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈
ℤ) |
| 58 | | fzn 13580 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ)
→ ((𝑀 − 1) <
𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅)) |
| 59 | 54, 57, 58 | syl2anc2 585 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅)) |
| 60 | 56, 59 | mpbid 232 |
. . . . . . . . 9
⊢ (𝜑 → (𝑀...(𝑀 − 1)) = ∅) |
| 61 | 60 | sumeq1d 15736 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = Σ𝑘 ∈ ∅ ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) |
| 62 | | sum0 15757 |
. . . . . . . 8
⊢
Σ𝑘 ∈
∅ ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = 0 |
| 63 | 61, 62 | eqtrdi 2793 |
. . . . . . 7
⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = 0) |
| 64 | 41, 52, 63 | 3brtr4d 5175 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) |
| 65 | 64 | a1d 25 |
. . . . 5
⊢ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) |
| 66 | 65 | a1i 11 |
. . . 4
⊢ (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))) |
| 67 | | peano2fzr 13577 |
. . . . . . . . . 10
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁)) |
| 68 | 67 | ex 412 |
. . . . . . . . 9
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁))) |
| 69 | 68 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁))) |
| 70 | 69 | imim1d 82 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))) |
| 71 | 42 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝐷 ∈ (Met‘𝑋)) |
| 72 | 50 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘𝑀) ∈ 𝑋) |
| 73 | | simp3 1139 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑛 + 1) ∈ (𝑀...𝑁)) |
| 74 | 46 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑋) |
| 75 | | fveq2 6906 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1))) |
| 76 | 75 | eleq1d 2826 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑋)) |
| 77 | 76 | rspcv 3618 |
. . . . . . . . . . . . 13
⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑋 → (𝐹‘(𝑛 + 1)) ∈ 𝑋)) |
| 78 | 73, 74, 77 | sylc 65 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑛 + 1)) ∈ 𝑋) |
| 79 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛)) |
| 80 | 79 | eleq1d 2826 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘𝑛) ∈ 𝑋)) |
| 81 | 80 | cbvralvw 3237 |
. . . . . . . . . . . . . 14
⊢
(∀𝑘 ∈
(𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑋 ↔ ∀𝑛 ∈ (𝑀...𝑁)(𝐹‘𝑛) ∈ 𝑋) |
| 82 | 74, 81 | sylib 218 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ∀𝑛 ∈ (𝑀...𝑁)(𝐹‘𝑛) ∈ 𝑋) |
| 83 | 69 | 3impia 1118 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁)) |
| 84 | | rsp 3247 |
. . . . . . . . . . . . 13
⊢
(∀𝑛 ∈
(𝑀...𝑁)(𝐹‘𝑛) ∈ 𝑋 → (𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑛) ∈ 𝑋)) |
| 85 | 82, 83, 84 | sylc 65 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘𝑛) ∈ 𝑋) |
| 86 | | mettri 24362 |
. . . . . . . . . . . 12
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ ((𝐹‘𝑀) ∈ 𝑋 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑋 ∧ (𝐹‘𝑛) ∈ 𝑋)) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1))))) |
| 87 | 71, 72, 78, 85, 86 | syl13anc 1374 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1))))) |
| 88 | | metcl 24342 |
. . . . . . . . . . . . 13
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑀) ∈ 𝑋 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑋) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ∈ ℝ) |
| 89 | 71, 72, 78, 88 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ∈ ℝ) |
| 90 | | metcl 24342 |
. . . . . . . . . . . . . 14
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑀) ∈ 𝑋 ∧ (𝐹‘𝑛) ∈ 𝑋) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ∈ ℝ) |
| 91 | 71, 72, 85, 90 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ∈ ℝ) |
| 92 | | metcl 24342 |
. . . . . . . . . . . . . 14
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑛) ∈ 𝑋 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑋) → ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1))) ∈ ℝ) |
| 93 | 71, 85, 78, 92 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1))) ∈ ℝ) |
| 94 | 91, 93 | readdcld 11290 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ∈ ℝ) |
| 95 | | fzfid 14014 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...𝑛) ∈ Fin) |
| 96 | 71 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐷 ∈ (Met‘𝑋)) |
| 97 | | elfzuz3 13561 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑛)) |
| 98 | 83, 97 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑁 ∈ (ℤ≥‘𝑛)) |
| 99 | | fzss2 13604 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁)) |
| 100 | 98, 99 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...𝑛) ⊆ (𝑀...𝑁)) |
| 101 | 100 | sselda 3983 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ (𝑀...𝑁)) |
| 102 | 45 | 3ad2antl1 1186 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑋) |
| 103 | 101, 102 | syldan 591 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ 𝑋) |
| 104 | | elfzuz 13560 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀)) |
| 105 | 104 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ (ℤ≥‘𝑀)) |
| 106 | | peano2uz 12943 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (𝑘 + 1) ∈
(ℤ≥‘𝑀)) |
| 107 | 105, 106 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝑘 + 1) ∈
(ℤ≥‘𝑀)) |
| 108 | | elfzuz3 13561 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1))) |
| 109 | 73, 108 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1))) |
| 110 | 109 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1))) |
| 111 | | elfzuz3 13561 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑛 ∈ (ℤ≥‘𝑘)) |
| 112 | 111 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑛 ∈ (ℤ≥‘𝑘)) |
| 113 | | eluzp1p1 12906 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈
(ℤ≥‘𝑘) → (𝑛 + 1) ∈
(ℤ≥‘(𝑘 + 1))) |
| 114 | 112, 113 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝑛 + 1) ∈
(ℤ≥‘(𝑘 + 1))) |
| 115 | | uztrn 12896 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈
(ℤ≥‘(𝑛 + 1)) ∧ (𝑛 + 1) ∈
(ℤ≥‘(𝑘 + 1))) → 𝑁 ∈ (ℤ≥‘(𝑘 + 1))) |
| 116 | 110, 114,
115 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑁 ∈ (ℤ≥‘(𝑘 + 1))) |
| 117 | | elfzuzb 13558 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) ↔ ((𝑘 + 1) ∈
(ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘(𝑘 + 1)))) |
| 118 | 107, 116,
117 | sylanbrc 583 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝑘 + 1) ∈ (𝑀...𝑁)) |
| 119 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1))) |
| 120 | 119 | eleq1d 2826 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ 𝑋 ↔ (𝐹‘(𝑘 + 1)) ∈ 𝑋)) |
| 121 | 120 | rspccva 3621 |
. . . . . . . . . . . . . . . 16
⊢
((∀𝑛 ∈
(𝑀...𝑁)(𝐹‘𝑛) ∈ 𝑋 ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑘 + 1)) ∈ 𝑋) |
| 122 | 82, 121 | sylan 580 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑘 + 1)) ∈ 𝑋) |
| 123 | 118, 122 | syldan 591 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘(𝑘 + 1)) ∈ 𝑋) |
| 124 | | metcl 24342 |
. . . . . . . . . . . . . 14
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘(𝑘 + 1)) ∈ 𝑋) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ) |
| 125 | 96, 103, 123, 124 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ) |
| 126 | 95, 125 | fsumrecl 15770 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ) |
| 127 | | letr 11355 |
. . . . . . . . . . . 12
⊢ ((((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ∈ ℝ ∧ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ∈ ℝ ∧ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ) → ((((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ∧ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) |
| 128 | 89, 94, 126, 127 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ∧ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) |
| 129 | 87, 128 | mpand 695 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) |
| 130 | | fzfid 14014 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...(𝑛 − 1)) ∈ Fin) |
| 131 | | fzssp1 13607 |
. . . . . . . . . . . . . . . 16
⊢ (𝑀...(𝑛 − 1)) ⊆ (𝑀...((𝑛 − 1) + 1)) |
| 132 | | eluzelz 12888 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → 𝑛 ∈ ℤ) |
| 133 | 132 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ ℤ) |
| 134 | 133 | zcnd 12723 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ ℂ) |
| 135 | | ax-1cn 11213 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
ℂ |
| 136 | | npcan 11517 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑛 −
1) + 1) = 𝑛) |
| 137 | 134, 135,
136 | sylancl 586 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝑛 − 1) + 1) = 𝑛) |
| 138 | 137 | oveq2d 7447 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...((𝑛 − 1) + 1)) = (𝑀...𝑛)) |
| 139 | 131, 138 | sseqtrid 4026 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...(𝑛 − 1)) ⊆ (𝑀...𝑛)) |
| 140 | 139 | sselda 3983 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑛 − 1))) → 𝑘 ∈ (𝑀...𝑛)) |
| 141 | 140, 125 | syldan 591 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑛 − 1))) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ) |
| 142 | 130, 141 | fsumrecl 15770 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ) |
| 143 | 91, 142, 93 | leadd1d 11857 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ≤ (Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))))) |
| 144 | | simp2 1138 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (ℤ≥‘𝑀)) |
| 145 | 125 | recnd 11289 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℂ) |
| 146 | | fvoveq1 7454 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1))) |
| 147 | 79, 146 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) |
| 148 | 144, 145,
147 | fsumm1 15787 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = (Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1))))) |
| 149 | 148 | breq2d 5155 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ≤ (Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))))) |
| 150 | 143, 149 | bitr4d 282 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) |
| 151 | | pncan 11514 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑛 + 1)
− 1) = 𝑛) |
| 152 | 134, 135,
151 | sylancl 586 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝑛 + 1) − 1) = 𝑛) |
| 153 | 152 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...((𝑛 + 1) − 1)) = (𝑀...𝑛)) |
| 154 | 153 | sumeq1d 15736 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) |
| 155 | 154 | breq2d 5155 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) |
| 156 | 129, 150,
155 | 3imtr4d 294 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) |
| 157 | 156 | 3expia 1122 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))) |
| 158 | 157 | a2d 29 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))) |
| 159 | 70, 158 | syld 47 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))) |
| 160 | 159 | expcom 413 |
. . . . 5
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (𝜑 → ((𝑛 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))) |
| 161 | 160 | a2d 29 |
. . . 4
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → ((𝜑 → (𝑛 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) → (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))) |
| 162 | 12, 21, 30, 39, 66, 161 | uzind4 12948 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))) |
| 163 | 1, 162 | mpcom 38 |
. 2
⊢ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) |
| 164 | 3, 163 | mpd 15 |
1
⊢ (𝜑 → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) |