| Step | Hyp | Ref
| Expression |
| 1 | | simprr 773 |
. . . . . . 7
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝐴‘𝑘) ≠ 0) |
| 2 | | ffun 6739 |
. . . . . . . . . . . 12
⊢ (𝐴:ℕ0⟶ℂ →
Fun 𝐴) |
| 3 | 2 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
Fun 𝐴) |
| 4 | | peano2nn0 12566 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℕ0) |
| 5 | 4 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
(𝑁 + 1) ∈
ℕ0) |
| 6 | | eluznn0 12959 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 + 1) ∈ ℕ0
∧ 𝑘 ∈
(ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ ℕ0) |
| 7 | 6 | ex 412 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 + 1) ∈ ℕ0
→ (𝑘 ∈
(ℤ≥‘(𝑁 + 1)) → 𝑘 ∈
ℕ0)) |
| 8 | 5, 7 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
(𝑘 ∈
(ℤ≥‘(𝑁 + 1)) → 𝑘 ∈
ℕ0)) |
| 9 | 8 | ssrdv 3989 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
(ℤ≥‘(𝑁 + 1)) ⊆
ℕ0) |
| 10 | | fdm 6745 |
. . . . . . . . . . . . 13
⊢ (𝐴:ℕ0⟶ℂ →
dom 𝐴 =
ℕ0) |
| 11 | 10 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
dom 𝐴 =
ℕ0) |
| 12 | 9, 11 | sseqtrrd 4021 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
(ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴) |
| 13 | | funfvima2 7251 |
. . . . . . . . . . 11
⊢ ((Fun
𝐴 ∧
(ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “
(ℤ≥‘(𝑁 + 1))))) |
| 14 | 3, 12, 13 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
(𝑘 ∈
(ℤ≥‘(𝑁 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “
(ℤ≥‘(𝑁 + 1))))) |
| 15 | 14 | ad2antrr 726 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “
(ℤ≥‘(𝑁 + 1))))) |
| 16 | | nn0z 12638 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℤ) |
| 17 | 16 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
𝑁 ∈
ℤ) |
| 18 | 17 | peano2zd 12725 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
(𝑁 + 1) ∈
ℤ) |
| 19 | 18 | ad2antrr 726 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑁 + 1) ∈ ℤ) |
| 20 | | nn0z 12638 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℤ) |
| 21 | 20 | ad2antrl 728 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ∈ ℤ) |
| 22 | | eluz 12892 |
. . . . . . . . . 10
⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈
(ℤ≥‘(𝑁 + 1)) ↔ (𝑁 + 1) ≤ 𝑘)) |
| 23 | 19, 21, 22 | syl2anc 584 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) ↔ (𝑁 + 1) ≤ 𝑘)) |
| 24 | | simplr 769 |
. . . . . . . . . . 11
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) |
| 25 | 24 | eleq2d 2827 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → ((𝐴‘𝑘) ∈ (𝐴 “
(ℤ≥‘(𝑁 + 1))) ↔ (𝐴‘𝑘) ∈ {0})) |
| 26 | | fvex 6919 |
. . . . . . . . . . 11
⊢ (𝐴‘𝑘) ∈ V |
| 27 | 26 | elsn 4641 |
. . . . . . . . . 10
⊢ ((𝐴‘𝑘) ∈ {0} ↔ (𝐴‘𝑘) = 0) |
| 28 | 25, 27 | bitrdi 287 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → ((𝐴‘𝑘) ∈ (𝐴 “
(ℤ≥‘(𝑁 + 1))) ↔ (𝐴‘𝑘) = 0)) |
| 29 | 15, 23, 28 | 3imtr3d 293 |
. . . . . . . 8
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → ((𝑁 + 1) ≤ 𝑘 → (𝐴‘𝑘) = 0)) |
| 30 | 29 | necon3ad 2953 |
. . . . . . 7
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → ((𝐴‘𝑘) ≠ 0 → ¬ (𝑁 + 1) ≤ 𝑘)) |
| 31 | 1, 30 | mpd 15 |
. . . . . 6
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → ¬ (𝑁 + 1) ≤ 𝑘) |
| 32 | | nn0re 12535 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℝ) |
| 33 | 32 | ad2antrl 728 |
. . . . . . 7
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ∈ ℝ) |
| 34 | 18 | zred 12722 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
(𝑁 + 1) ∈
ℝ) |
| 35 | 34 | ad2antrr 726 |
. . . . . . 7
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑁 + 1) ∈ ℝ) |
| 36 | 33, 35 | ltnled 11408 |
. . . . . 6
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑘 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑘)) |
| 37 | 31, 36 | mpbird 257 |
. . . . 5
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 < (𝑁 + 1)) |
| 38 | 17 | ad2antrr 726 |
. . . . . 6
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑁 ∈ ℤ) |
| 39 | | zleltp1 12668 |
. . . . . 6
⊢ ((𝑘 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑘 ≤ 𝑁 ↔ 𝑘 < (𝑁 + 1))) |
| 40 | 21, 38, 39 | syl2anc 584 |
. . . . 5
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑘 ≤ 𝑁 ↔ 𝑘 < (𝑁 + 1))) |
| 41 | 37, 40 | mpbird 257 |
. . . 4
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ≤ 𝑁) |
| 42 | 41 | expr 456 |
. . 3
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
| 43 | 42 | ralrimiva 3146 |
. 2
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) → ∀𝑘 ∈ ℕ0
((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
| 44 | | simpr 484 |
. . . . . . . 8
⊢
((∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑛 ∈
(ℤ≥‘(𝑁 + 1))) |
| 45 | | eluznn0 12959 |
. . . . . . . 8
⊢ (((𝑁 + 1) ∈ ℕ0
∧ 𝑛 ∈
(ℤ≥‘(𝑁 + 1))) → 𝑛 ∈ ℕ0) |
| 46 | 5, 44, 45 | syl2an 596 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈
ℕ0) |
| 47 | | nn0re 12535 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℝ) |
| 48 | 47 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
𝑁 ∈
ℝ) |
| 49 | 48 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑁 ∈
ℝ) |
| 50 | 34 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (𝑁 + 1) ∈
ℝ) |
| 51 | 46 | nn0red 12588 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈
ℝ) |
| 52 | 49 | ltp1d 12198 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑁 < (𝑁 + 1)) |
| 53 | | eluzle 12891 |
. . . . . . . . . . 11
⊢ (𝑛 ∈
(ℤ≥‘(𝑁 + 1)) → (𝑁 + 1) ≤ 𝑛) |
| 54 | 53 | ad2antll 729 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (𝑁 + 1) ≤ 𝑛) |
| 55 | 49, 50, 51, 52, 54 | ltletrd 11421 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑁 < 𝑛) |
| 56 | 49, 51 | ltnled 11408 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (𝑁 < 𝑛 ↔ ¬ 𝑛 ≤ 𝑁)) |
| 57 | 55, 56 | mpbid 232 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → ¬ 𝑛 ≤ 𝑁) |
| 58 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑛 → (𝐴‘𝑘) = (𝐴‘𝑛)) |
| 59 | 58 | neeq1d 3000 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑛 → ((𝐴‘𝑘) ≠ 0 ↔ (𝐴‘𝑛) ≠ 0)) |
| 60 | | breq1 5146 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑛 → (𝑘 ≤ 𝑁 ↔ 𝑛 ≤ 𝑁)) |
| 61 | 59, 60 | imbi12d 344 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑛 → (((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ↔ ((𝐴‘𝑛) ≠ 0 → 𝑛 ≤ 𝑁))) |
| 62 | | simprl 771 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → ∀𝑘 ∈ ℕ0
((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
| 63 | 61, 62, 46 | rspcdva 3623 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → ((𝐴‘𝑛) ≠ 0 → 𝑛 ≤ 𝑁)) |
| 64 | 63 | necon1bd 2958 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (¬ 𝑛 ≤ 𝑁 → (𝐴‘𝑛) = 0)) |
| 65 | 57, 64 | mpd 15 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (𝐴‘𝑛) = 0) |
| 66 | | ffn 6736 |
. . . . . . . . 9
⊢ (𝐴:ℕ0⟶ℂ →
𝐴 Fn
ℕ0) |
| 67 | 66 | ad2antlr 727 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝐴 Fn
ℕ0) |
| 68 | | fniniseg 7080 |
. . . . . . . 8
⊢ (𝐴 Fn ℕ0 →
(𝑛 ∈ (◡𝐴 “ {0}) ↔ (𝑛 ∈ ℕ0 ∧ (𝐴‘𝑛) = 0))) |
| 69 | 67, 68 | syl 17 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (𝑛 ∈ (◡𝐴 “ {0}) ↔ (𝑛 ∈ ℕ0 ∧ (𝐴‘𝑛) = 0))) |
| 70 | 46, 65, 69 | mpbir2and 713 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
(∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈ (◡𝐴 “ {0})) |
| 71 | 70 | expr 456 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → 𝑛 ∈ (◡𝐴 “ {0}))) |
| 72 | 71 | ssrdv 3989 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) →
(ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0})) |
| 73 | | funimass3 7074 |
. . . . . 6
⊢ ((Fun
𝐴 ∧
(ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴) → ((𝐴 “
(ℤ≥‘(𝑁 + 1))) ⊆ {0} ↔
(ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0}))) |
| 74 | 3, 12, 73 | syl2anc 584 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
((𝐴 “
(ℤ≥‘(𝑁 + 1))) ⊆ {0} ↔
(ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0}))) |
| 75 | 74 | adantr 480 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ((𝐴 “
(ℤ≥‘(𝑁 + 1))) ⊆ {0} ↔
(ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0}))) |
| 76 | 72, 75 | mpbird 257 |
. . 3
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴 “
(ℤ≥‘(𝑁 + 1))) ⊆ {0}) |
| 77 | 48 | ltp1d 12198 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
𝑁 < (𝑁 + 1)) |
| 78 | 48, 34 | ltnled 11408 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
(𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁)) |
| 79 | 77, 78 | mpbid 232 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
¬ (𝑁 + 1) ≤ 𝑁) |
| 80 | 79 | adantr 480 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ¬ (𝑁 + 1) ≤ 𝑁) |
| 81 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑁 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑁 + 1))) |
| 82 | 81 | neeq1d 3000 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑁 + 1) → ((𝐴‘𝑘) ≠ 0 ↔ (𝐴‘(𝑁 + 1)) ≠ 0)) |
| 83 | | breq1 5146 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑁 + 1) → (𝑘 ≤ 𝑁 ↔ (𝑁 + 1) ≤ 𝑁)) |
| 84 | 82, 83 | imbi12d 344 |
. . . . . . . . 9
⊢ (𝑘 = (𝑁 + 1) → (((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ↔ ((𝐴‘(𝑁 + 1)) ≠ 0 → (𝑁 + 1) ≤ 𝑁))) |
| 85 | 84 | rspcva 3620 |
. . . . . . . 8
⊢ (((𝑁 + 1) ∈ ℕ0
∧ ∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ((𝐴‘(𝑁 + 1)) ≠ 0 → (𝑁 + 1) ≤ 𝑁)) |
| 86 | 5, 85 | sylan 580 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ((𝐴‘(𝑁 + 1)) ≠ 0 → (𝑁 + 1) ≤ 𝑁)) |
| 87 | 86 | necon1bd 2958 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (¬ (𝑁 + 1) ≤ 𝑁 → (𝐴‘(𝑁 + 1)) = 0)) |
| 88 | 80, 87 | mpd 15 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴‘(𝑁 + 1)) = 0) |
| 89 | | uzid 12893 |
. . . . . . . 8
⊢ ((𝑁 + 1) ∈ ℤ →
(𝑁 + 1) ∈
(ℤ≥‘(𝑁 + 1))) |
| 90 | 18, 89 | syl 17 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
(𝑁 + 1) ∈
(ℤ≥‘(𝑁 + 1))) |
| 91 | | funfvima2 7251 |
. . . . . . . 8
⊢ ((Fun
𝐴 ∧
(ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴) → ((𝑁 + 1) ∈
(ℤ≥‘(𝑁 + 1)) → (𝐴‘(𝑁 + 1)) ∈ (𝐴 “
(ℤ≥‘(𝑁 + 1))))) |
| 92 | 3, 12, 91 | syl2anc 584 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
((𝑁 + 1) ∈
(ℤ≥‘(𝑁 + 1)) → (𝐴‘(𝑁 + 1)) ∈ (𝐴 “
(ℤ≥‘(𝑁 + 1))))) |
| 93 | 90, 92 | mpd 15 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
(𝐴‘(𝑁 + 1)) ∈ (𝐴 “
(ℤ≥‘(𝑁 + 1)))) |
| 94 | 93 | adantr 480 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴‘(𝑁 + 1)) ∈ (𝐴 “
(ℤ≥‘(𝑁 + 1)))) |
| 95 | 88, 94 | eqeltrrd 2842 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → 0 ∈ (𝐴 “
(ℤ≥‘(𝑁 + 1)))) |
| 96 | 95 | snssd 4809 |
. . 3
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → {0} ⊆ (𝐴 “
(ℤ≥‘(𝑁 + 1)))) |
| 97 | 76, 96 | eqssd 4001 |
. 2
⊢ (((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) ∧
∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) |
| 98 | 43, 97 | impbida 801 |
1
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
((𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0
((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) |