Step | Hyp | Ref
| Expression |
1 | | algcvga.5 |
. . 3
⊢ 𝑁 = (𝐶‘𝐴) |
2 | | algcvga.3 |
. . . 4
⊢ 𝐶:𝑆⟶ℕ0 |
3 | 2 | ffvelrni 5628 |
. . 3
⊢ (𝐴 ∈ 𝑆 → (𝐶‘𝐴) ∈
ℕ0) |
4 | 1, 3 | eqeltrid 2257 |
. 2
⊢ (𝐴 ∈ 𝑆 → 𝑁 ∈
ℕ0) |
5 | | nn0z 9225 |
. . . 4
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℤ) |
6 | | eluz1 9484 |
. . . . 5
⊢ (𝑁 ∈ ℤ → (𝐾 ∈
(ℤ≥‘𝑁) ↔ (𝐾 ∈ ℤ ∧ 𝑁 ≤ 𝐾))) |
7 | | 2fveq3 5499 |
. . . . . . . . 9
⊢ (𝑚 = 𝑁 → (𝐶‘(𝑅‘𝑚)) = (𝐶‘(𝑅‘𝑁))) |
8 | 7 | eqeq1d 2179 |
. . . . . . . 8
⊢ (𝑚 = 𝑁 → ((𝐶‘(𝑅‘𝑚)) = 0 ↔ (𝐶‘(𝑅‘𝑁)) = 0)) |
9 | 8 | imbi2d 229 |
. . . . . . 7
⊢ (𝑚 = 𝑁 → ((𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑚)) = 0) ↔ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑁)) = 0))) |
10 | | 2fveq3 5499 |
. . . . . . . . 9
⊢ (𝑚 = 𝑘 → (𝐶‘(𝑅‘𝑚)) = (𝐶‘(𝑅‘𝑘))) |
11 | 10 | eqeq1d 2179 |
. . . . . . . 8
⊢ (𝑚 = 𝑘 → ((𝐶‘(𝑅‘𝑚)) = 0 ↔ (𝐶‘(𝑅‘𝑘)) = 0)) |
12 | 11 | imbi2d 229 |
. . . . . . 7
⊢ (𝑚 = 𝑘 → ((𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑚)) = 0) ↔ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑘)) = 0))) |
13 | | 2fveq3 5499 |
. . . . . . . . 9
⊢ (𝑚 = (𝑘 + 1) → (𝐶‘(𝑅‘𝑚)) = (𝐶‘(𝑅‘(𝑘 + 1)))) |
14 | 13 | eqeq1d 2179 |
. . . . . . . 8
⊢ (𝑚 = (𝑘 + 1) → ((𝐶‘(𝑅‘𝑚)) = 0 ↔ (𝐶‘(𝑅‘(𝑘 + 1))) = 0)) |
15 | 14 | imbi2d 229 |
. . . . . . 7
⊢ (𝑚 = (𝑘 + 1) → ((𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑚)) = 0) ↔ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘(𝑘 + 1))) = 0))) |
16 | | 2fveq3 5499 |
. . . . . . . . 9
⊢ (𝑚 = 𝐾 → (𝐶‘(𝑅‘𝑚)) = (𝐶‘(𝑅‘𝐾))) |
17 | 16 | eqeq1d 2179 |
. . . . . . . 8
⊢ (𝑚 = 𝐾 → ((𝐶‘(𝑅‘𝑚)) = 0 ↔ (𝐶‘(𝑅‘𝐾)) = 0)) |
18 | 17 | imbi2d 229 |
. . . . . . 7
⊢ (𝑚 = 𝐾 → ((𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑚)) = 0) ↔ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝐾)) = 0))) |
19 | | algcvga.1 |
. . . . . . . . 9
⊢ 𝐹:𝑆⟶𝑆 |
20 | | algcvga.2 |
. . . . . . . . 9
⊢ 𝑅 = seq0((𝐹 ∘ 1st ),
(ℕ0 × {𝐴})) |
21 | | algcvga.4 |
. . . . . . . . 9
⊢ (𝑧 ∈ 𝑆 → ((𝐶‘(𝐹‘𝑧)) ≠ 0 → (𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧))) |
22 | 19, 20, 2, 21, 1 | algcvg 11995 |
. . . . . . . 8
⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑁)) = 0) |
23 | 22 | a1i 9 |
. . . . . . 7
⊢ (𝑁 ∈ ℤ → (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑁)) = 0)) |
24 | | nn0ge0 9153 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ0
→ 0 ≤ 𝑁) |
25 | 24 | adantr 274 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ 0 ≤ 𝑁) |
26 | | nn0re 9137 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℝ) |
27 | | zre 9209 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℤ → 𝑘 ∈
ℝ) |
28 | | 0re 7913 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
ℝ |
29 | | letr 7995 |
. . . . . . . . . . . . . . . . . 18
⊢ ((0
∈ ℝ ∧ 𝑁
∈ ℝ ∧ 𝑘
∈ ℝ) → ((0 ≤ 𝑁 ∧ 𝑁 ≤ 𝑘) → 0 ≤ 𝑘)) |
30 | 28, 29 | mp3an1 1319 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((0 ≤
𝑁 ∧ 𝑁 ≤ 𝑘) → 0 ≤ 𝑘)) |
31 | 26, 27, 30 | syl2an 287 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ ((0 ≤ 𝑁 ∧
𝑁 ≤ 𝑘) → 0 ≤ 𝑘)) |
32 | 25, 31 | mpand 427 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ (𝑁 ≤ 𝑘 → 0 ≤ 𝑘)) |
33 | | elnn0z 9218 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ0
↔ (𝑘 ∈ ℤ
∧ 0 ≤ 𝑘)) |
34 | 33 | simplbi2 383 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℤ → (0 ≤
𝑘 → 𝑘 ∈
ℕ0)) |
35 | 34 | adantl 275 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ (0 ≤ 𝑘 →
𝑘 ∈
ℕ0)) |
36 | 32, 35 | syld 45 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ (𝑁 ≤ 𝑘 → 𝑘 ∈
ℕ0)) |
37 | 4, 36 | sylan 281 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℤ) → (𝑁 ≤ 𝑘 → 𝑘 ∈
ℕ0)) |
38 | 37 | impr 377 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ 𝑆 ∧ (𝑘 ∈ ℤ ∧ 𝑁 ≤ 𝑘)) → 𝑘 ∈ ℕ0) |
39 | 38 | expcom 115 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℤ ∧ 𝑁 ≤ 𝑘) → (𝐴 ∈ 𝑆 → 𝑘 ∈
ℕ0)) |
40 | 39 | 3adant1 1010 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑁 ≤ 𝑘) → (𝐴 ∈ 𝑆 → 𝑘 ∈
ℕ0)) |
41 | 40 | ancld 323 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑁 ≤ 𝑘) → (𝐴 ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝑘 ∈
ℕ0))) |
42 | | nn0uz 9514 |
. . . . . . . . . . . . 13
⊢
ℕ0 = (ℤ≥‘0) |
43 | | 0zd 9217 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ 𝑆 → 0 ∈ ℤ) |
44 | | id 19 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝑆) |
45 | 19 | a1i 9 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ 𝑆 → 𝐹:𝑆⟶𝑆) |
46 | 42, 20, 43, 44, 45 | algrf 11992 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ 𝑆 → 𝑅:ℕ0⟶𝑆) |
47 | 46 | ffvelrnda 5629 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘𝑘) ∈ 𝑆) |
48 | | 2fveq3 5499 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘(𝐹‘𝑧)) = (𝐶‘(𝐹‘(𝑅‘𝑘)))) |
49 | 48 | neeq1d 2358 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0)) |
50 | | fveq2 5494 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘𝑧) = (𝐶‘(𝑅‘𝑘))) |
51 | 48, 50 | breq12d 4000 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧) ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘)))) |
52 | 49, 51 | imbi12d 233 |
. . . . . . . . . . . . 13
⊢ (𝑧 = (𝑅‘𝑘) → (((𝐶‘(𝐹‘𝑧)) ≠ 0 → (𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧)) ↔ ((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))))) |
53 | 52, 21 | vtoclga 2796 |
. . . . . . . . . . . 12
⊢ ((𝑅‘𝑘) ∈ 𝑆 → ((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘)))) |
54 | 19, 2 | algcvgb 11997 |
. . . . . . . . . . . . 13
⊢ ((𝑅‘𝑘) ∈ 𝑆 → (((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))) ↔ (((𝐶‘(𝑅‘𝑘)) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))) ∧ ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) = 0)))) |
55 | | simpr 109 |
. . . . . . . . . . . . 13
⊢ ((((𝐶‘(𝑅‘𝑘)) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))) ∧ ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) = 0)) → ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) = 0)) |
56 | 54, 55 | syl6bi 162 |
. . . . . . . . . . . 12
⊢ ((𝑅‘𝑘) ∈ 𝑆 → (((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))) → ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) = 0))) |
57 | 53, 56 | mpd 13 |
. . . . . . . . . . 11
⊢ ((𝑅‘𝑘) ∈ 𝑆 → ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) = 0)) |
58 | 47, 57 | syl 14 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) = 0)) |
59 | 42, 20, 43, 44, 45 | algrp1 11993 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅‘𝑘))) |
60 | 59 | fveqeq2d 5502 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘(𝑅‘(𝑘 + 1))) = 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) = 0)) |
61 | 58, 60 | sylibrd 168 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐶‘(𝑅‘(𝑘 + 1))) = 0)) |
62 | 41, 61 | syl6 33 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑁 ≤ 𝑘) → (𝐴 ∈ 𝑆 → ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐶‘(𝑅‘(𝑘 + 1))) = 0))) |
63 | 62 | a2d 26 |
. . . . . . 7
⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑁 ≤ 𝑘) → ((𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑘)) = 0) → (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘(𝑘 + 1))) = 0))) |
64 | 9, 12, 15, 18, 23, 63 | uzind 9316 |
. . . . . 6
⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ≤ 𝐾) → (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝐾)) = 0)) |
65 | 64 | 3expib 1201 |
. . . . 5
⊢ (𝑁 ∈ ℤ → ((𝐾 ∈ ℤ ∧ 𝑁 ≤ 𝐾) → (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝐾)) = 0))) |
66 | 6, 65 | sylbid 149 |
. . . 4
⊢ (𝑁 ∈ ℤ → (𝐾 ∈
(ℤ≥‘𝑁) → (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝐾)) = 0))) |
67 | 5, 66 | syl 14 |
. . 3
⊢ (𝑁 ∈ ℕ0
→ (𝐾 ∈
(ℤ≥‘𝑁) → (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝐾)) = 0))) |
68 | 67 | com3r 79 |
. 2
⊢ (𝐴 ∈ 𝑆 → (𝑁 ∈ ℕ0 → (𝐾 ∈
(ℤ≥‘𝑁) → (𝐶‘(𝑅‘𝐾)) = 0))) |
69 | 4, 68 | mpd 13 |
1
⊢ (𝐴 ∈ 𝑆 → (𝐾 ∈ (ℤ≥‘𝑁) → (𝐶‘(𝑅‘𝐾)) = 0)) |