Step | Hyp | Ref
| Expression |
1 | | algcvga.5 |
. . 3
⊢ 𝑁 = (𝐶‘𝐴) |
2 | | algcvga.3 |
. . . 4
⊢ 𝐶:𝑆⟶ℕ0 |
3 | 2 | ffvelrni 5322 |
. . 3
⊢ (𝐴 ∈ 𝑆 → (𝐶‘𝐴) ∈
ℕ0) |
4 | 1, 3 | syl5eqel 2165 |
. 2
⊢ (𝐴 ∈ 𝑆 → 𝑁 ∈
ℕ0) |
5 | | nn0z 8371 |
. . . 4
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℤ) |
6 | | eluz1 8623 |
. . . . 5
⊢ (𝑁 ∈ ℤ → (𝐾 ∈
(ℤ≥‘𝑁) ↔ (𝐾 ∈ ℤ ∧ 𝑁 ≤ 𝐾))) |
7 | | fveq2 5198 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑁 → (𝑅‘𝑚) = (𝑅‘𝑁)) |
8 | 7 | fveq2d 5202 |
. . . . . . . . 9
⊢ (𝑚 = 𝑁 → (𝐶‘(𝑅‘𝑚)) = (𝐶‘(𝑅‘𝑁))) |
9 | 8 | eqeq1d 2089 |
. . . . . . . 8
⊢ (𝑚 = 𝑁 → ((𝐶‘(𝑅‘𝑚)) = 0 ↔ (𝐶‘(𝑅‘𝑁)) = 0)) |
10 | 9 | imbi2d 228 |
. . . . . . 7
⊢ (𝑚 = 𝑁 → ((𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑚)) = 0) ↔ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑁)) = 0))) |
11 | | fveq2 5198 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑘 → (𝑅‘𝑚) = (𝑅‘𝑘)) |
12 | 11 | fveq2d 5202 |
. . . . . . . . 9
⊢ (𝑚 = 𝑘 → (𝐶‘(𝑅‘𝑚)) = (𝐶‘(𝑅‘𝑘))) |
13 | 12 | eqeq1d 2089 |
. . . . . . . 8
⊢ (𝑚 = 𝑘 → ((𝐶‘(𝑅‘𝑚)) = 0 ↔ (𝐶‘(𝑅‘𝑘)) = 0)) |
14 | 13 | imbi2d 228 |
. . . . . . 7
⊢ (𝑚 = 𝑘 → ((𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑚)) = 0) ↔ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑘)) = 0))) |
15 | | fveq2 5198 |
. . . . . . . . . 10
⊢ (𝑚 = (𝑘 + 1) → (𝑅‘𝑚) = (𝑅‘(𝑘 + 1))) |
16 | 15 | fveq2d 5202 |
. . . . . . . . 9
⊢ (𝑚 = (𝑘 + 1) → (𝐶‘(𝑅‘𝑚)) = (𝐶‘(𝑅‘(𝑘 + 1)))) |
17 | 16 | eqeq1d 2089 |
. . . . . . . 8
⊢ (𝑚 = (𝑘 + 1) → ((𝐶‘(𝑅‘𝑚)) = 0 ↔ (𝐶‘(𝑅‘(𝑘 + 1))) = 0)) |
18 | 17 | imbi2d 228 |
. . . . . . 7
⊢ (𝑚 = (𝑘 + 1) → ((𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑚)) = 0) ↔ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘(𝑘 + 1))) = 0))) |
19 | | fveq2 5198 |
. . . . . . . . . 10
⊢ (𝑚 = 𝐾 → (𝑅‘𝑚) = (𝑅‘𝐾)) |
20 | 19 | fveq2d 5202 |
. . . . . . . . 9
⊢ (𝑚 = 𝐾 → (𝐶‘(𝑅‘𝑚)) = (𝐶‘(𝑅‘𝐾))) |
21 | 20 | eqeq1d 2089 |
. . . . . . . 8
⊢ (𝑚 = 𝐾 → ((𝐶‘(𝑅‘𝑚)) = 0 ↔ (𝐶‘(𝑅‘𝐾)) = 0)) |
22 | 21 | imbi2d 228 |
. . . . . . 7
⊢ (𝑚 = 𝐾 → ((𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑚)) = 0) ↔ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝐾)) = 0))) |
23 | | algcvga.1 |
. . . . . . . . 9
⊢ 𝐹:𝑆⟶𝑆 |
24 | | algcvga.2 |
. . . . . . . . 9
⊢ 𝑅 = seq0((𝐹 ∘ 1st ),
(ℕ0 × {𝐴}), 𝑆) |
25 | | algcvga.4 |
. . . . . . . . 9
⊢ (𝑧 ∈ 𝑆 → ((𝐶‘(𝐹‘𝑧)) ≠ 0 → (𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧))) |
26 | | ialgcvga.s |
. . . . . . . . 9
⊢ 𝑆 ∈ 𝑉 |
27 | 23, 24, 2, 25, 1, 26 | ialgcvg 10430 |
. . . . . . . 8
⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑁)) = 0) |
28 | 27 | a1i 9 |
. . . . . . 7
⊢ (𝑁 ∈ ℤ → (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑁)) = 0)) |
29 | | nn0ge0 8313 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ0
→ 0 ≤ 𝑁) |
30 | 29 | adantr 270 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ 0 ≤ 𝑁) |
31 | | nn0re 8297 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℝ) |
32 | | zre 8355 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℤ → 𝑘 ∈
ℝ) |
33 | | 0re 7119 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
ℝ |
34 | | letr 7194 |
. . . . . . . . . . . . . . . . . 18
⊢ ((0
∈ ℝ ∧ 𝑁
∈ ℝ ∧ 𝑘
∈ ℝ) → ((0 ≤ 𝑁 ∧ 𝑁 ≤ 𝑘) → 0 ≤ 𝑘)) |
35 | 33, 34 | mp3an1 1255 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((0 ≤
𝑁 ∧ 𝑁 ≤ 𝑘) → 0 ≤ 𝑘)) |
36 | 31, 32, 35 | syl2an 283 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ ((0 ≤ 𝑁 ∧
𝑁 ≤ 𝑘) → 0 ≤ 𝑘)) |
37 | 30, 36 | mpand 419 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ (𝑁 ≤ 𝑘 → 0 ≤ 𝑘)) |
38 | | elnn0z 8364 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ0
↔ (𝑘 ∈ ℤ
∧ 0 ≤ 𝑘)) |
39 | 38 | simplbi2 377 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℤ → (0 ≤
𝑘 → 𝑘 ∈
ℕ0)) |
40 | 39 | adantl 271 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ (0 ≤ 𝑘 →
𝑘 ∈
ℕ0)) |
41 | 37, 40 | syld 44 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ (𝑁 ≤ 𝑘 → 𝑘 ∈
ℕ0)) |
42 | 4, 41 | sylan 277 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℤ) → (𝑁 ≤ 𝑘 → 𝑘 ∈
ℕ0)) |
43 | 42 | impr 371 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ 𝑆 ∧ (𝑘 ∈ ℤ ∧ 𝑁 ≤ 𝑘)) → 𝑘 ∈ ℕ0) |
44 | 43 | expcom 114 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℤ ∧ 𝑁 ≤ 𝑘) → (𝐴 ∈ 𝑆 → 𝑘 ∈
ℕ0)) |
45 | 44 | 3adant1 956 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑁 ≤ 𝑘) → (𝐴 ∈ 𝑆 → 𝑘 ∈
ℕ0)) |
46 | 45 | ancld 318 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑁 ≤ 𝑘) → (𝐴 ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝑘 ∈
ℕ0))) |
47 | | nn0uz 8653 |
. . . . . . . . . . . . 13
⊢
ℕ0 = (ℤ≥‘0) |
48 | | 0zd 8363 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ 𝑆 → 0 ∈ ℤ) |
49 | | id 19 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝑆) |
50 | 23 | a1i 9 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ 𝑆 → 𝐹:𝑆⟶𝑆) |
51 | 26 | a1i 9 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ 𝑆 → 𝑆 ∈ 𝑉) |
52 | 47, 24, 48, 49, 50, 51 | ialgrf 10427 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ 𝑆 → 𝑅:ℕ0⟶𝑆) |
53 | 52 | ffvelrnda 5323 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘𝑘) ∈ 𝑆) |
54 | | fveq2 5198 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = (𝑅‘𝑘) → (𝐹‘𝑧) = (𝐹‘(𝑅‘𝑘))) |
55 | 54 | fveq2d 5202 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘(𝐹‘𝑧)) = (𝐶‘(𝐹‘(𝑅‘𝑘)))) |
56 | 55 | neeq1d 2263 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0)) |
57 | | fveq2 5198 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘𝑧) = (𝐶‘(𝑅‘𝑘))) |
58 | 55, 57 | breq12d 3798 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧) ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘)))) |
59 | 56, 58 | imbi12d 232 |
. . . . . . . . . . . . 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))))) |
60 | 59, 25 | vtoclga 2664 |
. . . . . . . . . . . 12
⊢ ((𝑅‘𝑘) ∈ 𝑆 → ((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘)))) |
61 | 23, 2 | algcvgb 10432 |
. . . . . . . . . . . . 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)))) |
62 | | simpr 108 |
. . . . . . . . . . . . 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)) |
63 | 61, 62 | syl6bi 161 |
. . . . . . . . . . . 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))) |
64 | 60, 63 | mpd 13 |
. . . . . . . . . . 11
⊢ ((𝑅‘𝑘) ∈ 𝑆 → ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) = 0)) |
65 | 53, 64 | syl 14 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) = 0)) |
66 | 47, 24, 48, 49, 50, 51 | ialgrp1 10428 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅‘𝑘))) |
67 | 66 | fveq2d 5202 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐶‘(𝑅‘(𝑘 + 1))) = (𝐶‘(𝐹‘(𝑅‘𝑘)))) |
68 | 67 | eqeq1d 2089 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘(𝑅‘(𝑘 + 1))) = 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) = 0)) |
69 | 65, 68 | sylibrd 167 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐶‘(𝑅‘(𝑘 + 1))) = 0)) |
70 | 46, 69 | syl6 33 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑁 ≤ 𝑘) → (𝐴 ∈ 𝑆 → ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐶‘(𝑅‘(𝑘 + 1))) = 0))) |
71 | 70 | a2d 26 |
. . . . . . 7
⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑁 ≤ 𝑘) → ((𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑘)) = 0) → (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘(𝑘 + 1))) = 0))) |
72 | 10, 14, 18, 22, 28, 71 | uzind 8458 |
. . . . . 6
⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ≤ 𝐾) → (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝐾)) = 0)) |
73 | 72 | 3expib 1141 |
. . . . 5
⊢ (𝑁 ∈ ℤ → ((𝐾 ∈ ℤ ∧ 𝑁 ≤ 𝐾) → (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝐾)) = 0))) |
74 | 6, 73 | sylbid 148 |
. . . 4
⊢ (𝑁 ∈ ℤ → (𝐾 ∈
(ℤ≥‘𝑁) → (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝐾)) = 0))) |
75 | 5, 74 | syl 14 |
. . 3
⊢ (𝑁 ∈ ℕ0
→ (𝐾 ∈
(ℤ≥‘𝑁) → (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝐾)) = 0))) |
76 | 75 | com3r 78 |
. 2
⊢ (𝐴 ∈ 𝑆 → (𝑁 ∈ ℕ0 → (𝐾 ∈
(ℤ≥‘𝑁) → (𝐶‘(𝑅‘𝐾)) = 0))) |
77 | 4, 76 | mpd 13 |
1
⊢ (𝐴 ∈ 𝑆 → (𝐾 ∈ (ℤ≥‘𝑁) → (𝐶‘(𝑅‘𝐾)) = 0)) |