Proof of Theorem seqz
Step | Hyp | Ref
| Expression |
1 | | seqz.5 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) |
2 | | elfzuz 12631 |
. . . 4
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) |
3 | 1, 2 | syl 17 |
. . 3
⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) |
4 | | eluzelz 11978 |
. . . . . . . . 9
⊢ (𝐾 ∈
(ℤ≥‘𝑀) → 𝐾 ∈ ℤ) |
5 | 3, 4 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ ℤ) |
6 | | seq1 13108 |
. . . . . . . 8
⊢ (𝐾 ∈ ℤ → (seq𝐾( + , 𝐹)‘𝐾) = (𝐹‘𝐾)) |
7 | 5, 6 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝐾) = (𝐹‘𝐾)) |
8 | | seqz.7 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘𝐾) = 𝑍) |
9 | 7, 8 | eqtrd 2861 |
. . . . . 6
⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝐾) = 𝑍) |
10 | | seqeq1 13098 |
. . . . . . . 8
⊢ (𝐾 = 𝑀 → seq𝐾( + , 𝐹) = seq𝑀( + , 𝐹)) |
11 | 10 | fveq1d 6435 |
. . . . . . 7
⊢ (𝐾 = 𝑀 → (seq𝐾( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾)) |
12 | 11 | eqeq1d 2827 |
. . . . . 6
⊢ (𝐾 = 𝑀 → ((seq𝐾( + , 𝐹)‘𝐾) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)) |
13 | 9, 12 | syl5ibcom 237 |
. . . . 5
⊢ (𝜑 → (𝐾 = 𝑀 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)) |
14 | | eluzel2 11973 |
. . . . . . . . 9
⊢ (𝐾 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
15 | 3, 14 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℤ) |
16 | | seqm1 13112 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈
(ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹‘𝐾))) |
17 | 15, 16 | sylan 577 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹‘𝐾))) |
18 | 8 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝐾) = 𝑍) |
19 | 18 | oveq2d 6921 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹‘𝐾)) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍)) |
20 | | oveq1 6912 |
. . . . . . . . . 10
⊢ (𝑥 = (seq𝑀( + , 𝐹)‘(𝐾 − 1)) → (𝑥 + 𝑍) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍)) |
21 | 20 | eqeq1d 2827 |
. . . . . . . . 9
⊢ (𝑥 = (seq𝑀( + , 𝐹)‘(𝐾 − 1)) → ((𝑥 + 𝑍) = 𝑍 ↔ ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍) = 𝑍)) |
22 | | seqz.4 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑍) |
23 | 22 | ralrimiva 3175 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑥 + 𝑍) = 𝑍) |
24 | 23 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → ∀𝑥 ∈ 𝑆 (𝑥 + 𝑍) = 𝑍) |
25 | | eluzp1m1 11992 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈
(ℤ≥‘(𝑀 + 1))) → (𝐾 − 1) ∈
(ℤ≥‘𝑀)) |
26 | 15, 25 | sylan 577 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐾 − 1) ∈
(ℤ≥‘𝑀)) |
27 | | fzssp1 12677 |
. . . . . . . . . . . . . . 15
⊢ (𝑀...(𝐾 − 1)) ⊆ (𝑀...((𝐾 − 1) + 1)) |
28 | 5 | zcnd 11811 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐾 ∈ ℂ) |
29 | | ax-1cn 10310 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
ℂ |
30 | | npcan 10611 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐾 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝐾 −
1) + 1) = 𝐾) |
31 | 28, 29, 30 | sylancl 582 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾) |
32 | 31 | oveq2d 6921 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑀...((𝐾 − 1) + 1)) = (𝑀...𝐾)) |
33 | 27, 32 | syl5sseq 3878 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑀...(𝐾 − 1)) ⊆ (𝑀...𝐾)) |
34 | | elfzuz3 12632 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) |
35 | 1, 34 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾)) |
36 | | fzss2 12674 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘𝐾) → (𝑀...𝐾) ⊆ (𝑀...𝑁)) |
37 | 35, 36 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑀...𝐾) ⊆ (𝑀...𝑁)) |
38 | 33, 37 | sstrd 3837 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑀...(𝐾 − 1)) ⊆ (𝑀...𝑁)) |
39 | 38 | adantr 474 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑀...(𝐾 − 1)) ⊆ (𝑀...𝑁)) |
40 | 39 | sselda 3827 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → 𝑥 ∈ (𝑀...𝑁)) |
41 | | seqhomo.2 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
42 | 41 | adantlr 708 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
43 | 40, 42 | syldan 587 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) ∧ 𝑥 ∈ (𝑀...(𝐾 − 1))) → (𝐹‘𝑥) ∈ 𝑆) |
44 | | seqhomo.1 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
45 | 44 | adantlr 708 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
46 | 26, 43, 45 | seqcl 13115 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘(𝐾 − 1)) ∈ 𝑆) |
47 | 21, 24, 46 | rspcdva 3532 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍) = 𝑍) |
48 | 19, 47 | eqtrd 2861 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹‘𝐾)) = 𝑍) |
49 | 17, 48 | eqtrd 2861 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍) |
50 | 49 | ex 403 |
. . . . 5
⊢ (𝜑 → (𝐾 ∈ (ℤ≥‘(𝑀 + 1)) → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)) |
51 | | uzp1 12003 |
. . . . . 6
⊢ (𝐾 ∈
(ℤ≥‘𝑀) → (𝐾 = 𝑀 ∨ 𝐾 ∈ (ℤ≥‘(𝑀 + 1)))) |
52 | 3, 51 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐾 = 𝑀 ∨ 𝐾 ∈ (ℤ≥‘(𝑀 + 1)))) |
53 | 13, 50, 52 | mpjaod 893 |
. . . 4
⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍) |
54 | 53, 8 | eqtr4d 2864 |
. . 3
⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐹‘𝐾)) |
55 | | eqidd 2826 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑥) = (𝐹‘𝑥)) |
56 | 3, 54, 35, 55 | seqfveq2 13117 |
. 2
⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐹)‘𝑁)) |
57 | | fvex 6446 |
. . . . . 6
⊢ (𝐹‘𝐾) ∈ V |
58 | 57 | elsn 4412 |
. . . . 5
⊢ ((𝐹‘𝐾) ∈ {𝑍} ↔ (𝐹‘𝐾) = 𝑍) |
59 | 8, 58 | sylibr 226 |
. . . 4
⊢ (𝜑 → (𝐹‘𝐾) ∈ {𝑍}) |
60 | | simprl 789 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ {𝑍}) |
61 | | velsn 4413 |
. . . . . . . 8
⊢ (𝑥 ∈ {𝑍} ↔ 𝑥 = 𝑍) |
62 | 60, 61 | sylib 210 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → 𝑥 = 𝑍) |
63 | 62 | oveq1d 6920 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑍 + 𝑦)) |
64 | | oveq2 6913 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑍 + 𝑥) = (𝑍 + 𝑦)) |
65 | 64 | eqeq1d 2827 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → ((𝑍 + 𝑥) = 𝑍 ↔ (𝑍 + 𝑦) = 𝑍)) |
66 | | seqz.3 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑍) |
67 | 66 | ralrimiva 3175 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑍 + 𝑥) = 𝑍) |
68 | 67 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → ∀𝑥 ∈ 𝑆 (𝑍 + 𝑥) = 𝑍) |
69 | | simprr 791 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆) |
70 | 65, 68, 69 | rspcdva 3532 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → (𝑍 + 𝑦) = 𝑍) |
71 | 63, 70 | eqtrd 2861 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = 𝑍) |
72 | | ovex 6937 |
. . . . . 6
⊢ (𝑥 + 𝑦) ∈ V |
73 | 72 | elsn 4412 |
. . . . 5
⊢ ((𝑥 + 𝑦) ∈ {𝑍} ↔ (𝑥 + 𝑦) = 𝑍) |
74 | 71, 73 | sylibr 226 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ {𝑍}) |
75 | | peano2uz 12023 |
. . . . . . . 8
⊢ (𝐾 ∈
(ℤ≥‘𝑀) → (𝐾 + 1) ∈
(ℤ≥‘𝑀)) |
76 | 3, 75 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝐾 + 1) ∈
(ℤ≥‘𝑀)) |
77 | | fzss1 12673 |
. . . . . . 7
⊢ ((𝐾 + 1) ∈
(ℤ≥‘𝑀) → ((𝐾 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
78 | 76, 77 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((𝐾 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
79 | 78 | sselda 3827 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → 𝑥 ∈ (𝑀...𝑁)) |
80 | 79, 41 | syldan 587 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
81 | 59, 74, 35, 80 | seqcl2 13113 |
. . 3
⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) ∈ {𝑍}) |
82 | | elsni 4414 |
. . 3
⊢
((seq𝐾( + , 𝐹)‘𝑁) ∈ {𝑍} → (seq𝐾( + , 𝐹)‘𝑁) = 𝑍) |
83 | 81, 82 | syl 17 |
. 2
⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = 𝑍) |
84 | 56, 83 | eqtrd 2861 |
1
⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍) |