Step | Hyp | Ref
| Expression |
1 | | uzsinds.2 |
. 2
⊢ (𝑥 = 𝑁 → (𝜑 ↔ 𝜒)) |
2 | | oveq2 5850 |
. . . 4
⊢ (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀)) |
3 | 2 | raleqdv 2667 |
. . 3
⊢ (𝑤 = 𝑀 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑀)𝜑)) |
4 | | oveq2 5850 |
. . . 4
⊢ (𝑤 = 𝑘 → (𝑀...𝑤) = (𝑀...𝑘)) |
5 | 4 | raleqdv 2667 |
. . 3
⊢ (𝑤 = 𝑘 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑘)𝜑)) |
6 | | oveq2 5850 |
. . . 4
⊢ (𝑤 = (𝑘 + 1) → (𝑀...𝑤) = (𝑀...(𝑘 + 1))) |
7 | 6 | raleqdv 2667 |
. . 3
⊢ (𝑤 = (𝑘 + 1) → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑)) |
8 | | oveq2 5850 |
. . . 4
⊢ (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁)) |
9 | 8 | raleqdv 2667 |
. . 3
⊢ (𝑤 = 𝑁 → (∀𝑥 ∈ (𝑀...𝑤)𝜑 ↔ ∀𝑥 ∈ (𝑀...𝑁)𝜑)) |
10 | | ral0 3510 |
. . . . . . 7
⊢
∀𝑦 ∈
∅ 𝜓 |
11 | | zre 9195 |
. . . . . . . . . 10
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
ℝ) |
12 | 11 | ltm1d 8827 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℤ → (𝑀 − 1) < 𝑀) |
13 | | peano2zm 9229 |
. . . . . . . . . 10
⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈
ℤ) |
14 | | fzn 9977 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ)
→ ((𝑀 − 1) <
𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅)) |
15 | 13, 14 | mpdan 418 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅)) |
16 | 12, 15 | mpbid 146 |
. . . . . . . 8
⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 − 1)) = ∅) |
17 | 16 | raleqdv 2667 |
. . . . . . 7
⊢ (𝑀 ∈ ℤ →
(∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 ↔ ∀𝑦 ∈ ∅ 𝜓)) |
18 | 10, 17 | mpbiri 167 |
. . . . . 6
⊢ (𝑀 ∈ ℤ →
∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓) |
19 | | uzid 9480 |
. . . . . . 7
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
(ℤ≥‘𝑀)) |
20 | | uzsinds.3 |
. . . . . . . 8
⊢ (𝑥 ∈
(ℤ≥‘𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑)) |
21 | 20 | rgen 2519 |
. . . . . . 7
⊢
∀𝑥 ∈
(ℤ≥‘𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑) |
22 | | nfv 1516 |
. . . . . . . . 9
⊢
Ⅎ𝑥∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 |
23 | | nfsbc1v 2969 |
. . . . . . . . 9
⊢
Ⅎ𝑥[𝑀 / 𝑥]𝜑 |
24 | 22, 23 | nfim 1560 |
. . . . . . . 8
⊢
Ⅎ𝑥(∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑) |
25 | | oveq1 5849 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑀 → (𝑥 − 1) = (𝑀 − 1)) |
26 | 25 | oveq2d 5858 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑀 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑀 − 1))) |
27 | 26 | raleqdv 2667 |
. . . . . . . . 9
⊢ (𝑥 = 𝑀 → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓)) |
28 | | sbceq1a 2960 |
. . . . . . . . 9
⊢ (𝑥 = 𝑀 → (𝜑 ↔ [𝑀 / 𝑥]𝜑)) |
29 | 27, 28 | imbi12d 233 |
. . . . . . . 8
⊢ (𝑥 = 𝑀 → ((∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑) ↔ (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑))) |
30 | 24, 29 | rspc 2824 |
. . . . . . 7
⊢ (𝑀 ∈
(ℤ≥‘𝑀) → (∀𝑥 ∈ (ℤ≥‘𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑) → (∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑))) |
31 | 19, 21, 30 | mpisyl 1434 |
. . . . . 6
⊢ (𝑀 ∈ ℤ →
(∀𝑦 ∈ (𝑀...(𝑀 − 1))𝜓 → [𝑀 / 𝑥]𝜑)) |
32 | 18, 31 | mpd 13 |
. . . . 5
⊢ (𝑀 ∈ ℤ →
[𝑀 / 𝑥]𝜑) |
33 | | ralsns 3614 |
. . . . 5
⊢ (𝑀 ∈ ℤ →
(∀𝑥 ∈ {𝑀}𝜑 ↔ [𝑀 / 𝑥]𝜑)) |
34 | 32, 33 | mpbird 166 |
. . . 4
⊢ (𝑀 ∈ ℤ →
∀𝑥 ∈ {𝑀}𝜑) |
35 | | fzsn 10001 |
. . . . 5
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
36 | 35 | raleqdv 2667 |
. . . 4
⊢ (𝑀 ∈ ℤ →
(∀𝑥 ∈ (𝑀...𝑀)𝜑 ↔ ∀𝑥 ∈ {𝑀}𝜑)) |
37 | 34, 36 | mpbird 166 |
. . 3
⊢ (𝑀 ∈ ℤ →
∀𝑥 ∈ (𝑀...𝑀)𝜑) |
38 | | simpr 109 |
. . . . . 6
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ (𝑀...𝑘)𝜑) |
39 | | uzsinds.1 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
40 | 39 | cbvralv 2692 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
(𝑀...𝑘)𝜑 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓) |
41 | 38, 40 | sylib 121 |
. . . . . . . 8
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑦 ∈ (𝑀...𝑘)𝜓) |
42 | | eluzelz 9475 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → 𝑘 ∈ ℤ) |
43 | 42 | adantr 274 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℤ) |
44 | 43 | zcnd 9314 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 𝑘 ∈ ℂ) |
45 | | 1cnd 7915 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → 1 ∈ ℂ) |
46 | 44, 45 | pncand 8210 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘) |
47 | 46 | oveq2d 5858 |
. . . . . . . . . 10
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑀...((𝑘 + 1) − 1)) = (𝑀...𝑘)) |
48 | 47 | raleqdv 2667 |
. . . . . . . . 9
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...𝑘)𝜓)) |
49 | | peano2uz 9521 |
. . . . . . . . . . 11
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (𝑘 + 1) ∈
(ℤ≥‘𝑀)) |
50 | 49 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈
(ℤ≥‘𝑀)) |
51 | | nfv 1516 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 |
52 | | nfsbc1v 2969 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥[(𝑘 + 1) / 𝑥]𝜑 |
53 | 51, 52 | nfim 1560 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥(∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑) |
54 | | oveq1 5849 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1)) |
55 | 54 | oveq2d 5858 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝑘 + 1) → (𝑀...(𝑥 − 1)) = (𝑀...((𝑘 + 1) − 1))) |
56 | 55 | raleqdv 2667 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 ↔ ∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓)) |
57 | | sbceq1a 2960 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑘 + 1) → (𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑)) |
58 | 56, 57 | imbi12d 233 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑘 + 1) → ((∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑) ↔ (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑))) |
59 | 53, 58 | rspc 2824 |
. . . . . . . . . 10
⊢ ((𝑘 + 1) ∈
(ℤ≥‘𝑀) → (∀𝑥 ∈ (ℤ≥‘𝑀)(∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑))) |
60 | 50, 21, 59 | mpisyl 1434 |
. . . . . . . . 9
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...((𝑘 + 1) − 1))𝜓 → [(𝑘 + 1) / 𝑥]𝜑)) |
61 | 48, 60 | sylbird 169 |
. . . . . . . 8
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑦 ∈ (𝑀...𝑘)𝜓 → [(𝑘 + 1) / 𝑥]𝜑)) |
62 | 41, 61 | mpd 13 |
. . . . . . 7
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → [(𝑘 + 1) / 𝑥]𝜑) |
63 | 42 | peano2zd 9316 |
. . . . . . . . 9
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (𝑘 + 1) ∈ ℤ) |
64 | 63 | adantr 274 |
. . . . . . . 8
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (𝑘 + 1) ∈ ℤ) |
65 | | ralsns 3614 |
. . . . . . . 8
⊢ ((𝑘 + 1) ∈ ℤ →
(∀𝑥 ∈ {(𝑘 + 1)}𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑)) |
66 | 64, 65 | syl 14 |
. . . . . . 7
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑥 ∈ {(𝑘 + 1)}𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑)) |
67 | 62, 66 | mpbird 166 |
. . . . . 6
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ {(𝑘 + 1)}𝜑) |
68 | | ralun 3304 |
. . . . . 6
⊢
((∀𝑥 ∈
(𝑀...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑) |
69 | 38, 67, 68 | syl2anc 409 |
. . . . 5
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑) |
70 | | fzsuc 10004 |
. . . . . . 7
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (𝑀...(𝑘 + 1)) = ((𝑀...𝑘) ∪ {(𝑘 + 1)})) |
71 | 70 | raleqdv 2667 |
. . . . . 6
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)) |
72 | 71 | adantr 274 |
. . . . 5
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → (∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((𝑀...𝑘) ∪ {(𝑘 + 1)})𝜑)) |
73 | 69, 72 | mpbird 166 |
. . . 4
⊢ ((𝑘 ∈
(ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (𝑀...𝑘)𝜑) → ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑) |
74 | 73 | ex 114 |
. . 3
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (∀𝑥 ∈ (𝑀...𝑘)𝜑 → ∀𝑥 ∈ (𝑀...(𝑘 + 1))𝜑)) |
75 | 3, 5, 7, 9, 37, 74 | uzind4 9526 |
. 2
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → ∀𝑥 ∈ (𝑀...𝑁)𝜑) |
76 | | eluzfz2 9967 |
. 2
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
77 | 1, 75, 76 | rspcdva 2835 |
1
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝜒) |