Step | Hyp | Ref
| Expression |
1 | | df-seqfrec 10345 |
. 2
⊢ seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
2 | | seq3val.m |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℤ) |
3 | | fveq2 5468 |
. . . . . . . 8
⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) |
4 | 3 | eleq1d 2226 |
. . . . . . 7
⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) |
5 | | seq3val.f |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
6 | 5 | ralrimiva 2530 |
. . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆) |
7 | | uzid 9453 |
. . . . . . . 8
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
(ℤ≥‘𝑀)) |
8 | 2, 7 | syl 14 |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
9 | 4, 6, 8 | rspcdva 2821 |
. . . . . 6
⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆) |
10 | | ssv 3150 |
. . . . . . 7
⊢ 𝑆 ⊆ V |
11 | 10 | a1i 9 |
. . . . . 6
⊢ (𝜑 → 𝑆 ⊆ V) |
12 | | seq3val.pl |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
13 | 5, 12 | iseqovex 10355 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆) |
14 | | seq3val.r |
. . . . . 6
⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) |
15 | 2, 9, 11, 13, 14 | frecuzrdgrclt 10314 |
. . . . 5
⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝑀) × 𝑆)) |
16 | | ffn 5319 |
. . . . 5
⊢ (𝑅:ω⟶((ℤ≥‘𝑀) × 𝑆) → 𝑅 Fn ω) |
17 | 15, 16 | syl 14 |
. . . 4
⊢ (𝜑 → 𝑅 Fn ω) |
18 | | 1st2nd2 6123 |
. . . . . . . . . . . 12
⊢ (𝑢 ∈
((ℤ≥‘𝑀) × 𝑆) → 𝑢 = 〈(1st ‘𝑢), (2nd ‘𝑢)〉) |
19 | 18 | adantl 275 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → 𝑢 = 〈(1st ‘𝑢), (2nd ‘𝑢)〉) |
20 | 19 | fveq2d 5472 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘𝑢) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘〈(1st
‘𝑢), (2nd
‘𝑢)〉)) |
21 | | df-ov 5827 |
. . . . . . . . . 10
⊢
((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)(2nd ‘𝑢)) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘〈(1st
‘𝑢), (2nd
‘𝑢)〉) |
22 | 20, 21 | eqtr4di 2208 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘𝑢) = ((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)(2nd ‘𝑢))) |
23 | | xp1st 6113 |
. . . . . . . . . . 11
⊢ (𝑢 ∈
((ℤ≥‘𝑀) × 𝑆) → (1st ‘𝑢) ∈
(ℤ≥‘𝑀)) |
24 | 23 | adantl 275 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → (1st ‘𝑢) ∈
(ℤ≥‘𝑀)) |
25 | | xp2nd 6114 |
. . . . . . . . . . . 12
⊢ (𝑢 ∈
((ℤ≥‘𝑀) × 𝑆) → (2nd ‘𝑢) ∈ 𝑆) |
26 | 25 | adantl 275 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → (2nd ‘𝑢) ∈ 𝑆) |
27 | 26 | elexd 2725 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → (2nd ‘𝑢) ∈ V) |
28 | | peano2uz 9494 |
. . . . . . . . . . . 12
⊢
((1st ‘𝑢) ∈ (ℤ≥‘𝑀) → ((1st
‘𝑢) + 1) ∈
(ℤ≥‘𝑀)) |
29 | 24, 28 | syl 14 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢) + 1) ∈
(ℤ≥‘𝑀)) |
30 | 12 | caovclg 5973 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎 + 𝑏) ∈ 𝑆) |
31 | 30 | adantlr 469 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎 + 𝑏) ∈ 𝑆) |
32 | | fveq2 5468 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = ((1st ‘𝑢) + 1) → (𝐹‘𝑥) = (𝐹‘((1st ‘𝑢) + 1))) |
33 | 32 | eleq1d 2226 |
. . . . . . . . . . . . 13
⊢ (𝑥 = ((1st ‘𝑢) + 1) → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘((1st ‘𝑢) + 1)) ∈ 𝑆)) |
34 | 6 | adantr 274 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆) |
35 | 33, 34, 29 | rspcdva 2821 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → (𝐹‘((1st ‘𝑢) + 1)) ∈ 𝑆) |
36 | 31, 26, 35 | caovcld 5974 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1))) ∈ 𝑆) |
37 | | opelxpi 4618 |
. . . . . . . . . . 11
⊢
((((1st ‘𝑢) + 1) ∈
(ℤ≥‘𝑀) ∧ ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1))) ∈ 𝑆) → 〈((1st
‘𝑢) + 1),
((2nd ‘𝑢)
+ (𝐹‘((1st
‘𝑢) + 1)))〉
∈ ((ℤ≥‘𝑀) × 𝑆)) |
38 | 29, 36, 37 | syl2anc 409 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → 〈((1st
‘𝑢) + 1),
((2nd ‘𝑢)
+ (𝐹‘((1st
‘𝑢) + 1)))〉
∈ ((ℤ≥‘𝑀) × 𝑆)) |
39 | | oveq1 5831 |
. . . . . . . . . . . 12
⊢ (𝑥 = (1st ‘𝑢) → (𝑥 + 1) = ((1st ‘𝑢) + 1)) |
40 | | fvoveq1 5847 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (1st ‘𝑢) → (𝐹‘(𝑥 + 1)) = (𝐹‘((1st ‘𝑢) + 1))) |
41 | 40 | oveq2d 5840 |
. . . . . . . . . . . 12
⊢ (𝑥 = (1st ‘𝑢) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘((1st ‘𝑢) + 1)))) |
42 | 39, 41 | opeq12d 3749 |
. . . . . . . . . . 11
⊢ (𝑥 = (1st ‘𝑢) → 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉 = 〈((1st
‘𝑢) + 1), (𝑦 + (𝐹‘((1st ‘𝑢) + 1)))〉) |
43 | | oveq1 5831 |
. . . . . . . . . . . 12
⊢ (𝑦 = (2nd ‘𝑢) → (𝑦 + (𝐹‘((1st ‘𝑢) + 1))) = ((2nd
‘𝑢) + (𝐹‘((1st
‘𝑢) +
1)))) |
44 | 43 | opeq2d 3748 |
. . . . . . . . . . 11
⊢ (𝑦 = (2nd ‘𝑢) → 〈((1st
‘𝑢) + 1), (𝑦 + (𝐹‘((1st ‘𝑢) + 1)))〉 =
〈((1st ‘𝑢) + 1), ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1)))〉) |
45 | | eqid 2157 |
. . . . . . . . . . 11
⊢ (𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉) = (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉) |
46 | 42, 44, 45 | ovmpog 5955 |
. . . . . . . . . 10
⊢
(((1st ‘𝑢) ∈ (ℤ≥‘𝑀) ∧ (2nd
‘𝑢) ∈ V ∧
〈((1st ‘𝑢) + 1), ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1)))〉 ∈
((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)(2nd ‘𝑢)) = 〈((1st
‘𝑢) + 1),
((2nd ‘𝑢)
+ (𝐹‘((1st
‘𝑢) +
1)))〉) |
47 | 24, 27, 38, 46 | syl3anc 1220 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)(2nd ‘𝑢)) = 〈((1st
‘𝑢) + 1),
((2nd ‘𝑢)
+ (𝐹‘((1st
‘𝑢) +
1)))〉) |
48 | 22, 47 | eqtrd 2190 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘𝑢) = 〈((1st ‘𝑢) + 1), ((2nd
‘𝑢) + (𝐹‘((1st
‘𝑢) +
1)))〉) |
49 | 48, 38 | eqeltrd 2234 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆)) |
50 | 49 | ralrimiva 2530 |
. . . . . 6
⊢ (𝜑 → ∀𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆)) |
51 | | opelxpi 4618 |
. . . . . . 7
⊢ ((𝑀 ∈
(ℤ≥‘𝑀) ∧ (𝐹‘𝑀) ∈ 𝑆) → 〈𝑀, (𝐹‘𝑀)〉 ∈
((ℤ≥‘𝑀) × 𝑆)) |
52 | 8, 9, 51 | syl2anc 409 |
. . . . . 6
⊢ (𝜑 → 〈𝑀, (𝐹‘𝑀)〉 ∈
((ℤ≥‘𝑀) × 𝑆)) |
53 | 50, 52 | jca 304 |
. . . . 5
⊢ (𝜑 → (∀𝑢 ∈
((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆) ∧ 〈𝑀, (𝐹‘𝑀)〉 ∈
((ℤ≥‘𝑀) × 𝑆))) |
54 | | frecfcl 6352 |
. . . . 5
⊢
((∀𝑢 ∈
((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆) ∧ 〈𝑀, (𝐹‘𝑀)〉 ∈
((ℤ≥‘𝑀) × 𝑆)) → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉):ω⟶((ℤ≥‘𝑀) × 𝑆)) |
55 | | ffn 5319 |
. . . . 5
⊢
(frec((𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉):ω⟶((ℤ≥‘𝑀) × 𝑆) → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦
∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) Fn ω) |
56 | 53, 54, 55 | 3syl 17 |
. . . 4
⊢ (𝜑 → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) Fn ω) |
57 | | fveq2 5468 |
. . . . . . . 8
⊢ (𝑐 = ∅ → (𝑅‘𝑐) = (𝑅‘∅)) |
58 | | fveq2 5468 |
. . . . . . . 8
⊢ (𝑐 = ∅ → (frec((𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘∅)) |
59 | 57, 58 | eqeq12d 2172 |
. . . . . . 7
⊢ (𝑐 = ∅ → ((𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑐) ↔ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘∅))) |
60 | 59 | imbi2d 229 |
. . . . . 6
⊢ (𝑐 = ∅ → ((𝜑 → (𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑐)) ↔ (𝜑 → (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘∅)))) |
61 | | fveq2 5468 |
. . . . . . . 8
⊢ (𝑐 = 𝑘 → (𝑅‘𝑐) = (𝑅‘𝑘)) |
62 | | fveq2 5468 |
. . . . . . . 8
⊢ (𝑐 = 𝑘 → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) |
63 | 61, 62 | eqeq12d 2172 |
. . . . . . 7
⊢ (𝑐 = 𝑘 → ((𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑐) ↔ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘))) |
64 | 63 | imbi2d 229 |
. . . . . 6
⊢ (𝑐 = 𝑘 → ((𝜑 → (𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑐)) ↔ (𝜑 → (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)))) |
65 | | fveq2 5468 |
. . . . . . . 8
⊢ (𝑐 = suc 𝑘 → (𝑅‘𝑐) = (𝑅‘suc 𝑘)) |
66 | | fveq2 5468 |
. . . . . . . 8
⊢ (𝑐 = suc 𝑘 → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘suc 𝑘)) |
67 | 65, 66 | eqeq12d 2172 |
. . . . . . 7
⊢ (𝑐 = suc 𝑘 → ((𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑐) ↔ (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘suc 𝑘))) |
68 | 67 | imbi2d 229 |
. . . . . 6
⊢ (𝑐 = suc 𝑘 → ((𝜑 → (𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑐)) ↔ (𝜑 → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘suc 𝑘)))) |
69 | | fveq2 5468 |
. . . . . . . 8
⊢ (𝑐 = 𝑛 → (𝑅‘𝑐) = (𝑅‘𝑛)) |
70 | | fveq2 5468 |
. . . . . . . 8
⊢ (𝑐 = 𝑛 → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑛)) |
71 | 69, 70 | eqeq12d 2172 |
. . . . . . 7
⊢ (𝑐 = 𝑛 → ((𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑐) ↔ (𝑅‘𝑛) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑛))) |
72 | 71 | imbi2d 229 |
. . . . . 6
⊢ (𝑐 = 𝑛 → ((𝜑 → (𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑐)) ↔ (𝜑 → (𝑅‘𝑛) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑛)))) |
73 | 14 | fveq1i 5469 |
. . . . . . . 8
⊢ (𝑅‘∅) = (frec((𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉)‘∅) |
74 | | frec0g 6344 |
. . . . . . . . 9
⊢
(〈𝑀, (𝐹‘𝑀)〉 ∈
((ℤ≥‘𝑀) × 𝑆) → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉)‘∅) = 〈𝑀, (𝐹‘𝑀)〉) |
75 | 52, 74 | syl 14 |
. . . . . . . 8
⊢ (𝜑 → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉)‘∅) = 〈𝑀, (𝐹‘𝑀)〉) |
76 | 73, 75 | syl5eq 2202 |
. . . . . . 7
⊢ (𝜑 → (𝑅‘∅) = 〈𝑀, (𝐹‘𝑀)〉) |
77 | | frec0g 6344 |
. . . . . . . 8
⊢
(〈𝑀, (𝐹‘𝑀)〉 ∈
((ℤ≥‘𝑀) × 𝑆) → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘∅) = 〈𝑀, (𝐹‘𝑀)〉) |
78 | 52, 77 | syl 14 |
. . . . . . 7
⊢ (𝜑 → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘∅) = 〈𝑀, (𝐹‘𝑀)〉) |
79 | 76, 78 | eqtr4d 2193 |
. . . . . 6
⊢ (𝜑 → (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘∅)) |
80 | 15 | ad2antlr 481 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → 𝑅:ω⟶((ℤ≥‘𝑀) × 𝑆)) |
81 | | simpll 519 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → 𝑘 ∈ ω) |
82 | 80, 81 | ffvelrnd 5603 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (𝑅‘𝑘) ∈ ((ℤ≥‘𝑀) × 𝑆)) |
83 | | xp1st 6113 |
. . . . . . . . . . 11
⊢ ((𝑅‘𝑘) ∈ ((ℤ≥‘𝑀) × 𝑆) → (1st ‘(𝑅‘𝑘)) ∈ (ℤ≥‘𝑀)) |
84 | 82, 83 | syl 14 |
. . . . . . . . . 10
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (1st ‘(𝑅‘𝑘)) ∈ (ℤ≥‘𝑀)) |
85 | | xp2nd 6114 |
. . . . . . . . . . . 12
⊢ ((𝑅‘𝑘) ∈ ((ℤ≥‘𝑀) × 𝑆) → (2nd ‘(𝑅‘𝑘)) ∈ 𝑆) |
86 | 82, 85 | syl 14 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (2nd ‘(𝑅‘𝑘)) ∈ 𝑆) |
87 | 86 | elexd 2725 |
. . . . . . . . . 10
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (2nd ‘(𝑅‘𝑘)) ∈ V) |
88 | 30 | adantll 468 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎 + 𝑏) ∈ 𝑆) |
89 | 88 | adantlr 469 |
. . . . . . . . . . . . . 14
⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎 + 𝑏) ∈ 𝑆) |
90 | | fveq2 5468 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = ((1st
‘(𝑅‘𝑘)) + 1) → (𝐹‘𝑎) = (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))) |
91 | 90 | eleq1d 2226 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = ((1st
‘(𝑅‘𝑘)) + 1) → ((𝐹‘𝑎) ∈ 𝑆 ↔ (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)) ∈ 𝑆)) |
92 | | fveq2 5468 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎)) |
93 | 92 | eleq1d 2226 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑎 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑎) ∈ 𝑆)) |
94 | 93 | cbvralv 2680 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑥 ∈
(ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆 ↔ ∀𝑎 ∈ (ℤ≥‘𝑀)(𝐹‘𝑎) ∈ 𝑆) |
95 | 6, 94 | sylib 121 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∀𝑎 ∈ (ℤ≥‘𝑀)(𝐹‘𝑎) ∈ 𝑆) |
96 | 95 | ad2antlr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → ∀𝑎 ∈ (ℤ≥‘𝑀)(𝐹‘𝑎) ∈ 𝑆) |
97 | | peano2uz 9494 |
. . . . . . . . . . . . . . . 16
⊢
((1st ‘(𝑅‘𝑘)) ∈ (ℤ≥‘𝑀) → ((1st
‘(𝑅‘𝑘)) + 1) ∈
(ℤ≥‘𝑀)) |
98 | 84, 97 | syl 14 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → ((1st ‘(𝑅‘𝑘)) + 1) ∈
(ℤ≥‘𝑀)) |
99 | 91, 96, 98 | rspcdva 2821 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)) ∈ 𝑆) |
100 | 89, 86, 99 | caovcld 5974 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))) ∈ 𝑆) |
101 | | fvoveq1 5847 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (1st ‘(𝑅‘𝑘)) → (𝐹‘(𝑧 + 1)) = (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))) |
102 | 101 | oveq2d 5840 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (1st ‘(𝑅‘𝑘)) → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))) |
103 | | oveq1 5831 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = (2nd ‘(𝑅‘𝑘)) → (𝑤 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))) = ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))) |
104 | | eqid 2157 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈
(ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) = (𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) |
105 | 102, 103,
104 | ovmpog 5955 |
. . . . . . . . . . . . 13
⊢
(((1st ‘(𝑅‘𝑘)) ∈ (ℤ≥‘𝑀) ∧ (2nd
‘(𝑅‘𝑘)) ∈ 𝑆 ∧ ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))) ∈ 𝑆) → ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘))) = ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))) |
106 | 84, 86, 100, 105 | syl3anc 1220 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘))) = ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))) |
107 | 106 | opeq2d 3748 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → 〈((1st
‘(𝑅‘𝑘)) + 1), ((1st
‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))〉 = 〈((1st
‘(𝑅‘𝑘)) + 1), ((2nd
‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))〉) |
108 | 106, 100 | eqeltrd 2234 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘))) ∈ 𝑆) |
109 | | opelxpi 4618 |
. . . . . . . . . . . 12
⊢
((((1st ‘(𝑅‘𝑘)) + 1) ∈
(ℤ≥‘𝑀) ∧ ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘))) ∈ 𝑆) → 〈((1st
‘(𝑅‘𝑘)) + 1), ((1st
‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))〉 ∈
((ℤ≥‘𝑀) × 𝑆)) |
110 | 98, 108, 109 | syl2anc 409 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → 〈((1st
‘(𝑅‘𝑘)) + 1), ((1st
‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))〉 ∈
((ℤ≥‘𝑀) × 𝑆)) |
111 | 107, 110 | eqeltrrd 2235 |
. . . . . . . . . 10
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → 〈((1st
‘(𝑅‘𝑘)) + 1), ((2nd
‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))〉 ∈
((ℤ≥‘𝑀) × 𝑆)) |
112 | | oveq1 5831 |
. . . . . . . . . . . 12
⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → (𝑥 + 1) = ((1st ‘(𝑅‘𝑘)) + 1)) |
113 | | fvoveq1 5847 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → (𝐹‘(𝑥 + 1)) = (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))) |
114 | 113 | oveq2d 5840 |
. . . . . . . . . . . 12
⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))) |
115 | 112, 114 | opeq12d 3749 |
. . . . . . . . . . 11
⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉 = 〈((1st
‘(𝑅‘𝑘)) + 1), (𝑦 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))〉) |
116 | | oveq1 5831 |
. . . . . . . . . . . 12
⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → (𝑦 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))) = ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))) |
117 | 116 | opeq2d 3748 |
. . . . . . . . . . 11
⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → 〈((1st
‘(𝑅‘𝑘)) + 1), (𝑦 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))〉 = 〈((1st
‘(𝑅‘𝑘)) + 1), ((2nd
‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))〉) |
118 | 115, 117,
45 | ovmpog 5955 |
. . . . . . . . . 10
⊢
(((1st ‘(𝑅‘𝑘)) ∈ (ℤ≥‘𝑀) ∧ (2nd
‘(𝑅‘𝑘)) ∈ V ∧
〈((1st ‘(𝑅‘𝑘)) + 1), ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))〉 ∈
((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)(2nd ‘(𝑅‘𝑘))) = 〈((1st ‘(𝑅‘𝑘)) + 1), ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))〉) |
119 | 84, 87, 111, 118 | syl3anc 1220 |
. . . . . . . . 9
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)(2nd ‘(𝑅‘𝑘))) = 〈((1st ‘(𝑅‘𝑘)) + 1), ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))〉) |
120 | 50 | ad2antlr 481 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → ∀𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆)) |
121 | 52 | ad2antlr 481 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → 〈𝑀, (𝐹‘𝑀)〉 ∈
((ℤ≥‘𝑀) × 𝑆)) |
122 | | frecsuc 6354 |
. . . . . . . . . . . 12
⊢
((∀𝑢 ∈
((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆) ∧ 〈𝑀, (𝐹‘𝑀)〉 ∈
((ℤ≥‘𝑀) × 𝑆) ∧ 𝑘 ∈ ω) → (frec((𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘(frec((𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘))) |
123 | 120, 121,
81, 122 | syl3anc 1220 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘(frec((𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘))) |
124 | | simpr 109 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) |
125 | 124 | fveq2d 5472 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘(𝑅‘𝑘)) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘(frec((𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘))) |
126 | 123, 125 | eqtr4d 2193 |
. . . . . . . . . 10
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘(𝑅‘𝑘))) |
127 | | 1st2nd2 6123 |
. . . . . . . . . . . . 13
⊢ ((𝑅‘𝑘) ∈ ((ℤ≥‘𝑀) × 𝑆) → (𝑅‘𝑘) = 〈(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))〉) |
128 | 82, 127 | syl 14 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (𝑅‘𝑘) = 〈(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))〉) |
129 | 128 | fveq2d 5472 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘(𝑅‘𝑘)) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘〈(1st
‘(𝑅‘𝑘)), (2nd
‘(𝑅‘𝑘))〉)) |
130 | | df-ov 5827 |
. . . . . . . . . . 11
⊢
((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)(2nd ‘(𝑅‘𝑘))) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘〈(1st
‘(𝑅‘𝑘)), (2nd
‘(𝑅‘𝑘))〉) |
131 | 129, 130 | eqtr4di 2208 |
. . . . . . . . . 10
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)‘(𝑅‘𝑘)) = ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)(2nd ‘(𝑅‘𝑘)))) |
132 | 126, 131 | eqtrd 2190 |
. . . . . . . . 9
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘suc 𝑘) = ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉)(2nd ‘(𝑅‘𝑘)))) |
133 | 14 | fveq1i 5469 |
. . . . . . . . . . . . . . 15
⊢ (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉)‘suc 𝑘) |
134 | 19 | fveq2d 5472 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘𝑢) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘〈(1st
‘𝑢), (2nd
‘𝑢)〉)) |
135 | | df-ov 5827 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)(2nd ‘𝑢)) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘〈(1st
‘𝑢), (2nd
‘𝑢)〉) |
136 | 134, 135 | eqtr4di 2208 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘𝑢) = ((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)(2nd ‘𝑢))) |
137 | | fvoveq1 5847 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧 = (1st ‘𝑢) → (𝐹‘(𝑧 + 1)) = (𝐹‘((1st ‘𝑢) + 1))) |
138 | 137 | oveq2d 5840 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑧 = (1st ‘𝑢) → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘((1st ‘𝑢) + 1)))) |
139 | | oveq1 5831 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑤 = (2nd ‘𝑢) → (𝑤 + (𝐹‘((1st ‘𝑢) + 1))) = ((2nd
‘𝑢) + (𝐹‘((1st
‘𝑢) +
1)))) |
140 | 138, 139,
104 | ovmpog 5955 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((1st ‘𝑢) ∈ (ℤ≥‘𝑀) ∧ (2nd
‘𝑢) ∈ 𝑆 ∧ ((2nd
‘𝑢) + (𝐹‘((1st
‘𝑢) + 1))) ∈
𝑆) → ((1st
‘𝑢)(𝑧 ∈
(ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢)) = ((2nd
‘𝑢) + (𝐹‘((1st
‘𝑢) +
1)))) |
141 | 24, 26, 36, 140 | syl3anc 1220 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢)) = ((2nd
‘𝑢) + (𝐹‘((1st
‘𝑢) +
1)))) |
142 | 141, 36 | eqeltrd 2234 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢)) ∈ 𝑆) |
143 | | opelxpi 4618 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((1st ‘𝑢) + 1) ∈
(ℤ≥‘𝑀) ∧ ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢)) ∈ 𝑆) → 〈((1st ‘𝑢) + 1), ((1st
‘𝑢)(𝑧 ∈
(ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))〉 ∈
((ℤ≥‘𝑀) × 𝑆)) |
144 | 29, 142, 143 | syl2anc 409 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → 〈((1st
‘𝑢) + 1),
((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))〉 ∈
((ℤ≥‘𝑀) × 𝑆)) |
145 | | oveq1 5831 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = (1st ‘𝑢) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)) |
146 | 39, 145 | opeq12d 3749 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = (1st ‘𝑢) → 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉 = 〈((1st ‘𝑢) + 1), ((1st
‘𝑢)(𝑧 ∈
(ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉) |
147 | | oveq2 5832 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = (2nd ‘𝑢) → ((1st
‘𝑢)(𝑧 ∈
(ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))) |
148 | 147 | opeq2d 3748 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = (2nd ‘𝑢) → 〈((1st
‘𝑢) + 1),
((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉 = 〈((1st ‘𝑢) + 1), ((1st
‘𝑢)(𝑧 ∈
(ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))〉) |
149 | | eqid 2157 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉) = (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉) |
150 | 146, 148,
149 | ovmpog 5955 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((1st ‘𝑢) ∈ (ℤ≥‘𝑀) ∧ (2nd
‘𝑢) ∈ V ∧
〈((1st ‘𝑢) + 1), ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))〉 ∈
((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)(2nd ‘𝑢)) = 〈((1st
‘𝑢) + 1),
((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))〉) |
151 | 24, 27, 144, 150 | syl3anc 1220 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)(2nd ‘𝑢)) = 〈((1st
‘𝑢) + 1),
((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))〉) |
152 | 136, 151 | eqtrd 2190 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘𝑢) = 〈((1st ‘𝑢) + 1), ((1st
‘𝑢)(𝑧 ∈
(ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))〉) |
153 | 152, 144 | eqeltrd 2234 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆)) |
154 | 153 | ralrimiva 2530 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ∀𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆)) |
155 | 154 | ad2antlr 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → ∀𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆)) |
156 | | frecsuc 6354 |
. . . . . . . . . . . . . . . 16
⊢
((∀𝑢 ∈
((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆) ∧ 〈𝑀, (𝐹‘𝑀)〉 ∈
((ℤ≥‘𝑀) × 𝑆) ∧ 𝑘 ∈ ω) → (frec((𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉)‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘(frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘))) |
157 | 155, 121,
81, 156 | syl3anc 1220 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉)‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘(frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘))) |
158 | 133, 157 | syl5eq 2202 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘(frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘))) |
159 | 14 | fveq1i 5469 |
. . . . . . . . . . . . . . 15
⊢ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘) |
160 | 159 | fveq2i 5471 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘(𝑅‘𝑘)) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘(frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) |
161 | 158, 160 | eqtr4di 2208 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘(𝑅‘𝑘))) |
162 | 128 | fveq2d 5472 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘(𝑅‘𝑘)) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘〈(1st
‘(𝑅‘𝑘)), (2nd
‘(𝑅‘𝑘))〉)) |
163 | 161, 162 | eqtrd 2190 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘〈(1st
‘(𝑅‘𝑘)), (2nd
‘(𝑅‘𝑘))〉)) |
164 | | df-ov 5827 |
. . . . . . . . . . . 12
⊢
((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)(2nd ‘(𝑅‘𝑘))) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)‘〈(1st
‘(𝑅‘𝑘)), (2nd
‘(𝑅‘𝑘))〉) |
165 | 163, 164 | eqtr4di 2208 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (𝑅‘suc 𝑘) = ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)(2nd ‘(𝑅‘𝑘)))) |
166 | | oveq1 5831 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)) |
167 | 112, 166 | opeq12d 3749 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉 = 〈((1st
‘(𝑅‘𝑘)) + 1), ((1st
‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉) |
168 | | oveq2 5832 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))) |
169 | 168 | opeq2d 3748 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → 〈((1st
‘(𝑅‘𝑘)) + 1), ((1st
‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉 = 〈((1st
‘(𝑅‘𝑘)) + 1), ((1st
‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))〉) |
170 | 167, 169,
149 | ovmpog 5955 |
. . . . . . . . . . . 12
⊢
(((1st ‘(𝑅‘𝑘)) ∈ (ℤ≥‘𝑀) ∧ (2nd
‘(𝑅‘𝑘)) ∈ V ∧
〈((1st ‘(𝑅‘𝑘)) + 1), ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))〉 ∈
((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)(2nd ‘(𝑅‘𝑘))) = 〈((1st ‘(𝑅‘𝑘)) + 1), ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))〉) |
171 | 84, 87, 110, 170 | syl3anc 1220 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉)(2nd ‘(𝑅‘𝑘))) = 〈((1st ‘(𝑅‘𝑘)) + 1), ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))〉) |
172 | 165, 171 | eqtrd 2190 |
. . . . . . . . . 10
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (𝑅‘suc 𝑘) = 〈((1st ‘(𝑅‘𝑘)) + 1), ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))〉) |
173 | 172, 107 | eqtrd 2190 |
. . . . . . . . 9
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (𝑅‘suc 𝑘) = 〈((1st ‘(𝑅‘𝑘)) + 1), ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))〉) |
174 | 119, 132,
173 | 3eqtr4rd 2201 |
. . . . . . . 8
⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘suc 𝑘)) |
175 | 174 | exp31 362 |
. . . . . . 7
⊢ (𝑘 ∈ ω → (𝜑 → ((𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘) → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘suc 𝑘)))) |
176 | 175 | a2d 26 |
. . . . . 6
⊢ (𝑘 ∈ ω → ((𝜑 → (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑘)) → (𝜑 → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘suc 𝑘)))) |
177 | 60, 64, 68, 72, 79, 176 | finds 4559 |
. . . . 5
⊢ (𝑛 ∈ ω → (𝜑 → (𝑅‘𝑛) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑛))) |
178 | 177 | impcom 124 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (𝑅‘𝑛) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)‘𝑛)) |
179 | 17, 56, 178 | eqfnfvd 5568 |
. . 3
⊢ (𝜑 → 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)) |
180 | 179 | rneqd 4815 |
. 2
⊢ (𝜑 → ran 𝑅 = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)) |
181 | 1, 180 | eqtr4id 2209 |
1
⊢ (𝜑 → seq𝑀( + , 𝐹) = ran 𝑅) |