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Theorem seq3val 10224
Description: Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 10227, seq3-1 10226 and seq3p1 10228, as further development can be done in terms of those. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 4-Nov-2022.)
Hypotheses
Ref Expression
seq3val.m (𝜑𝑀 ∈ ℤ)
seq3val.r 𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)
seq3val.f ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
seq3val.pl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
Assertion
Ref Expression
seq3val (𝜑 → seq𝑀( + , 𝐹) = ran 𝑅)
Distinct variable groups:   𝑥, + ,𝑦,𝑤,𝑧   𝑥,𝐹,𝑦,𝑤,𝑧   𝑥,𝑀,𝑦,𝑤,𝑧   𝑥,𝑅,𝑦,𝑤,𝑧   𝑥,𝑆,𝑦,𝑤,𝑧   𝜑,𝑥,𝑦,𝑤,𝑧

Proof of Theorem seq3val
Dummy variables 𝑎 𝑏 𝑘 𝑐 𝑛 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seq3val.m . . . . . 6 (𝜑𝑀 ∈ ℤ)
2 fveq2 5414 . . . . . . . 8 (𝑥 = 𝑀 → (𝐹𝑥) = (𝐹𝑀))
32eleq1d 2206 . . . . . . 7 (𝑥 = 𝑀 → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹𝑀) ∈ 𝑆))
4 seq3val.f . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
54ralrimiva 2503 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
6 uzid 9333 . . . . . . . 8 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
71, 6syl 14 . . . . . . 7 (𝜑𝑀 ∈ (ℤ𝑀))
83, 5, 7rspcdva 2789 . . . . . 6 (𝜑 → (𝐹𝑀) ∈ 𝑆)
9 ssv 3114 . . . . . . 7 𝑆 ⊆ V
109a1i 9 . . . . . 6 (𝜑𝑆 ⊆ V)
11 seq3val.pl . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
124, 11iseqovex 10222 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝑆)) → (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆)
13 seq3val.r . . . . . 6 𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)
141, 8, 10, 12, 13frecuzrdgrclt 10181 . . . . 5 (𝜑𝑅:ω⟶((ℤ𝑀) × 𝑆))
15 ffn 5267 . . . . 5 (𝑅:ω⟶((ℤ𝑀) × 𝑆) → 𝑅 Fn ω)
1614, 15syl 14 . . . 4 (𝜑𝑅 Fn ω)
17 1st2nd2 6066 . . . . . . . . . . . 12 (𝑢 ∈ ((ℤ𝑀) × 𝑆) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
1817adantl 275 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
1918fveq2d 5418 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st𝑢), (2nd𝑢)⟩))
20 df-ov 5770 . . . . . . . . . 10 ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd𝑢)) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st𝑢), (2nd𝑢)⟩)
2119, 20syl6eqr 2188 . . . . . . . . 9 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) = ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd𝑢)))
22 xp1st 6056 . . . . . . . . . . 11 (𝑢 ∈ ((ℤ𝑀) × 𝑆) → (1st𝑢) ∈ (ℤ𝑀))
2322adantl 275 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → (1st𝑢) ∈ (ℤ𝑀))
24 xp2nd 6057 . . . . . . . . . . . 12 (𝑢 ∈ ((ℤ𝑀) × 𝑆) → (2nd𝑢) ∈ 𝑆)
2524adantl 275 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → (2nd𝑢) ∈ 𝑆)
2625elexd 2694 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → (2nd𝑢) ∈ V)
27 peano2uz 9371 . . . . . . . . . . . 12 ((1st𝑢) ∈ (ℤ𝑀) → ((1st𝑢) + 1) ∈ (ℤ𝑀))
2823, 27syl 14 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((1st𝑢) + 1) ∈ (ℤ𝑀))
2911caovclg 5916 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝑆𝑏𝑆)) → (𝑎 + 𝑏) ∈ 𝑆)
3029adantlr 468 . . . . . . . . . . . 12 (((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) ∧ (𝑎𝑆𝑏𝑆)) → (𝑎 + 𝑏) ∈ 𝑆)
31 fveq2 5414 . . . . . . . . . . . . . 14 (𝑥 = ((1st𝑢) + 1) → (𝐹𝑥) = (𝐹‘((1st𝑢) + 1)))
3231eleq1d 2206 . . . . . . . . . . . . 13 (𝑥 = ((1st𝑢) + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘((1st𝑢) + 1)) ∈ 𝑆))
335adantr 274 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
3432, 33, 28rspcdva 2789 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → (𝐹‘((1st𝑢) + 1)) ∈ 𝑆)
3530, 25, 34caovcld 5917 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))) ∈ 𝑆)
36 opelxpi 4566 . . . . . . . . . . 11 ((((1st𝑢) + 1) ∈ (ℤ𝑀) ∧ ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))) ∈ 𝑆) → ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩ ∈ ((ℤ𝑀) × 𝑆))
3728, 35, 36syl2anc 408 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩ ∈ ((ℤ𝑀) × 𝑆))
38 oveq1 5774 . . . . . . . . . . . 12 (𝑥 = (1st𝑢) → (𝑥 + 1) = ((1st𝑢) + 1))
39 fvoveq1 5790 . . . . . . . . . . . . 13 (𝑥 = (1st𝑢) → (𝐹‘(𝑥 + 1)) = (𝐹‘((1st𝑢) + 1)))
4039oveq2d 5783 . . . . . . . . . . . 12 (𝑥 = (1st𝑢) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘((1st𝑢) + 1))))
4138, 40opeq12d 3708 . . . . . . . . . . 11 (𝑥 = (1st𝑢) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨((1st𝑢) + 1), (𝑦 + (𝐹‘((1st𝑢) + 1)))⟩)
42 oveq1 5774 . . . . . . . . . . . 12 (𝑦 = (2nd𝑢) → (𝑦 + (𝐹‘((1st𝑢) + 1))) = ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))))
4342opeq2d 3707 . . . . . . . . . . 11 (𝑦 = (2nd𝑢) → ⟨((1st𝑢) + 1), (𝑦 + (𝐹‘((1st𝑢) + 1)))⟩ = ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩)
44 eqid 2137 . . . . . . . . . . 11 (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)
4541, 43, 44ovmpog 5898 . . . . . . . . . 10 (((1st𝑢) ∈ (ℤ𝑀) ∧ (2nd𝑢) ∈ V ∧ ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩ ∈ ((ℤ𝑀) × 𝑆)) → ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd𝑢)) = ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩)
4623, 26, 37, 45syl3anc 1216 . . . . . . . . 9 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd𝑢)) = ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩)
4721, 46eqtrd 2170 . . . . . . . 8 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) = ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩)
4847, 37eqeltrd 2214 . . . . . . 7 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆))
4948ralrimiva 2503 . . . . . 6 (𝜑 → ∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆))
50 opelxpi 4566 . . . . . . 7 ((𝑀 ∈ (ℤ𝑀) ∧ (𝐹𝑀) ∈ 𝑆) → ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆))
517, 8, 50syl2anc 408 . . . . . 6 (𝜑 → ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆))
5249, 51jca 304 . . . . 5 (𝜑 → (∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆) ∧ ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆)))
53 frecfcl 6295 . . . . 5 ((∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆) ∧ ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆)) → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩):ω⟶((ℤ𝑀) × 𝑆))
54 ffn 5267 . . . . 5 (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩):ω⟶((ℤ𝑀) × 𝑆) → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) Fn ω)
5552, 53, 543syl 17 . . . 4 (𝜑 → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) Fn ω)
56 fveq2 5414 . . . . . . . 8 (𝑐 = ∅ → (𝑅𝑐) = (𝑅‘∅))
57 fveq2 5414 . . . . . . . 8 (𝑐 = ∅ → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅))
5856, 57eqeq12d 2152 . . . . . . 7 (𝑐 = ∅ → ((𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) ↔ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅)))
5958imbi2d 229 . . . . . 6 (𝑐 = ∅ → ((𝜑 → (𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅))))
60 fveq2 5414 . . . . . . . 8 (𝑐 = 𝑘 → (𝑅𝑐) = (𝑅𝑘))
61 fveq2 5414 . . . . . . . 8 (𝑐 = 𝑘 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘))
6260, 61eqeq12d 2152 . . . . . . 7 (𝑐 = 𝑘 → ((𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) ↔ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
6362imbi2d 229 . . . . . 6 (𝑐 = 𝑘 → ((𝜑 → (𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘))))
64 fveq2 5414 . . . . . . . 8 (𝑐 = suc 𝑘 → (𝑅𝑐) = (𝑅‘suc 𝑘))
65 fveq2 5414 . . . . . . . 8 (𝑐 = suc 𝑘 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))
6664, 65eqeq12d 2152 . . . . . . 7 (𝑐 = suc 𝑘 → ((𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) ↔ (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘)))
6766imbi2d 229 . . . . . 6 (𝑐 = suc 𝑘 → ((𝜑 → (𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))))
68 fveq2 5414 . . . . . . . 8 (𝑐 = 𝑛 → (𝑅𝑐) = (𝑅𝑛))
69 fveq2 5414 . . . . . . . 8 (𝑐 = 𝑛 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛))
7068, 69eqeq12d 2152 . . . . . . 7 (𝑐 = 𝑛 → ((𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) ↔ (𝑅𝑛) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛)))
7170imbi2d 229 . . . . . 6 (𝑐 = 𝑛 → ((𝜑 → (𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅𝑛) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛))))
7213fveq1i 5415 . . . . . . . 8 (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅)
73 frec0g 6287 . . . . . . . . 9 (⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
7451, 73syl 14 . . . . . . . 8 (𝜑 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
7572, 74syl5eq 2182 . . . . . . 7 (𝜑 → (𝑅‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
76 frec0g 6287 . . . . . . . 8 (⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
7751, 76syl 14 . . . . . . 7 (𝜑 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
7875, 77eqtr4d 2173 . . . . . 6 (𝜑 → (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅))
7914ad2antlr 480 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → 𝑅:ω⟶((ℤ𝑀) × 𝑆))
80 simpll 518 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → 𝑘 ∈ ω)
8179, 80ffvelrnd 5549 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅𝑘) ∈ ((ℤ𝑀) × 𝑆))
82 xp1st 6056 . . . . . . . . . . 11 ((𝑅𝑘) ∈ ((ℤ𝑀) × 𝑆) → (1st ‘(𝑅𝑘)) ∈ (ℤ𝑀))
8381, 82syl 14 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (1st ‘(𝑅𝑘)) ∈ (ℤ𝑀))
84 xp2nd 6057 . . . . . . . . . . . 12 ((𝑅𝑘) ∈ ((ℤ𝑀) × 𝑆) → (2nd ‘(𝑅𝑘)) ∈ 𝑆)
8581, 84syl 14 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (2nd ‘(𝑅𝑘)) ∈ 𝑆)
8685elexd 2694 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (2nd ‘(𝑅𝑘)) ∈ V)
8729adantll 467 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑎𝑆𝑏𝑆)) → (𝑎 + 𝑏) ∈ 𝑆)
8887adantlr 468 . . . . . . . . . . . . . 14 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) ∧ (𝑎𝑆𝑏𝑆)) → (𝑎 + 𝑏) ∈ 𝑆)
89 fveq2 5414 . . . . . . . . . . . . . . . 16 (𝑎 = ((1st ‘(𝑅𝑘)) + 1) → (𝐹𝑎) = (𝐹‘((1st ‘(𝑅𝑘)) + 1)))
9089eleq1d 2206 . . . . . . . . . . . . . . 15 (𝑎 = ((1st ‘(𝑅𝑘)) + 1) → ((𝐹𝑎) ∈ 𝑆 ↔ (𝐹‘((1st ‘(𝑅𝑘)) + 1)) ∈ 𝑆))
91 fveq2 5414 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
9291eleq1d 2206 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹𝑎) ∈ 𝑆))
9392cbvralv 2652 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆 ↔ ∀𝑎 ∈ (ℤ𝑀)(𝐹𝑎) ∈ 𝑆)
945, 93sylib 121 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑎 ∈ (ℤ𝑀)(𝐹𝑎) ∈ 𝑆)
9594ad2antlr 480 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ∀𝑎 ∈ (ℤ𝑀)(𝐹𝑎) ∈ 𝑆)
96 peano2uz 9371 . . . . . . . . . . . . . . . 16 ((1st ‘(𝑅𝑘)) ∈ (ℤ𝑀) → ((1st ‘(𝑅𝑘)) + 1) ∈ (ℤ𝑀))
9783, 96syl 14 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘)) + 1) ∈ (ℤ𝑀))
9890, 95, 97rspcdva 2789 . . . . . . . . . . . . . 14 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝐹‘((1st ‘(𝑅𝑘)) + 1)) ∈ 𝑆)
9988, 85, 98caovcld 5917 . . . . . . . . . . . . 13 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))) ∈ 𝑆)
100 fvoveq1 5790 . . . . . . . . . . . . . . 15 (𝑧 = (1st ‘(𝑅𝑘)) → (𝐹‘(𝑧 + 1)) = (𝐹‘((1st ‘(𝑅𝑘)) + 1)))
101100oveq2d 5783 . . . . . . . . . . . . . 14 (𝑧 = (1st ‘(𝑅𝑘)) → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
102 oveq1 5774 . . . . . . . . . . . . . 14 (𝑤 = (2nd ‘(𝑅𝑘)) → (𝑤 + (𝐹‘((1st ‘(𝑅𝑘)) + 1))) = ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
103 eqid 2137 . . . . . . . . . . . . . 14 (𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) = (𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))
104101, 102, 103ovmpog 5898 . . . . . . . . . . . . 13 (((1st ‘(𝑅𝑘)) ∈ (ℤ𝑀) ∧ (2nd ‘(𝑅𝑘)) ∈ 𝑆 ∧ ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))) ∈ 𝑆) → ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘))) = ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
10583, 85, 99, 104syl3anc 1216 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘))) = ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
106105opeq2d 3707 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩ = ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
107105, 99eqeltrd 2214 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘))) ∈ 𝑆)
108 opelxpi 4566 . . . . . . . . . . . 12 ((((1st ‘(𝑅𝑘)) + 1) ∈ (ℤ𝑀) ∧ ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘))) ∈ 𝑆) → ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩ ∈ ((ℤ𝑀) × 𝑆))
10997, 107, 108syl2anc 408 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩ ∈ ((ℤ𝑀) × 𝑆))
110106, 109eqeltrrd 2215 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩ ∈ ((ℤ𝑀) × 𝑆))
111 oveq1 5774 . . . . . . . . . . . 12 (𝑥 = (1st ‘(𝑅𝑘)) → (𝑥 + 1) = ((1st ‘(𝑅𝑘)) + 1))
112 fvoveq1 5790 . . . . . . . . . . . . 13 (𝑥 = (1st ‘(𝑅𝑘)) → (𝐹‘(𝑥 + 1)) = (𝐹‘((1st ‘(𝑅𝑘)) + 1)))
113112oveq2d 5783 . . . . . . . . . . . 12 (𝑥 = (1st ‘(𝑅𝑘)) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
114111, 113opeq12d 3708 . . . . . . . . . . 11 (𝑥 = (1st ‘(𝑅𝑘)) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨((1st ‘(𝑅𝑘)) + 1), (𝑦 + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
115 oveq1 5774 . . . . . . . . . . . 12 (𝑦 = (2nd ‘(𝑅𝑘)) → (𝑦 + (𝐹‘((1st ‘(𝑅𝑘)) + 1))) = ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
116115opeq2d 3707 . . . . . . . . . . 11 (𝑦 = (2nd ‘(𝑅𝑘)) → ⟨((1st ‘(𝑅𝑘)) + 1), (𝑦 + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩ = ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
117114, 116, 44ovmpog 5898 . . . . . . . . . 10 (((1st ‘(𝑅𝑘)) ∈ (ℤ𝑀) ∧ (2nd ‘(𝑅𝑘)) ∈ V ∧ ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩ ∈ ((ℤ𝑀) × 𝑆)) → ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))) = ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
11883, 86, 110, 117syl3anc 1216 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))) = ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
11949ad2antlr 480 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆))
12051ad2antlr 480 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆))
121 frecsuc 6297 . . . . . . . . . . . 12 ((∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆) ∧ ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆) ∧ 𝑘 ∈ ω) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
122119, 120, 80, 121syl3anc 1216 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
123 simpr 109 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘))
124123fveq2d 5418 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅𝑘)) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
125122, 124eqtr4d 2173 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅𝑘)))
126 1st2nd2 6066 . . . . . . . . . . . . 13 ((𝑅𝑘) ∈ ((ℤ𝑀) × 𝑆) → (𝑅𝑘) = ⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩)
12781, 126syl 14 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅𝑘) = ⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩)
128127fveq2d 5418 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅𝑘)) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩))
129 df-ov 5770 . . . . . . . . . . 11 ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩)
130128, 129syl6eqr 2188 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅𝑘)) = ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))))
131125, 130eqtrd 2170 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))))
13213fveq1i 5415 . . . . . . . . . . . . . . 15 (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘)
13318fveq2d 5418 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st𝑢), (2nd𝑢)⟩))
134 df-ov 5770 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd𝑢)) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st𝑢), (2nd𝑢)⟩)
135133, 134syl6eqr 2188 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) = ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd𝑢)))
136 fvoveq1 5790 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = (1st𝑢) → (𝐹‘(𝑧 + 1)) = (𝐹‘((1st𝑢) + 1)))
137136oveq2d 5783 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = (1st𝑢) → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘((1st𝑢) + 1))))
138 oveq1 5774 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (2nd𝑢) → (𝑤 + (𝐹‘((1st𝑢) + 1))) = ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))))
139137, 138, 103ovmpog 5898 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1st𝑢) ∈ (ℤ𝑀) ∧ (2nd𝑢) ∈ 𝑆 ∧ ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))) ∈ 𝑆) → ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢)) = ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))))
14023, 25, 35, 139syl3anc 1216 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢)) = ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))))
141140, 35eqeltrd 2214 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢)) ∈ 𝑆)
142 opelxpi 4566 . . . . . . . . . . . . . . . . . . . . . 22 ((((1st𝑢) + 1) ∈ (ℤ𝑀) ∧ ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢)) ∈ 𝑆) → ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩ ∈ ((ℤ𝑀) × 𝑆))
14328, 141, 142syl2anc 408 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩ ∈ ((ℤ𝑀) × 𝑆))
144 oveq1 5774 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (1st𝑢) → (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦))
14538, 144opeq12d 3708 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (1st𝑢) → ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ = ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)
146 oveq2 5775 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (2nd𝑢) → ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢)))
147146opeq2d 3707 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (2nd𝑢) → ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ = ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩)
148 eqid 2137 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)
149145, 147, 148ovmpog 5898 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑢) ∈ (ℤ𝑀) ∧ (2nd𝑢) ∈ V ∧ ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩ ∈ ((ℤ𝑀) × 𝑆)) → ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd𝑢)) = ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩)
15023, 26, 143, 149syl3anc 1216 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd𝑢)) = ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩)
151135, 150eqtrd 2170 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) = ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩)
152151, 143eqeltrd 2214 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆))
153152ralrimiva 2503 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆))
154153ad2antlr 480 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆))
155 frecsuc 6297 . . . . . . . . . . . . . . . 16 ((∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆) ∧ ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆) ∧ 𝑘 ∈ ω) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
156154, 120, 80, 155syl3anc 1216 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
157132, 156syl5eq 2182 . . . . . . . . . . . . . 14 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
15813fveq1i 5415 . . . . . . . . . . . . . . 15 (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)
159158fveq2i 5417 . . . . . . . . . . . . . 14 ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(𝑅𝑘)) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘))
160157, 159syl6eqr 2188 . . . . . . . . . . . . 13 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(𝑅𝑘)))
161127fveq2d 5418 . . . . . . . . . . . . 13 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(𝑅𝑘)) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩))
162160, 161eqtrd 2170 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩))
163 df-ov 5770 . . . . . . . . . . . 12 ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘(𝑅𝑘))) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩)
164162, 163syl6eqr 2188 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘(𝑅𝑘))))
165 oveq1 5774 . . . . . . . . . . . . . 14 (𝑥 = (1st ‘(𝑅𝑘)) → (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦))
166111, 165opeq12d 3708 . . . . . . . . . . . . 13 (𝑥 = (1st ‘(𝑅𝑘)) → ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ = ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)
167 oveq2 5775 . . . . . . . . . . . . . 14 (𝑦 = (2nd ‘(𝑅𝑘)) → ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘))))
168167opeq2d 3707 . . . . . . . . . . . . 13 (𝑦 = (2nd ‘(𝑅𝑘)) → ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ = ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩)
169166, 168, 148ovmpog 5898 . . . . . . . . . . . 12 (((1st ‘(𝑅𝑘)) ∈ (ℤ𝑀) ∧ (2nd ‘(𝑅𝑘)) ∈ V ∧ ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩ ∈ ((ℤ𝑀) × 𝑆)) → ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘(𝑅𝑘))) = ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩)
17083, 86, 109, 169syl3anc 1216 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘(𝑅𝑘))) = ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩)
171164, 170eqtrd 2170 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩)
172171, 106eqtrd 2170 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
173118, 131, 1723eqtr4rd 2181 . . . . . . . 8 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))
174173exp31 361 . . . . . . 7 (𝑘 ∈ ω → (𝜑 → ((𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘) → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))))
175174a2d 26 . . . . . 6 (𝑘 ∈ ω → ((𝜑 → (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝜑 → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))))
17659, 63, 67, 71, 78, 175finds 4509 . . . . 5 (𝑛 ∈ ω → (𝜑 → (𝑅𝑛) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛)))
177176impcom 124 . . . 4 ((𝜑𝑛 ∈ ω) → (𝑅𝑛) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛))
17816, 55, 177eqfnfvd 5514 . . 3 (𝜑𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩))
179178rneqd 4763 . 2 (𝜑 → ran 𝑅 = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩))
180 df-seqfrec 10212 . 2 seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
181179, 180syl6reqr 2189 1 (𝜑 → seq𝑀( + , 𝐹) = ran 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  wral 2414  Vcvv 2681  wss 3066  c0 3358  cop 3525  suc csuc 4282  ωcom 4499   × cxp 4532  ran crn 4535   Fn wfn 5113  wf 5114  cfv 5118  (class class class)co 5767  cmpo 5769  1st c1st 6029  2nd c2nd 6030  freccfrec 6280  1c1 7614   + caddc 7616  cz 9047  cuz 9319  seqcseq 10211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-ltadd 7729
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-seqfrec 10212
This theorem is referenced by:  seq3-1  10226  seqf  10227  seq3p1  10228
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