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Theorem iseqvalt 9838
 Description: Value of the sequence builder function. There should be no need for new usages of this theorem because once we have proved theorems seqf 9845, seq3-1 9842 and seq3p1 9849 future development can be done in terms of those. (Contributed by Jim Kingdon, 27-Apr-2022.) (New usage is discouraged.)
Hypotheses
Ref Expression
iseqvalt.m (𝜑𝑀 ∈ ℤ)
iseqvalt.r 𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)
iseqvalt.f ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
iseqvalt.pl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqvalt.t (𝜑𝑆𝑇)
Assertion
Ref Expression
iseqvalt (𝜑 → seq𝑀( + , 𝐹, 𝑇) = ran 𝑅)
Distinct variable groups:   𝑤, + ,𝑥,𝑦,𝑧   𝑤,𝐹,𝑥,𝑦,𝑧   𝑤,𝑀,𝑥,𝑦,𝑧   𝑤,𝑅,𝑥,𝑦,𝑧   𝑤,𝑆,𝑥,𝑦,𝑧   𝑥,𝑇,𝑦   𝜑,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝑇(𝑧,𝑤)

Proof of Theorem iseqvalt
Dummy variables 𝑎 𝑏 𝑘 𝑐 𝑛 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqvalt.m . . . . . 6 (𝜑𝑀 ∈ ℤ)
2 fveq2 5289 . . . . . . . 8 (𝑥 = 𝑀 → (𝐹𝑥) = (𝐹𝑀))
32eleq1d 2156 . . . . . . 7 (𝑥 = 𝑀 → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹𝑀) ∈ 𝑆))
4 iseqvalt.f . . . . . . . 8 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
54ralrimiva 2446 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
6 uzid 9002 . . . . . . . 8 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
71, 6syl 14 . . . . . . 7 (𝜑𝑀 ∈ (ℤ𝑀))
83, 5, 7rspcdva 2727 . . . . . 6 (𝜑 → (𝐹𝑀) ∈ 𝑆)
9 iseqvalt.t . . . . . 6 (𝜑𝑆𝑇)
10 iseqvalt.pl . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
114, 10iseqovex 9835 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝑆)) → (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆)
12 iseqvalt.r . . . . . 6 𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)
131, 8, 9, 11, 12frecuzrdgrclt 9787 . . . . 5 (𝜑𝑅:ω⟶((ℤ𝑀) × 𝑆))
14 ffn 5147 . . . . 5 (𝑅:ω⟶((ℤ𝑀) × 𝑆) → 𝑅 Fn ω)
1513, 14syl 14 . . . 4 (𝜑𝑅 Fn ω)
16 1st2nd2 5927 . . . . . . . . . . . 12 (𝑢 ∈ ((ℤ𝑀) × 𝑆) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
1716adantl 271 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
1817fveq2d 5293 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st𝑢), (2nd𝑢)⟩))
19 df-ov 5637 . . . . . . . . . 10 ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd𝑢)) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st𝑢), (2nd𝑢)⟩)
2018, 19syl6eqr 2138 . . . . . . . . 9 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) = ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd𝑢)))
21 xp1st 5918 . . . . . . . . . . 11 (𝑢 ∈ ((ℤ𝑀) × 𝑆) → (1st𝑢) ∈ (ℤ𝑀))
2221adantl 271 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → (1st𝑢) ∈ (ℤ𝑀))
239adantr 270 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → 𝑆𝑇)
24 xp2nd 5919 . . . . . . . . . . . 12 (𝑢 ∈ ((ℤ𝑀) × 𝑆) → (2nd𝑢) ∈ 𝑆)
2524adantl 271 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → (2nd𝑢) ∈ 𝑆)
2623, 25sseldd 3024 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → (2nd𝑢) ∈ 𝑇)
27 peano2uz 9040 . . . . . . . . . . . 12 ((1st𝑢) ∈ (ℤ𝑀) → ((1st𝑢) + 1) ∈ (ℤ𝑀))
2822, 27syl 14 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((1st𝑢) + 1) ∈ (ℤ𝑀))
2910caovclg 5779 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝑆𝑏𝑆)) → (𝑎 + 𝑏) ∈ 𝑆)
3029adantlr 461 . . . . . . . . . . . 12 (((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) ∧ (𝑎𝑆𝑏𝑆)) → (𝑎 + 𝑏) ∈ 𝑆)
31 fveq2 5289 . . . . . . . . . . . . . 14 (𝑥 = ((1st𝑢) + 1) → (𝐹𝑥) = (𝐹‘((1st𝑢) + 1)))
3231eleq1d 2156 . . . . . . . . . . . . 13 (𝑥 = ((1st𝑢) + 1) → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹‘((1st𝑢) + 1)) ∈ 𝑆))
335adantr 270 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆)
3432, 33, 28rspcdva 2727 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → (𝐹‘((1st𝑢) + 1)) ∈ 𝑆)
3530, 25, 34caovcld 5780 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))) ∈ 𝑆)
36 opelxpi 4459 . . . . . . . . . . 11 ((((1st𝑢) + 1) ∈ (ℤ𝑀) ∧ ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))) ∈ 𝑆) → ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩ ∈ ((ℤ𝑀) × 𝑆))
3728, 35, 36syl2anc 403 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩ ∈ ((ℤ𝑀) × 𝑆))
38 oveq1 5641 . . . . . . . . . . . 12 (𝑥 = (1st𝑢) → (𝑥 + 1) = ((1st𝑢) + 1))
3938fveq2d 5293 . . . . . . . . . . . . 13 (𝑥 = (1st𝑢) → (𝐹‘(𝑥 + 1)) = (𝐹‘((1st𝑢) + 1)))
4039oveq2d 5650 . . . . . . . . . . . 12 (𝑥 = (1st𝑢) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘((1st𝑢) + 1))))
4138, 40opeq12d 3625 . . . . . . . . . . 11 (𝑥 = (1st𝑢) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨((1st𝑢) + 1), (𝑦 + (𝐹‘((1st𝑢) + 1)))⟩)
42 oveq1 5641 . . . . . . . . . . . 12 (𝑦 = (2nd𝑢) → (𝑦 + (𝐹‘((1st𝑢) + 1))) = ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))))
4342opeq2d 3624 . . . . . . . . . . 11 (𝑦 = (2nd𝑢) → ⟨((1st𝑢) + 1), (𝑦 + (𝐹‘((1st𝑢) + 1)))⟩ = ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩)
44 eqid 2088 . . . . . . . . . . 11 (𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)
4541, 43, 44ovmpt2g 5761 . . . . . . . . . 10 (((1st𝑢) ∈ (ℤ𝑀) ∧ (2nd𝑢) ∈ 𝑇 ∧ ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩ ∈ ((ℤ𝑀) × 𝑆)) → ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd𝑢)) = ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩)
4622, 26, 37, 45syl3anc 1174 . . . . . . . . 9 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd𝑢)) = ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩)
4720, 46eqtrd 2120 . . . . . . . 8 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) = ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩)
4847, 37eqeltrd 2164 . . . . . . 7 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆))
4948ralrimiva 2446 . . . . . 6 (𝜑 → ∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆))
50 opelxpi 4459 . . . . . . 7 ((𝑀 ∈ (ℤ𝑀) ∧ (𝐹𝑀) ∈ 𝑆) → ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆))
517, 8, 50syl2anc 403 . . . . . 6 (𝜑 → ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆))
5249, 51jca 300 . . . . 5 (𝜑 → (∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆) ∧ ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆)))
53 frecfcl 6152 . . . . 5 ((∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆) ∧ ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆)) → frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩):ω⟶((ℤ𝑀) × 𝑆))
54 ffn 5147 . . . . 5 (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩):ω⟶((ℤ𝑀) × 𝑆) → frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) Fn ω)
5552, 53, 543syl 17 . . . 4 (𝜑 → frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) Fn ω)
56 fveq2 5289 . . . . . . . 8 (𝑐 = ∅ → (𝑅𝑐) = (𝑅‘∅))
57 fveq2 5289 . . . . . . . 8 (𝑐 = ∅ → (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅))
5856, 57eqeq12d 2102 . . . . . . 7 (𝑐 = ∅ → ((𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) ↔ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅)))
5958imbi2d 228 . . . . . 6 (𝑐 = ∅ → ((𝜑 → (𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅))))
60 fveq2 5289 . . . . . . . 8 (𝑐 = 𝑘 → (𝑅𝑐) = (𝑅𝑘))
61 fveq2 5289 . . . . . . . 8 (𝑐 = 𝑘 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘))
6260, 61eqeq12d 2102 . . . . . . 7 (𝑐 = 𝑘 → ((𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) ↔ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
6362imbi2d 228 . . . . . 6 (𝑐 = 𝑘 → ((𝜑 → (𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘))))
64 fveq2 5289 . . . . . . . 8 (𝑐 = suc 𝑘 → (𝑅𝑐) = (𝑅‘suc 𝑘))
65 fveq2 5289 . . . . . . . 8 (𝑐 = suc 𝑘 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))
6664, 65eqeq12d 2102 . . . . . . 7 (𝑐 = suc 𝑘 → ((𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) ↔ (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘)))
6766imbi2d 228 . . . . . 6 (𝑐 = suc 𝑘 → ((𝜑 → (𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))))
68 fveq2 5289 . . . . . . . 8 (𝑐 = 𝑛 → (𝑅𝑐) = (𝑅𝑛))
69 fveq2 5289 . . . . . . . 8 (𝑐 = 𝑛 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛))
7068, 69eqeq12d 2102 . . . . . . 7 (𝑐 = 𝑛 → ((𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) ↔ (𝑅𝑛) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛)))
7170imbi2d 228 . . . . . 6 (𝑐 = 𝑛 → ((𝜑 → (𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅𝑛) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛))))
7212fveq1i 5290 . . . . . . . 8 (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅)
73 frec0g 6144 . . . . . . . . 9 (⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
7451, 73syl 14 . . . . . . . 8 (𝜑 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
7572, 74syl5eq 2132 . . . . . . 7 (𝜑 → (𝑅‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
76 frec0g 6144 . . . . . . . 8 (⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
7751, 76syl 14 . . . . . . 7 (𝜑 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
7875, 77eqtr4d 2123 . . . . . 6 (𝜑 → (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅))
7913ad2antlr 473 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → 𝑅:ω⟶((ℤ𝑀) × 𝑆))
80 simpll 496 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → 𝑘 ∈ ω)
8179, 80ffvelrnd 5419 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅𝑘) ∈ ((ℤ𝑀) × 𝑆))
82 xp1st 5918 . . . . . . . . . . 11 ((𝑅𝑘) ∈ ((ℤ𝑀) × 𝑆) → (1st ‘(𝑅𝑘)) ∈ (ℤ𝑀))
8381, 82syl 14 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (1st ‘(𝑅𝑘)) ∈ (ℤ𝑀))
849ad2antlr 473 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → 𝑆𝑇)
85 xp2nd 5919 . . . . . . . . . . . 12 ((𝑅𝑘) ∈ ((ℤ𝑀) × 𝑆) → (2nd ‘(𝑅𝑘)) ∈ 𝑆)
8681, 85syl 14 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (2nd ‘(𝑅𝑘)) ∈ 𝑆)
8784, 86sseldd 3024 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (2nd ‘(𝑅𝑘)) ∈ 𝑇)
8829adantll 460 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑎𝑆𝑏𝑆)) → (𝑎 + 𝑏) ∈ 𝑆)
8988adantlr 461 . . . . . . . . . . . . . 14 ((((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) ∧ (𝑎𝑆𝑏𝑆)) → (𝑎 + 𝑏) ∈ 𝑆)
90 fveq2 5289 . . . . . . . . . . . . . . . 16 (𝑎 = ((1st ‘(𝑅𝑘)) + 1) → (𝐹𝑎) = (𝐹‘((1st ‘(𝑅𝑘)) + 1)))
9190eleq1d 2156 . . . . . . . . . . . . . . 15 (𝑎 = ((1st ‘(𝑅𝑘)) + 1) → ((𝐹𝑎) ∈ 𝑆 ↔ (𝐹‘((1st ‘(𝑅𝑘)) + 1)) ∈ 𝑆))
92 fveq2 5289 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
9392eleq1d 2156 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹𝑎) ∈ 𝑆))
9493cbvralv 2590 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ (ℤ𝑀)(𝐹𝑥) ∈ 𝑆 ↔ ∀𝑎 ∈ (ℤ𝑀)(𝐹𝑎) ∈ 𝑆)
955, 94sylib 120 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑎 ∈ (ℤ𝑀)(𝐹𝑎) ∈ 𝑆)
9695ad2antlr 473 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ∀𝑎 ∈ (ℤ𝑀)(𝐹𝑎) ∈ 𝑆)
97 peano2uz 9040 . . . . . . . . . . . . . . . 16 ((1st ‘(𝑅𝑘)) ∈ (ℤ𝑀) → ((1st ‘(𝑅𝑘)) + 1) ∈ (ℤ𝑀))
9883, 97syl 14 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘)) + 1) ∈ (ℤ𝑀))
9991, 96, 98rspcdva 2727 . . . . . . . . . . . . . 14 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝐹‘((1st ‘(𝑅𝑘)) + 1)) ∈ 𝑆)
10089, 86, 99caovcld 5780 . . . . . . . . . . . . 13 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))) ∈ 𝑆)
101 oveq1 5641 . . . . . . . . . . . . . . . 16 (𝑧 = (1st ‘(𝑅𝑘)) → (𝑧 + 1) = ((1st ‘(𝑅𝑘)) + 1))
102101fveq2d 5293 . . . . . . . . . . . . . . 15 (𝑧 = (1st ‘(𝑅𝑘)) → (𝐹‘(𝑧 + 1)) = (𝐹‘((1st ‘(𝑅𝑘)) + 1)))
103102oveq2d 5650 . . . . . . . . . . . . . 14 (𝑧 = (1st ‘(𝑅𝑘)) → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
104 oveq1 5641 . . . . . . . . . . . . . 14 (𝑤 = (2nd ‘(𝑅𝑘)) → (𝑤 + (𝐹‘((1st ‘(𝑅𝑘)) + 1))) = ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
105 eqid 2088 . . . . . . . . . . . . . 14 (𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) = (𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))
106103, 104, 105ovmpt2g 5761 . . . . . . . . . . . . 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))))
10783, 86, 100, 106syl3anc 1174 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(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))))
108107opeq2d 3624 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(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)))⟩)
109107, 100eqeltrd 2164 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘))) ∈ 𝑆)
110 opelxpi 4459 . . . . . . . . . . . 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))
11198, 109, 110syl2anc 403 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(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))
112108, 111eqeltrrd 2165 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩ ∈ ((ℤ𝑀) × 𝑆))
113 oveq1 5641 . . . . . . . . . . . 12 (𝑥 = (1st ‘(𝑅𝑘)) → (𝑥 + 1) = ((1st ‘(𝑅𝑘)) + 1))
114113fveq2d 5293 . . . . . . . . . . . . 13 (𝑥 = (1st ‘(𝑅𝑘)) → (𝐹‘(𝑥 + 1)) = (𝐹‘((1st ‘(𝑅𝑘)) + 1)))
115114oveq2d 5650 . . . . . . . . . . . 12 (𝑥 = (1st ‘(𝑅𝑘)) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
116113, 115opeq12d 3625 . . . . . . . . . . 11 (𝑥 = (1st ‘(𝑅𝑘)) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨((1st ‘(𝑅𝑘)) + 1), (𝑦 + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
117 oveq1 5641 . . . . . . . . . . . 12 (𝑦 = (2nd ‘(𝑅𝑘)) → (𝑦 + (𝐹‘((1st ‘(𝑅𝑘)) + 1))) = ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
118117opeq2d 3624 . . . . . . . . . . 11 (𝑦 = (2nd ‘(𝑅𝑘)) → ⟨((1st ‘(𝑅𝑘)) + 1), (𝑦 + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩ = ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
119116, 118, 44ovmpt2g 5761 . . . . . . . . . 10 (((1st ‘(𝑅𝑘)) ∈ (ℤ𝑀) ∧ (2nd ‘(𝑅𝑘)) ∈ 𝑇 ∧ ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩ ∈ ((ℤ𝑀) × 𝑆)) → ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))) = ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
12083, 87, 112, 119syl3anc 1174 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))) = ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
12149ad2antlr 473 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆))
12251ad2antlr 473 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆))
123 frecsuc 6154 . . . . . . . . . . . 12 ((∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆) ∧ ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆) ∧ 𝑘 ∈ ω) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
124121, 122, 80, 123syl3anc 1174 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
125 simpr 108 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘))
126125fveq2d 5293 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅𝑘)) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
127124, 126eqtr4d 2123 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅𝑘)))
128 1st2nd2 5927 . . . . . . . . . . . . 13 ((𝑅𝑘) ∈ ((ℤ𝑀) × 𝑆) → (𝑅𝑘) = ⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩)
12981, 128syl 14 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅𝑘) = ⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩)
130129fveq2d 5293 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅𝑘)) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩))
131 df-ov 5637 . . . . . . . . . . 11 ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩)
132130, 131syl6eqr 2138 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅𝑘)) = ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))))
133127, 132eqtrd 2120 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))))
13412fveq1i 5290 . . . . . . . . . . . . . . 15 (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘)
13517fveq2d 5293 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st𝑢), (2nd𝑢)⟩))
136 df-ov 5637 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd𝑢)) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st𝑢), (2nd𝑢)⟩)
137135, 136syl6eqr 2138 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) = ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd𝑢)))
138 oveq1 5641 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = (1st𝑢) → (𝑧 + 1) = ((1st𝑢) + 1))
139138fveq2d 5293 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = (1st𝑢) → (𝐹‘(𝑧 + 1)) = (𝐹‘((1st𝑢) + 1)))
140139oveq2d 5650 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = (1st𝑢) → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘((1st𝑢) + 1))))
141 oveq1 5641 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (2nd𝑢) → (𝑤 + (𝐹‘((1st𝑢) + 1))) = ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))))
142140, 141, 105ovmpt2g 5761 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1st𝑢) ∈ (ℤ𝑀) ∧ (2nd𝑢) ∈ 𝑆 ∧ ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))) ∈ 𝑆) → ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢)) = ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))))
14322, 25, 35, 142syl3anc 1174 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢)) = ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))))
144143, 35eqeltrd 2164 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢)) ∈ 𝑆)
145 opelxpi 4459 . . . . . . . . . . . . . . . . . . . . . 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))
14628, 144, 145syl2anc 403 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩ ∈ ((ℤ𝑀) × 𝑆))
147 oveq1 5641 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (1st𝑢) → (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦))
14838, 147opeq12d 3625 . . . . . . . . . . . . . . . . . . . . . 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)⟩)
149 oveq2 5642 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (2nd𝑢) → ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢)))
150149opeq2d 3624 . . . . . . . . . . . . . . . . . . . . . 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))⟩)
151 eqid 2088 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)
152148, 150, 151ovmpt2g 5761 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑢) ∈ (ℤ𝑀) ∧ (2nd𝑢) ∈ 𝑇 ∧ ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩ ∈ ((ℤ𝑀) × 𝑆)) → ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd𝑢)) = ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩)
15322, 26, 146, 152syl3anc 1174 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd𝑢)) = ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩)
154137, 153eqtrd 2120 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) = ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩)
155154, 146eqeltrd 2164 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆))
156155ralrimiva 2446 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆))
157156ad2antlr 473 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆))
158 frecsuc 6154 . . . . . . . . . . . . . . . 16 ((∀𝑢 ∈ ((ℤ𝑀) × 𝑆)((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝑆) ∧ ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝑆) ∧ 𝑘 ∈ ω) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
159157, 122, 80, 158syl3anc 1174 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
160134, 159syl5eq 2132 . . . . . . . . . . . . . 14 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
16112fveq1i 5290 . . . . . . . . . . . . . . 15 (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)
162161fveq2i 5292 . . . . . . . . . . . . . 14 ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(𝑅𝑘)) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘))
163160, 162syl6eqr 2138 . . . . . . . . . . . . 13 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(𝑅𝑘)))
164129fveq2d 5293 . . . . . . . . . . . . 13 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(𝑅𝑘)) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩))
165163, 164eqtrd 2120 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩))
166 df-ov 5637 . . . . . . . . . . . 12 ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘(𝑅𝑘))) = ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩)
167165, 166syl6eqr 2138 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘(𝑅𝑘))))
168 oveq1 5641 . . . . . . . . . . . . . 14 (𝑥 = (1st ‘(𝑅𝑘)) → (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦))
169113, 168opeq12d 3625 . . . . . . . . . . . . 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)⟩)
170 oveq2 5642 . . . . . . . . . . . . . 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))))
171170opeq2d 3624 . . . . . . . . . . . . 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)))⟩)
172169, 171, 151ovmpt2g 5761 . . . . . . . . . . . 12 (((1st ‘(𝑅𝑘)) ∈ (ℤ𝑀) ∧ (2nd ‘(𝑅𝑘)) ∈ 𝑇 ∧ ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩ ∈ ((ℤ𝑀) × 𝑆)) → ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(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)))⟩)
17383, 87, 111, 172syl3anc 1174 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(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)))⟩)
174167, 173eqtrd 2120 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(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)))⟩)
175174, 108eqtrd 2120 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
176120, 133, 1753eqtr4rd 2131 . . . . . . . 8 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))
177176exp31 356 . . . . . . 7 (𝑘 ∈ ω → (𝜑 → ((𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘) → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))))
178177a2d 26 . . . . . 6 (𝑘 ∈ ω → ((𝜑 → (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝜑 → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))))
17959, 63, 67, 71, 78, 178finds 4405 . . . . 5 (𝑛 ∈ ω → (𝜑 → (𝑅𝑛) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛)))
180179impcom 123 . . . 4 ((𝜑𝑛 ∈ ω) → (𝑅𝑛) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛))
18115, 55, 180eqfnfvd 5384 . . 3 (𝜑𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩))
182181rneqd 4652 . 2 (𝜑 → ran 𝑅 = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩))
183 df-iseq 9818 . 2 seq𝑀( + , 𝐹, 𝑇) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
184182, 183syl6reqr 2139 1 (𝜑 → seq𝑀( + , 𝐹, 𝑇) = ran 𝑅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   = wceq 1289   ∈ wcel 1438  ∀wral 2359   ⊆ wss 2997  ∅c0 3284  ⟨cop 3444  suc csuc 4183  ωcom 4395   × cxp 4426  ran crn 4429   Fn wfn 4997  ⟶wf 4998  ‘cfv 5002  (class class class)co 5634   ↦ cmpt2 5636  1st c1st 5891  2nd c2nd 5892  freccfrec 6137  1c1 7330   + caddc 7332  ℤcz 8720  ℤ≥cuz 8988  seqcseq4 9816 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440 This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989  df-iseq 9818 This theorem is referenced by:  iseq1t  9841  iseqfclt  9844  iseqp1t  9848
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