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Theorem frec2uzrdg 9703
Description: A helper lemma for the value of a recursive definition generator on upper integers (typically either or 0) with characteristic function 𝐹(𝑥, 𝑦) and initial value 𝐴. This lemma shows that evaluating 𝑅 at an element of ω gives an ordered pair whose first element is the index (translated from ω to (ℤ𝐶)). See comment in frec2uz0d 9693 which describes 𝐺 and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
Hypotheses
Ref Expression
frec2uz.1 (𝜑𝐶 ∈ ℤ)
frec2uz.2 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
frecuzrdgrrn.a (𝜑𝐴𝑆)
frecuzrdgrrn.f ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
frecuzrdgrrn.2 𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
frec2uzrdg.b (𝜑𝐵 ∈ ω)
Assertion
Ref Expression
frec2uzrdg (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
Distinct variable groups:   𝑦,𝐴   𝑥,𝐶,𝑦   𝑦,𝐺   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐺(𝑥)

Proof of Theorem frec2uzrdg
Dummy variables 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frec2uzrdg.b . 2 (𝜑𝐵 ∈ ω)
2 fveq2 5251 . . . . 5 (𝑧 = 𝐵 → (𝑅𝑧) = (𝑅𝐵))
3 fveq2 5251 . . . . . 6 (𝑧 = 𝐵 → (𝐺𝑧) = (𝐺𝐵))
42fveq2d 5255 . . . . . 6 (𝑧 = 𝐵 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅𝐵)))
53, 4opeq12d 3604 . . . . 5 (𝑧 = 𝐵 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
62, 5eqeq12d 2097 . . . 4 (𝑧 = 𝐵 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩))
76imbi2d 228 . . 3 (𝑧 = 𝐵 → ((𝜑 → (𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩) ↔ (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)))
8 fveq2 5251 . . . . 5 (𝑧 = ∅ → (𝑅𝑧) = (𝑅‘∅))
9 fveq2 5251 . . . . . 6 (𝑧 = ∅ → (𝐺𝑧) = (𝐺‘∅))
108fveq2d 5255 . . . . . 6 (𝑧 = ∅ → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅‘∅)))
119, 10opeq12d 3604 . . . . 5 (𝑧 = ∅ → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
128, 11eqeq12d 2097 . . . 4 (𝑧 = ∅ → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩))
13 fveq2 5251 . . . . 5 (𝑧 = 𝑣 → (𝑅𝑧) = (𝑅𝑣))
14 fveq2 5251 . . . . . 6 (𝑧 = 𝑣 → (𝐺𝑧) = (𝐺𝑣))
1513fveq2d 5255 . . . . . 6 (𝑧 = 𝑣 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅𝑣)))
1614, 15opeq12d 3604 . . . . 5 (𝑧 = 𝑣 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)
1713, 16eqeq12d 2097 . . . 4 (𝑧 = 𝑣 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩))
18 fveq2 5251 . . . . 5 (𝑧 = suc 𝑣 → (𝑅𝑧) = (𝑅‘suc 𝑣))
19 fveq2 5251 . . . . . 6 (𝑧 = suc 𝑣 → (𝐺𝑧) = (𝐺‘suc 𝑣))
2018fveq2d 5255 . . . . . 6 (𝑧 = suc 𝑣 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅‘suc 𝑣)))
2119, 20opeq12d 3604 . . . . 5 (𝑧 = suc 𝑣 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
2218, 21eqeq12d 2097 . . . 4 (𝑧 = suc 𝑣 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
23 frecuzrdgrrn.2 . . . . . . 7 𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
2423fveq1i 5252 . . . . . 6 (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅)
25 frec2uz.1 . . . . . . . 8 (𝜑𝐶 ∈ ℤ)
26 frecuzrdgrrn.a . . . . . . . 8 (𝜑𝐴𝑆)
27 opexg 4018 . . . . . . . 8 ((𝐶 ∈ ℤ ∧ 𝐴𝑆) → ⟨𝐶, 𝐴⟩ ∈ V)
2825, 26, 27syl2anc 403 . . . . . . 7 (𝜑 → ⟨𝐶, 𝐴⟩ ∈ V)
29 frec0g 6092 . . . . . . 7 (⟨𝐶, 𝐴⟩ ∈ V → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅) = ⟨𝐶, 𝐴⟩)
3028, 29syl 14 . . . . . 6 (𝜑 → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅) = ⟨𝐶, 𝐴⟩)
3124, 30syl5eq 2127 . . . . 5 (𝜑 → (𝑅‘∅) = ⟨𝐶, 𝐴⟩)
32 frec2uz.2 . . . . . . 7 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
3325, 32frec2uz0d 9693 . . . . . 6 (𝜑 → (𝐺‘∅) = 𝐶)
3431fveq2d 5255 . . . . . . 7 (𝜑 → (2nd ‘(𝑅‘∅)) = (2nd ‘⟨𝐶, 𝐴⟩))
35 uzid 8926 . . . . . . . . 9 (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ𝐶))
3625, 35syl 14 . . . . . . . 8 (𝜑𝐶 ∈ (ℤ𝐶))
37 op2ndg 5855 . . . . . . . 8 ((𝐶 ∈ (ℤ𝐶) ∧ 𝐴𝑆) → (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴)
3836, 26, 37syl2anc 403 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴)
3934, 38eqtrd 2115 . . . . . 6 (𝜑 → (2nd ‘(𝑅‘∅)) = 𝐴)
4033, 39opeq12d 3604 . . . . 5 (𝜑 → ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩ = ⟨𝐶, 𝐴⟩)
4131, 40eqtr4d 2118 . . . 4 (𝜑 → (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
42 1st2nd2 5878 . . . . . . . . . . . . . . 15 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
4342adantl 271 . . . . . . . . . . . . . 14 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
4443fveq2d 5255 . . . . . . . . . . . . 13 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩))
45 df-ov 5592 . . . . . . . . . . . . . . 15 ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩)
46 xp1st 5869 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → (1st𝑧) ∈ (ℤ𝐶))
4746adantl 271 . . . . . . . . . . . . . . . 16 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (1st𝑧) ∈ (ℤ𝐶))
48 xp2nd 5870 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → (2nd𝑧) ∈ 𝑆)
4948adantl 271 . . . . . . . . . . . . . . . 16 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (2nd𝑧) ∈ 𝑆)
50 peano2uz 8964 . . . . . . . . . . . . . . . . . 18 ((1st𝑧) ∈ (ℤ𝐶) → ((1st𝑧) + 1) ∈ (ℤ𝐶))
5147, 50syl 14 . . . . . . . . . . . . . . . . 17 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧) + 1) ∈ (ℤ𝐶))
52 frecuzrdgrrn.f . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
5352ralrimivva 2449 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
5453ad2antrr 472 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
55 oveq1 5596 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = (1st𝑧) → (𝑥𝐹𝑦) = ((1st𝑧)𝐹𝑦))
5655eleq1d 2151 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (1st𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st𝑧)𝐹𝑦) ∈ 𝑆))
57 oveq2 5597 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (2nd𝑧) → ((1st𝑧)𝐹𝑦) = ((1st𝑧)𝐹(2nd𝑧)))
5857eleq1d 2151 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (2nd𝑧) → (((1st𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
5956, 58rspc2v 2723 . . . . . . . . . . . . . . . . . . 19 (((1st𝑧) ∈ (ℤ𝐶) ∧ (2nd𝑧) ∈ 𝑆) → (∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆 → ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
6047, 49, 59syl2anc 403 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆 → ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
6154, 60mpd 13 . . . . . . . . . . . . . . . . 17 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆)
62 opelxp 4428 . . . . . . . . . . . . . . . . 17 (⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (((1st𝑧) + 1) ∈ (ℤ𝐶) ∧ ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
6351, 61, 62sylanbrc 408 . . . . . . . . . . . . . . . 16 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆))
64 oveq1 5596 . . . . . . . . . . . . . . . . . 18 (𝑥 = (1st𝑧) → (𝑥 + 1) = ((1st𝑧) + 1))
6564, 55opeq12d 3604 . . . . . . . . . . . . . . . . 17 (𝑥 = (1st𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹𝑦)⟩)
6657opeq2d 3603 . . . . . . . . . . . . . . . . 17 (𝑦 = (2nd𝑧) → ⟨((1st𝑧) + 1), ((1st𝑧)𝐹𝑦)⟩ = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
67 eqid 2083 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
6865, 66, 67ovmpt2g 5712 . . . . . . . . . . . . . . . 16 (((1st𝑧) ∈ (ℤ𝐶) ∧ (2nd𝑧) ∈ 𝑆 ∧ ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
6947, 49, 63, 68syl3anc 1170 . . . . . . . . . . . . . . 15 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
7045, 69syl5eqr 2129 . . . . . . . . . . . . . 14 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
7170, 63eqeltrd 2159 . . . . . . . . . . . . 13 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩) ∈ ((ℤ𝐶) × 𝑆))
7244, 71eqeltrd 2159 . . . . . . . . . . . 12 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆))
7372ralrimiva 2440 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ∀𝑧 ∈ ((ℤ𝐶) × 𝑆)((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆))
7436adantr 270 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → 𝐶 ∈ (ℤ𝐶))
7526adantr 270 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → 𝐴𝑆)
76 opelxp 4428 . . . . . . . . . . . 12 (⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (𝐶 ∈ (ℤ𝐶) ∧ 𝐴𝑆))
7774, 75, 76sylanbrc 408 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆))
78 simpr 108 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → 𝑣 ∈ ω)
79 frecsuc 6102 . . . . . . . . . . 11 ((∀𝑧 ∈ ((ℤ𝐶) × 𝑆)((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆) ∧ ⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆) ∧ 𝑣 ∈ ω) → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑣) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑣)))
8073, 77, 78, 79syl3anc 1170 . . . . . . . . . 10 ((𝜑𝑣 ∈ ω) → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑣) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑣)))
8123fveq1i 5252 . . . . . . . . . 10 (𝑅‘suc 𝑣) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑣)
8223fveq1i 5252 . . . . . . . . . . 11 (𝑅𝑣) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑣)
8382fveq2i 5254 . . . . . . . . . 10 ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑣))
8480, 81, 833eqtr4g 2140 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → (𝑅‘suc 𝑣) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)))
8584adantr 270 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)))
86 fveq2 5251 . . . . . . . . 9 ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩))
87 df-ov 5592 . . . . . . . . . 10 ((𝐺𝑣)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)
8825adantr 270 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → 𝐶 ∈ ℤ)
8988, 32, 78frec2uzuzd 9696 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → (𝐺𝑣) ∈ (ℤ𝐶))
9025, 32, 26, 52, 23frecuzrdgrrn 9702 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → (𝑅𝑣) ∈ ((ℤ𝐶) × 𝑆))
91 xp2nd 5870 . . . . . . . . . . . 12 ((𝑅𝑣) ∈ ((ℤ𝐶) × 𝑆) → (2nd ‘(𝑅𝑣)) ∈ 𝑆)
9290, 91syl 14 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → (2nd ‘(𝑅𝑣)) ∈ 𝑆)
93 peano2uz 8964 . . . . . . . . . . . . 13 ((𝐺𝑣) ∈ (ℤ𝐶) → ((𝐺𝑣) + 1) ∈ (ℤ𝐶))
9489, 93syl 14 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣) + 1) ∈ (ℤ𝐶))
9552caovclg 5730 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧 ∈ (ℤ𝐶) ∧ 𝑤𝑆)) → (𝑧𝐹𝑤) ∈ 𝑆)
9695adantlr 461 . . . . . . . . . . . . 13 (((𝜑𝑣 ∈ ω) ∧ (𝑧 ∈ (ℤ𝐶) ∧ 𝑤𝑆)) → (𝑧𝐹𝑤) ∈ 𝑆)
9796, 89, 92caovcld 5731 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆)
98 opelxp 4428 . . . . . . . . . . . 12 (⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (((𝐺𝑣) + 1) ∈ (ℤ𝐶) ∧ ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆))
9994, 97, 98sylanbrc 408 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩ ∈ ((ℤ𝐶) × 𝑆))
100 oveq1 5596 . . . . . . . . . . . . 13 (𝑤 = (𝐺𝑣) → (𝑤 + 1) = ((𝐺𝑣) + 1))
101 oveq1 5596 . . . . . . . . . . . . 13 (𝑤 = (𝐺𝑣) → (𝑤𝐹𝑧) = ((𝐺𝑣)𝐹𝑧))
102100, 101opeq12d 3604 . . . . . . . . . . . 12 (𝑤 = (𝐺𝑣) → ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹𝑧)⟩)
103 oveq2 5597 . . . . . . . . . . . . 13 (𝑧 = (2nd ‘(𝑅𝑣)) → ((𝐺𝑣)𝐹𝑧) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
104103opeq2d 3603 . . . . . . . . . . . 12 (𝑧 = (2nd ‘(𝑅𝑣)) → ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹𝑧)⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
105 oveq1 5596 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑥 + 1) = (𝑤 + 1))
106 oveq1 5596 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑥𝐹𝑦) = (𝑤𝐹𝑦))
107105, 106opeq12d 3604 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩)
108 oveq2 5597 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑤𝐹𝑦) = (𝑤𝐹𝑧))
109108opeq2d 3603 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
110107, 109cbvmpt2v 5661 . . . . . . . . . . . 12 (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑤 ∈ (ℤ𝐶), 𝑧𝑆 ↦ ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
111102, 104, 110ovmpt2g 5712 . . . . . . . . . . 11 (((𝐺𝑣) ∈ (ℤ𝐶) ∧ (2nd ‘(𝑅𝑣)) ∈ 𝑆 ∧ ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩ ∈ ((ℤ𝐶) × 𝑆)) → ((𝐺𝑣)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
11289, 92, 99, 111syl3anc 1170 . . . . . . . . . 10 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
11387, 112syl5eqr 2129 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
11486, 113sylan9eqr 2137 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
11585, 114eqtrd 2115 . . . . . . 7 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
11688, 32, 78frec2uzsucd 9695 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → (𝐺‘suc 𝑣) = ((𝐺𝑣) + 1))
117116adantr 270 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝐺‘suc 𝑣) = ((𝐺𝑣) + 1))
118115fveq2d 5255 . . . . . . . . 9 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩))
11988, 32, 78frec2uzzd 9694 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → (𝐺𝑣) ∈ ℤ)
120119peano2zd 8765 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣) + 1) ∈ ℤ)
121120adantr 270 . . . . . . . . . 10 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ((𝐺𝑣) + 1) ∈ ℤ)
12297adantr 270 . . . . . . . . . 10 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆)
123 op2ndg 5855 . . . . . . . . . 10 ((((𝐺𝑣) + 1) ∈ ℤ ∧ ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆) → (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
124121, 122, 123syl2anc 403 . . . . . . . . 9 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
125118, 124eqtrd 2115 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
126117, 125opeq12d 3604 . . . . . . 7 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
127115, 126eqtr4d 2118 . . . . . 6 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
128127ex 113 . . . . 5 ((𝜑𝑣 ∈ ω) → ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
129128expcom 114 . . . 4 (𝑣 ∈ ω → (𝜑 → ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)))
13012, 17, 22, 41, 129finds2 4378 . . 3 (𝑧 ∈ ω → (𝜑 → (𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩))
1317, 130vtoclga 2675 . 2 (𝐵 ∈ ω → (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩))
1321, 131mpcom 36 1 (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  wral 2353  Vcvv 2612  c0 3269  cop 3425  cmpt 3865  suc csuc 4155  ωcom 4367   × cxp 4397  cfv 4967  (class class class)co 5589  cmpt2 5591  1st c1st 5842  2nd c2nd 5843  freccfrec 6085  1c1 7252   + caddc 7254  cz 8644  cuz 8912
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365  ax-cnex 7337  ax-resscn 7338  ax-1cn 7339  ax-1re 7340  ax-icn 7341  ax-addcl 7342  ax-addrcl 7343  ax-mulcl 7344  ax-addcom 7346  ax-addass 7348  ax-distr 7350  ax-i2m1 7351  ax-0lt1 7352  ax-0id 7354  ax-rnegex 7355  ax-cnre 7357  ax-pre-ltirr 7358  ax-pre-ltwlin 7359  ax-pre-lttrn 7360  ax-pre-ltadd 7362
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-ilim 4159  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-riota 5545  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-frec 6086  df-pnf 7425  df-mnf 7426  df-xr 7427  df-ltxr 7428  df-le 7429  df-sub 7556  df-neg 7557  df-inn 8315  df-n0 8564  df-z 8645  df-uz 8913
This theorem is referenced by:  frecuzrdglem  9705  frecuzrdgtcl  9706  frecuzrdgsuc  9708
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