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Theorem frec2uzrdg 10435
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 10425 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 5531 . . . . 5 (𝑧 = 𝐵 → (𝑅𝑧) = (𝑅𝐵))
3 fveq2 5531 . . . . . 6 (𝑧 = 𝐵 → (𝐺𝑧) = (𝐺𝐵))
42fveq2d 5535 . . . . . 6 (𝑧 = 𝐵 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅𝐵)))
53, 4opeq12d 3801 . . . . 5 (𝑧 = 𝐵 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
62, 5eqeq12d 2204 . . . 4 (𝑧 = 𝐵 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩))
76imbi2d 230 . . 3 (𝑧 = 𝐵 → ((𝜑 → (𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩) ↔ (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)))
8 fveq2 5531 . . . . 5 (𝑧 = ∅ → (𝑅𝑧) = (𝑅‘∅))
9 fveq2 5531 . . . . . 6 (𝑧 = ∅ → (𝐺𝑧) = (𝐺‘∅))
108fveq2d 5535 . . . . . 6 (𝑧 = ∅ → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅‘∅)))
119, 10opeq12d 3801 . . . . 5 (𝑧 = ∅ → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
128, 11eqeq12d 2204 . . . 4 (𝑧 = ∅ → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩))
13 fveq2 5531 . . . . 5 (𝑧 = 𝑣 → (𝑅𝑧) = (𝑅𝑣))
14 fveq2 5531 . . . . . 6 (𝑧 = 𝑣 → (𝐺𝑧) = (𝐺𝑣))
1513fveq2d 5535 . . . . . 6 (𝑧 = 𝑣 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅𝑣)))
1614, 15opeq12d 3801 . . . . 5 (𝑧 = 𝑣 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)
1713, 16eqeq12d 2204 . . . 4 (𝑧 = 𝑣 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩))
18 fveq2 5531 . . . . 5 (𝑧 = suc 𝑣 → (𝑅𝑧) = (𝑅‘suc 𝑣))
19 fveq2 5531 . . . . . 6 (𝑧 = suc 𝑣 → (𝐺𝑧) = (𝐺‘suc 𝑣))
2018fveq2d 5535 . . . . . 6 (𝑧 = suc 𝑣 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅‘suc 𝑣)))
2119, 20opeq12d 3801 . . . . 5 (𝑧 = suc 𝑣 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
2218, 21eqeq12d 2204 . . . 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 5532 . . . . . 6 (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅)
25 frec2uz.1 . . . . . . . 8 (𝜑𝐶 ∈ ℤ)
26 frecuzrdgrrn.a . . . . . . . 8 (𝜑𝐴𝑆)
27 opexg 4243 . . . . . . . 8 ((𝐶 ∈ ℤ ∧ 𝐴𝑆) → ⟨𝐶, 𝐴⟩ ∈ V)
2825, 26, 27syl2anc 411 . . . . . . 7 (𝜑 → ⟨𝐶, 𝐴⟩ ∈ V)
29 frec0g 6417 . . . . . . 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, 30eqtrid 2234 . . . . 5 (𝜑 → (𝑅‘∅) = ⟨𝐶, 𝐴⟩)
32 frec2uz.2 . . . . . . 7 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
3325, 32frec2uz0d 10425 . . . . . 6 (𝜑 → (𝐺‘∅) = 𝐶)
3431fveq2d 5535 . . . . . . 7 (𝜑 → (2nd ‘(𝑅‘∅)) = (2nd ‘⟨𝐶, 𝐴⟩))
35 uzid 9567 . . . . . . . . 9 (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ𝐶))
3625, 35syl 14 . . . . . . . 8 (𝜑𝐶 ∈ (ℤ𝐶))
37 op2ndg 6171 . . . . . . . 8 ((𝐶 ∈ (ℤ𝐶) ∧ 𝐴𝑆) → (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴)
3836, 26, 37syl2anc 411 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴)
3934, 38eqtrd 2222 . . . . . 6 (𝜑 → (2nd ‘(𝑅‘∅)) = 𝐴)
4033, 39opeq12d 3801 . . . . 5 (𝜑 → ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩ = ⟨𝐶, 𝐴⟩)
4131, 40eqtr4d 2225 . . . 4 (𝜑 → (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
42 1st2nd2 6195 . . . . . . . . . . . . . . 15 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
4342adantl 277 . . . . . . . . . . . . . 14 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
4443fveq2d 5535 . . . . . . . . . . . . 13 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩))
45 df-ov 5895 . . . . . . . . . . . . . . 15 ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩)
46 xp1st 6185 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → (1st𝑧) ∈ (ℤ𝐶))
4746adantl 277 . . . . . . . . . . . . . . . 16 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (1st𝑧) ∈ (ℤ𝐶))
48 xp2nd 6186 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → (2nd𝑧) ∈ 𝑆)
4948adantl 277 . . . . . . . . . . . . . . . 16 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (2nd𝑧) ∈ 𝑆)
50 peano2uz 9608 . . . . . . . . . . . . . . . . . 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 2572 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
5453ad2antrr 488 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
55 oveq1 5899 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = (1st𝑧) → (𝑥𝐹𝑦) = ((1st𝑧)𝐹𝑦))
5655eleq1d 2258 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (1st𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st𝑧)𝐹𝑦) ∈ 𝑆))
57 oveq2 5900 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (2nd𝑧) → ((1st𝑧)𝐹𝑦) = ((1st𝑧)𝐹(2nd𝑧)))
5857eleq1d 2258 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (2nd𝑧) → (((1st𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
5956, 58rspc2v 2869 . . . . . . . . . . . . . . . . . . 19 (((1st𝑧) ∈ (ℤ𝐶) ∧ (2nd𝑧) ∈ 𝑆) → (∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆 → ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
6047, 49, 59syl2anc 411 . . . . . . . . . . . . . . . . . 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 4671 . . . . . . . . . . . . . . . . 17 (⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (((1st𝑧) + 1) ∈ (ℤ𝐶) ∧ ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
6351, 61, 62sylanbrc 417 . . . . . . . . . . . . . . . 16 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆))
64 oveq1 5899 . . . . . . . . . . . . . . . . . 18 (𝑥 = (1st𝑧) → (𝑥 + 1) = ((1st𝑧) + 1))
6564, 55opeq12d 3801 . . . . . . . . . . . . . . . . 17 (𝑥 = (1st𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹𝑦)⟩)
6657opeq2d 3800 . . . . . . . . . . . . . . . . 17 (𝑦 = (2nd𝑧) → ⟨((1st𝑧) + 1), ((1st𝑧)𝐹𝑦)⟩ = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
67 eqid 2189 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
6865, 66, 67ovmpog 6027 . . . . . . . . . . . . . . . 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 1249 . . . . . . . . . . . . . . 15 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
7045, 69eqtr3id 2236 . . . . . . . . . . . . . 14 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
7170, 63eqeltrd 2266 . . . . . . . . . . . . 13 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩) ∈ ((ℤ𝐶) × 𝑆))
7244, 71eqeltrd 2266 . . . . . . . . . . . 12 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆))
7372ralrimiva 2563 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ∀𝑧 ∈ ((ℤ𝐶) × 𝑆)((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆))
7436adantr 276 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → 𝐶 ∈ (ℤ𝐶))
7526adantr 276 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → 𝐴𝑆)
76 opelxp 4671 . . . . . . . . . . . 12 (⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (𝐶 ∈ (ℤ𝐶) ∧ 𝐴𝑆))
7774, 75, 76sylanbrc 417 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆))
78 simpr 110 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → 𝑣 ∈ ω)
79 frecsuc 6427 . . . . . . . . . . 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 1249 . . . . . . . . . 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 5532 . . . . . . . . . 10 (𝑅‘suc 𝑣) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑣)
8223fveq1i 5532 . . . . . . . . . . 11 (𝑅𝑣) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑣)
8382fveq2i 5534 . . . . . . . . . 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 2247 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → (𝑅‘suc 𝑣) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)))
8584adantr 276 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)))
86 fveq2 5531 . . . . . . . . 9 ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩))
87 df-ov 5895 . . . . . . . . . 10 ((𝐺𝑣)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)
8825adantr 276 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → 𝐶 ∈ ℤ)
8988, 32, 78frec2uzuzd 10428 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → (𝐺𝑣) ∈ (ℤ𝐶))
9025, 32, 26, 52, 23frecuzrdgrrn 10434 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → (𝑅𝑣) ∈ ((ℤ𝐶) × 𝑆))
91 xp2nd 6186 . . . . . . . . . . . 12 ((𝑅𝑣) ∈ ((ℤ𝐶) × 𝑆) → (2nd ‘(𝑅𝑣)) ∈ 𝑆)
9290, 91syl 14 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → (2nd ‘(𝑅𝑣)) ∈ 𝑆)
93 peano2uz 9608 . . . . . . . . . . . . 13 ((𝐺𝑣) ∈ (ℤ𝐶) → ((𝐺𝑣) + 1) ∈ (ℤ𝐶))
9489, 93syl 14 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣) + 1) ∈ (ℤ𝐶))
9552caovclg 6045 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧 ∈ (ℤ𝐶) ∧ 𝑤𝑆)) → (𝑧𝐹𝑤) ∈ 𝑆)
9695adantlr 477 . . . . . . . . . . . . 13 (((𝜑𝑣 ∈ ω) ∧ (𝑧 ∈ (ℤ𝐶) ∧ 𝑤𝑆)) → (𝑧𝐹𝑤) ∈ 𝑆)
9796, 89, 92caovcld 6046 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆)
98 opelxp 4671 . . . . . . . . . . . 12 (⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (((𝐺𝑣) + 1) ∈ (ℤ𝐶) ∧ ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆))
9994, 97, 98sylanbrc 417 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩ ∈ ((ℤ𝐶) × 𝑆))
100 oveq1 5899 . . . . . . . . . . . . 13 (𝑤 = (𝐺𝑣) → (𝑤 + 1) = ((𝐺𝑣) + 1))
101 oveq1 5899 . . . . . . . . . . . . 13 (𝑤 = (𝐺𝑣) → (𝑤𝐹𝑧) = ((𝐺𝑣)𝐹𝑧))
102100, 101opeq12d 3801 . . . . . . . . . . . 12 (𝑤 = (𝐺𝑣) → ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹𝑧)⟩)
103 oveq2 5900 . . . . . . . . . . . . 13 (𝑧 = (2nd ‘(𝑅𝑣)) → ((𝐺𝑣)𝐹𝑧) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
104103opeq2d 3800 . . . . . . . . . . . 12 (𝑧 = (2nd ‘(𝑅𝑣)) → ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹𝑧)⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
105 oveq1 5899 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑥 + 1) = (𝑤 + 1))
106 oveq1 5899 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑥𝐹𝑦) = (𝑤𝐹𝑦))
107105, 106opeq12d 3801 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩)
108 oveq2 5900 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑤𝐹𝑦) = (𝑤𝐹𝑧))
109108opeq2d 3800 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
110107, 109cbvmpov 5972 . . . . . . . . . . . 12 (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑤 ∈ (ℤ𝐶), 𝑧𝑆 ↦ ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
111102, 104, 110ovmpog 6027 . . . . . . . . . . 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 1249 . . . . . . . . . 10 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
11387, 112eqtr3id 2236 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
11486, 113sylan9eqr 2244 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
11585, 114eqtrd 2222 . . . . . . 7 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
11688, 32, 78frec2uzsucd 10427 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → (𝐺‘suc 𝑣) = ((𝐺𝑣) + 1))
117116adantr 276 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝐺‘suc 𝑣) = ((𝐺𝑣) + 1))
118115fveq2d 5535 . . . . . . . . 9 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩))
11988, 32, 78frec2uzzd 10426 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → (𝐺𝑣) ∈ ℤ)
120119peano2zd 9403 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣) + 1) ∈ ℤ)
121120adantr 276 . . . . . . . . . 10 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ((𝐺𝑣) + 1) ∈ ℤ)
12297adantr 276 . . . . . . . . . 10 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆)
123 op2ndg 6171 . . . . . . . . . 10 ((((𝐺𝑣) + 1) ∈ ℤ ∧ ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆) → (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
124121, 122, 123syl2anc 411 . . . . . . . . 9 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
125118, 124eqtrd 2222 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
126117, 125opeq12d 3801 . . . . . . 7 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
127115, 126eqtr4d 2225 . . . . . 6 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
128127ex 115 . . . . 5 ((𝜑𝑣 ∈ ω) → ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
129128expcom 116 . . . 4 (𝑣 ∈ ω → (𝜑 → ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)))
13012, 17, 22, 41, 129finds2 4615 . . 3 (𝑧 ∈ ω → (𝜑 → (𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩))
1317, 130vtoclga 2818 . 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 104   = wceq 1364  wcel 2160  wral 2468  Vcvv 2752  c0 3437  cop 3610  cmpt 4079  suc csuc 4380  ωcom 4604   × cxp 4639  cfv 5232  (class class class)co 5892  cmpo 5894  1st c1st 6158  2nd c2nd 6159  freccfrec 6410  1c1 7837   + caddc 7839  cz 9278  cuz 9553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-addcom 7936  ax-addass 7938  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-0id 7944  ax-rnegex 7945  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-ltadd 7952
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-frec 6411  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-inn 8945  df-n0 9202  df-z 9279  df-uz 9554
This theorem is referenced by:  frecuzrdglem  10437  frecuzrdgtcl  10438  frecuzrdgsuc  10440
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