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Theorem frec2uzrdg 10352
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 10342 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 5494 . . . . 5 (𝑧 = 𝐵 → (𝑅𝑧) = (𝑅𝐵))
3 fveq2 5494 . . . . . 6 (𝑧 = 𝐵 → (𝐺𝑧) = (𝐺𝐵))
42fveq2d 5498 . . . . . 6 (𝑧 = 𝐵 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅𝐵)))
53, 4opeq12d 3771 . . . . 5 (𝑧 = 𝐵 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
62, 5eqeq12d 2185 . . . 4 (𝑧 = 𝐵 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩))
76imbi2d 229 . . 3 (𝑧 = 𝐵 → ((𝜑 → (𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩) ↔ (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)))
8 fveq2 5494 . . . . 5 (𝑧 = ∅ → (𝑅𝑧) = (𝑅‘∅))
9 fveq2 5494 . . . . . 6 (𝑧 = ∅ → (𝐺𝑧) = (𝐺‘∅))
108fveq2d 5498 . . . . . 6 (𝑧 = ∅ → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅‘∅)))
119, 10opeq12d 3771 . . . . 5 (𝑧 = ∅ → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
128, 11eqeq12d 2185 . . . 4 (𝑧 = ∅ → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩))
13 fveq2 5494 . . . . 5 (𝑧 = 𝑣 → (𝑅𝑧) = (𝑅𝑣))
14 fveq2 5494 . . . . . 6 (𝑧 = 𝑣 → (𝐺𝑧) = (𝐺𝑣))
1513fveq2d 5498 . . . . . 6 (𝑧 = 𝑣 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅𝑣)))
1614, 15opeq12d 3771 . . . . 5 (𝑧 = 𝑣 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)
1713, 16eqeq12d 2185 . . . 4 (𝑧 = 𝑣 → ((𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ ↔ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩))
18 fveq2 5494 . . . . 5 (𝑧 = suc 𝑣 → (𝑅𝑧) = (𝑅‘suc 𝑣))
19 fveq2 5494 . . . . . 6 (𝑧 = suc 𝑣 → (𝐺𝑧) = (𝐺‘suc 𝑣))
2018fveq2d 5498 . . . . . 6 (𝑧 = suc 𝑣 → (2nd ‘(𝑅𝑧)) = (2nd ‘(𝑅‘suc 𝑣)))
2119, 20opeq12d 3771 . . . . 5 (𝑧 = suc 𝑣 → ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩ = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
2218, 21eqeq12d 2185 . . . 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 5495 . . . . . 6 (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅)
25 frec2uz.1 . . . . . . . 8 (𝜑𝐶 ∈ ℤ)
26 frecuzrdgrrn.a . . . . . . . 8 (𝜑𝐴𝑆)
27 opexg 4211 . . . . . . . 8 ((𝐶 ∈ ℤ ∧ 𝐴𝑆) → ⟨𝐶, 𝐴⟩ ∈ V)
2825, 26, 27syl2anc 409 . . . . . . 7 (𝜑 → ⟨𝐶, 𝐴⟩ ∈ V)
29 frec0g 6373 . . . . . . 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 2215 . . . . 5 (𝜑 → (𝑅‘∅) = ⟨𝐶, 𝐴⟩)
32 frec2uz.2 . . . . . . 7 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
3325, 32frec2uz0d 10342 . . . . . 6 (𝜑 → (𝐺‘∅) = 𝐶)
3431fveq2d 5498 . . . . . . 7 (𝜑 → (2nd ‘(𝑅‘∅)) = (2nd ‘⟨𝐶, 𝐴⟩))
35 uzid 9488 . . . . . . . . 9 (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ𝐶))
3625, 35syl 14 . . . . . . . 8 (𝜑𝐶 ∈ (ℤ𝐶))
37 op2ndg 6127 . . . . . . . 8 ((𝐶 ∈ (ℤ𝐶) ∧ 𝐴𝑆) → (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴)
3836, 26, 37syl2anc 409 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴)
3934, 38eqtrd 2203 . . . . . 6 (𝜑 → (2nd ‘(𝑅‘∅)) = 𝐴)
4033, 39opeq12d 3771 . . . . 5 (𝜑 → ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩ = ⟨𝐶, 𝐴⟩)
4131, 40eqtr4d 2206 . . . 4 (𝜑 → (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
42 1st2nd2 6151 . . . . . . . . . . . . . . 15 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
4342adantl 275 . . . . . . . . . . . . . 14 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
4443fveq2d 5498 . . . . . . . . . . . . 13 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩))
45 df-ov 5853 . . . . . . . . . . . . . . 15 ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩)
46 xp1st 6141 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → (1st𝑧) ∈ (ℤ𝐶))
4746adantl 275 . . . . . . . . . . . . . . . 16 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (1st𝑧) ∈ (ℤ𝐶))
48 xp2nd 6142 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → (2nd𝑧) ∈ 𝑆)
4948adantl 275 . . . . . . . . . . . . . . . 16 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (2nd𝑧) ∈ 𝑆)
50 peano2uz 9529 . . . . . . . . . . . . . . . . . 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 2552 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
5453ad2antrr 485 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
55 oveq1 5857 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = (1st𝑧) → (𝑥𝐹𝑦) = ((1st𝑧)𝐹𝑦))
5655eleq1d 2239 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (1st𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st𝑧)𝐹𝑦) ∈ 𝑆))
57 oveq2 5858 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (2nd𝑧) → ((1st𝑧)𝐹𝑦) = ((1st𝑧)𝐹(2nd𝑧)))
5857eleq1d 2239 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (2nd𝑧) → (((1st𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
5956, 58rspc2v 2847 . . . . . . . . . . . . . . . . . . 19 (((1st𝑧) ∈ (ℤ𝐶) ∧ (2nd𝑧) ∈ 𝑆) → (∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆 → ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
6047, 49, 59syl2anc 409 . . . . . . . . . . . . . . . . . 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 4639 . . . . . . . . . . . . . . . . 17 (⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (((1st𝑧) + 1) ∈ (ℤ𝐶) ∧ ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
6351, 61, 62sylanbrc 415 . . . . . . . . . . . . . . . 16 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆))
64 oveq1 5857 . . . . . . . . . . . . . . . . . 18 (𝑥 = (1st𝑧) → (𝑥 + 1) = ((1st𝑧) + 1))
6564, 55opeq12d 3771 . . . . . . . . . . . . . . . . 17 (𝑥 = (1st𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹𝑦)⟩)
6657opeq2d 3770 . . . . . . . . . . . . . . . . 17 (𝑦 = (2nd𝑧) → ⟨((1st𝑧) + 1), ((1st𝑧)𝐹𝑦)⟩ = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
67 eqid 2170 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
6865, 66, 67ovmpog 5984 . . . . . . . . . . . . . . . 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 1233 . . . . . . . . . . . . . . 15 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
7045, 69eqtr3id 2217 . . . . . . . . . . . . . 14 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
7170, 63eqeltrd 2247 . . . . . . . . . . . . 13 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩) ∈ ((ℤ𝐶) × 𝑆))
7244, 71eqeltrd 2247 . . . . . . . . . . . 12 (((𝜑𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆))
7372ralrimiva 2543 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ∀𝑧 ∈ ((ℤ𝐶) × 𝑆)((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆))
7436adantr 274 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → 𝐶 ∈ (ℤ𝐶))
7526adantr 274 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → 𝐴𝑆)
76 opelxp 4639 . . . . . . . . . . . 12 (⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (𝐶 ∈ (ℤ𝐶) ∧ 𝐴𝑆))
7774, 75, 76sylanbrc 415 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆))
78 simpr 109 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → 𝑣 ∈ ω)
79 frecsuc 6383 . . . . . . . . . . 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 1233 . . . . . . . . . 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 5495 . . . . . . . . . 10 (𝑅‘suc 𝑣) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑣)
8223fveq1i 5495 . . . . . . . . . . 11 (𝑅𝑣) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑣)
8382fveq2i 5497 . . . . . . . . . 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 2228 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → (𝑅‘suc 𝑣) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)))
8584adantr 274 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)))
86 fveq2 5494 . . . . . . . . 9 ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩))
87 df-ov 5853 . . . . . . . . . 10 ((𝐺𝑣)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩)
8825adantr 274 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → 𝐶 ∈ ℤ)
8988, 32, 78frec2uzuzd 10345 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → (𝐺𝑣) ∈ (ℤ𝐶))
9025, 32, 26, 52, 23frecuzrdgrrn 10351 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → (𝑅𝑣) ∈ ((ℤ𝐶) × 𝑆))
91 xp2nd 6142 . . . . . . . . . . . 12 ((𝑅𝑣) ∈ ((ℤ𝐶) × 𝑆) → (2nd ‘(𝑅𝑣)) ∈ 𝑆)
9290, 91syl 14 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → (2nd ‘(𝑅𝑣)) ∈ 𝑆)
93 peano2uz 9529 . . . . . . . . . . . . 13 ((𝐺𝑣) ∈ (ℤ𝐶) → ((𝐺𝑣) + 1) ∈ (ℤ𝐶))
9489, 93syl 14 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣) + 1) ∈ (ℤ𝐶))
9552caovclg 6002 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧 ∈ (ℤ𝐶) ∧ 𝑤𝑆)) → (𝑧𝐹𝑤) ∈ 𝑆)
9695adantlr 474 . . . . . . . . . . . . 13 (((𝜑𝑣 ∈ ω) ∧ (𝑧 ∈ (ℤ𝐶) ∧ 𝑤𝑆)) → (𝑧𝐹𝑤) ∈ 𝑆)
9796, 89, 92caovcld 6003 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆)
98 opelxp 4639 . . . . . . . . . . . 12 (⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (((𝐺𝑣) + 1) ∈ (ℤ𝐶) ∧ ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆))
9994, 97, 98sylanbrc 415 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩ ∈ ((ℤ𝐶) × 𝑆))
100 oveq1 5857 . . . . . . . . . . . . 13 (𝑤 = (𝐺𝑣) → (𝑤 + 1) = ((𝐺𝑣) + 1))
101 oveq1 5857 . . . . . . . . . . . . 13 (𝑤 = (𝐺𝑣) → (𝑤𝐹𝑧) = ((𝐺𝑣)𝐹𝑧))
102100, 101opeq12d 3771 . . . . . . . . . . . 12 (𝑤 = (𝐺𝑣) → ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹𝑧)⟩)
103 oveq2 5858 . . . . . . . . . . . . 13 (𝑧 = (2nd ‘(𝑅𝑣)) → ((𝐺𝑣)𝐹𝑧) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
104103opeq2d 3770 . . . . . . . . . . . 12 (𝑧 = (2nd ‘(𝑅𝑣)) → ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹𝑧)⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
105 oveq1 5857 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑥 + 1) = (𝑤 + 1))
106 oveq1 5857 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑥𝐹𝑦) = (𝑤𝐹𝑦))
107105, 106opeq12d 3771 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩)
108 oveq2 5858 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑤𝐹𝑦) = (𝑤𝐹𝑧))
109108opeq2d 3770 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
110107, 109cbvmpov 5930 . . . . . . . . . . . 12 (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑤 ∈ (ℤ𝐶), 𝑧𝑆 ↦ ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
111102, 104, 110ovmpog 5984 . . . . . . . . . . 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 1233 . . . . . . . . . 10 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅𝑣))) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
11387, 112eqtr3id 2217 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
11486, 113sylan9eqr 2225 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅𝑣)) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
11585, 114eqtrd 2203 . . . . . . 7 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
11688, 32, 78frec2uzsucd 10344 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → (𝐺‘suc 𝑣) = ((𝐺𝑣) + 1))
117116adantr 274 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝐺‘suc 𝑣) = ((𝐺𝑣) + 1))
118115fveq2d 5498 . . . . . . . . 9 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩))
11988, 32, 78frec2uzzd 10343 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ ω) → (𝐺𝑣) ∈ ℤ)
120119peano2zd 9324 . . . . . . . . . . 11 ((𝜑𝑣 ∈ ω) → ((𝐺𝑣) + 1) ∈ ℤ)
121120adantr 274 . . . . . . . . . 10 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ((𝐺𝑣) + 1) ∈ ℤ)
12297adantr 274 . . . . . . . . . 10 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆)
123 op2ndg 6127 . . . . . . . . . 10 ((((𝐺𝑣) + 1) ∈ ℤ ∧ ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))) ∈ 𝑆) → (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
124121, 122, 123syl2anc 409 . . . . . . . . 9 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
125118, 124eqtrd 2203 . . . . . . . 8 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣))))
126117, 125opeq12d 3771 . . . . . . 7 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩ = ⟨((𝐺𝑣) + 1), ((𝐺𝑣)𝐹(2nd ‘(𝑅𝑣)))⟩)
127115, 126eqtr4d 2206 . . . . . 6 (((𝜑𝑣 ∈ ω) ∧ (𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
128127ex 114 . . . . 5 ((𝜑𝑣 ∈ ω) → ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
129128expcom 115 . . . 4 (𝑣 ∈ ω → (𝜑 → ((𝑅𝑣) = ⟨(𝐺𝑣), (2nd ‘(𝑅𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)))
13012, 17, 22, 41, 129finds2 4583 . . 3 (𝑧 ∈ ω → (𝜑 → (𝑅𝑧) = ⟨(𝐺𝑧), (2nd ‘(𝑅𝑧))⟩))
1317, 130vtoclga 2796 . 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 103   = wceq 1348  wcel 2141  wral 2448  Vcvv 2730  c0 3414  cop 3584  cmpt 4048  suc csuc 4348  ωcom 4572   × cxp 4607  cfv 5196  (class class class)co 5850  cmpo 5852  1st c1st 6114  2nd c2nd 6115  freccfrec 6366  1c1 7762   + caddc 7764  cz 9199  cuz 9474
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-addass 7863  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-ltadd 7877
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-inn 8866  df-n0 9123  df-z 9200  df-uz 9475
This theorem is referenced by:  frecuzrdglem  10354  frecuzrdgtcl  10355  frecuzrdgsuc  10357
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