Theorem frec2uzrdg 10435

Description: A helper lemma for the value of a recursive definition generator on upper integers (typically either ℕ or ℕ₀) 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 (ℤ_≥‘𝐶)). See comment in frec2uz0d 10425 which describes 𝐺 and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)

Hypotheses

Ref	Expression
frec2uz.1	⊢ (𝜑 → 𝐶 ∈ ℤ)
frec2uz.2	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
frecuzrdgrrn.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
frecuzrdgrrn.f	⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
frecuzrdgrrn.2	⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
frec2uzrdg.b	⊢ (𝜑 → 𝐵 ∈ ω)

Assertion

Ref	Expression
frec2uzrdg	⊢ (𝜑 → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)

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.

Step	Hyp	Ref	Expression
1		frec2uzrdg.b	. 2 ⊢ (𝜑 → 𝐵 ∈ ω)
2		fveq2 5531	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑅‘𝑧) = (𝑅‘𝐵))
3		fveq2 5531	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝐺‘𝑧) = (𝐺‘𝐵))
4	2	fveq2d 5535	. . . . . 6 ⊢ (𝑧 = 𝐵 → (2nd ‘(𝑅‘𝑧)) = (2nd ‘(𝑅‘𝐵)))
5	3, 4	opeq12d 3801	. . . . 5 ⊢ (𝑧 = 𝐵 → ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)
6	2, 5	eqeq12d 2204	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ ↔ (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩))
7	6	imbi2d 230	. . 3 ⊢ (𝑧 = 𝐵 → ((𝜑 → (𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩) ↔ (𝜑 → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)))
8		fveq2 5531	. . . . 5 ⊢ (𝑧 = ∅ → (𝑅‘𝑧) = (𝑅‘∅))
9		fveq2 5531	. . . . . 6 ⊢ (𝑧 = ∅ → (𝐺‘𝑧) = (𝐺‘∅))
10	8	fveq2d 5535	. . . . . 6 ⊢ (𝑧 = ∅ → (2nd ‘(𝑅‘𝑧)) = (2nd ‘(𝑅‘∅)))
11	9, 10	opeq12d 3801	. . . . 5 ⊢ (𝑧 = ∅ → ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
12	8, 11	eqeq12d 2204	. . . 4 ⊢ (𝑧 = ∅ → ((𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ ↔ (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩))
13		fveq2 5531	. . . . 5 ⊢ (𝑧 = 𝑣 → (𝑅‘𝑧) = (𝑅‘𝑣))
14		fveq2 5531	. . . . . 6 ⊢ (𝑧 = 𝑣 → (𝐺‘𝑧) = (𝐺‘𝑣))
15	13	fveq2d 5535	. . . . . 6 ⊢ (𝑧 = 𝑣 → (2nd ‘(𝑅‘𝑧)) = (2nd ‘(𝑅‘𝑣)))
16	14, 15	opeq12d 3801	. . . . 5 ⊢ (𝑧 = 𝑣 → ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)
17	13, 16	eqeq12d 2204	. . . 4 ⊢ (𝑧 = 𝑣 → ((𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ ↔ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩))
18		fveq2 5531	. . . . 5 ⊢ (𝑧 = suc 𝑣 → (𝑅‘𝑧) = (𝑅‘suc 𝑣))
19		fveq2 5531	. . . . . 6 ⊢ (𝑧 = suc 𝑣 → (𝐺‘𝑧) = (𝐺‘suc 𝑣))
20	18	fveq2d 5535	. . . . . 6 ⊢ (𝑧 = suc 𝑣 → (2nd ‘(𝑅‘𝑧)) = (2nd ‘(𝑅‘suc 𝑣)))
21	19, 20	opeq12d 3801	. . . . 5 ⊢ (𝑧 = suc 𝑣 → ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
22	18, 21	eqeq12d 2204	. . . 4 ⊢ (𝑧 = suc 𝑣 → ((𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ ↔ (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
23		frecuzrdgrrn.2	. . . . . . 7 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
24	23	fveq1i 5532	. . . . . 6 ⊢ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅)
25		frec2uz.1	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℤ)
26		frecuzrdgrrn.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
27		opexg 4243	. . . . . . . 8 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ 𝑆) → ⟨𝐶, 𝐴⟩ ∈ V)
28	25, 26, 27	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ ∈ V)
29		frec0g 6417	. . . . . . 7 ⊢ (⟨𝐶, 𝐴⟩ ∈ V → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅) = ⟨𝐶, 𝐴⟩)
30	28, 29	syl 14	. . . . . 6 ⊢ (𝜑 → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅) = ⟨𝐶, 𝐴⟩)
31	24, 30	eqtrid 2234	. . . . 5 ⊢ (𝜑 → (𝑅‘∅) = ⟨𝐶, 𝐴⟩)
32		frec2uz.2	. . . . . . 7 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
33	25, 32	frec2uz0d 10425	. . . . . 6 ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
34	31	fveq2d 5535	. . . . . . 7 ⊢ (𝜑 → (2nd ‘(𝑅‘∅)) = (2nd ‘⟨𝐶, 𝐴⟩))
35		uzid 9567	. . . . . . . . 9 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶))
36	25, 35	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐶))
37		op2ndg 6171	. . . . . . . 8 ⊢ ((𝐶 ∈ (ℤ≥‘𝐶) ∧ 𝐴 ∈ 𝑆) → (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴)
38	36, 26, 37	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴)
39	34, 38	eqtrd 2222	. . . . . 6 ⊢ (𝜑 → (2nd ‘(𝑅‘∅)) = 𝐴)
40	33, 39	opeq12d 3801	. . . . 5 ⊢ (𝜑 → ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩ = ⟨𝐶, 𝐴⟩)
41	31, 40	eqtr4d 2225	. . . 4 ⊢ (𝜑 → (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
42		1st2nd2 6195	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
43	42	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
44	43	fveq2d 5535	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
45		df-ov 5895	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
46		xp1st 6185	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
47	46	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
48		xp2nd 6186	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘𝑧) ∈ 𝑆)
49	48	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑆)
50		peano2uz 9608	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑧) ∈ (ℤ≥‘𝐶) → ((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶))
51	47, 50	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶))
52		frecuzrdgrrn.f	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
53	52	ralrimivva 2572	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
54	53	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
55		oveq1 5899	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥𝐹𝑦) = ((1st ‘𝑧)𝐹𝑦))
56	55	eleq1d 2258	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (1st ‘𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
57		oveq2 5900	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧)𝐹𝑦) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
58	57	eleq1d 2258	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (2nd ‘𝑧) → (((1st ‘𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
59	56, 58	rspc2v 2869	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘𝑧) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘𝑧) ∈ 𝑆) → (∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆 → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
60	47, 49, 59	syl2anc 411	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆 → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
61	54, 60	mpd 13	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆)
62		opelxp 4671	. . . . . . . . . . . . . . . . 17 ⊢ (⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ (((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶) ∧ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
63	51, 61, 62	sylanbrc 417	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
64		oveq1 5899	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + 1) = ((1st ‘𝑧) + 1))
65	64, 55	opeq12d 3801	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (1st ‘𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩)
66	57	opeq2d 3800	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
67		eqid 2189	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
68	65, 66, 67	ovmpog 6027	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘𝑧) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘𝑧) ∈ 𝑆 ∧ ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
69	47, 49, 63, 68	syl3anc 1249	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
70	45, 69	eqtr3id 2236	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
71	70, 63	eqeltrd 2266	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩) ∈ ((ℤ≥‘𝐶) × 𝑆))
72	44, 71	eqeltrd 2266	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
73	72	ralrimiva 2563	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → ∀𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
74	36	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → 𝐶 ∈ (ℤ≥‘𝐶))
75	26	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → 𝐴 ∈ 𝑆)
76		opelxp 4671	. . . . . . . . . . . 12 ⊢ (⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ (𝐶 ∈ (ℤ≥‘𝐶) ∧ 𝐴 ∈ 𝑆))
77	74, 75, 76	sylanbrc 417	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
78		simpr 110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → 𝑣 ∈ ω)
79		frecsuc 6427	. . . . . . . . . . 11 ⊢ ((∀𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆) ∧ ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ∧ 𝑣 ∈ ω) → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑣) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑣)))
80	73, 77, 78, 79	syl3anc 1249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑣) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑣)))
81	23	fveq1i 5532	. . . . . . . . . 10 ⊢ (𝑅‘suc 𝑣) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑣)
82	23	fveq1i 5532	. . . . . . . . . . 11 ⊢ (𝑅‘𝑣) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑣)
83	82	fveq2i 5534	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑣))
84	80, 81, 83	3eqtr4g 2247	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → (𝑅‘suc 𝑣) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)))
85	84	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → (𝑅‘suc 𝑣) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)))
86		fveq2 5531	. . . . . . . . 9 ⊢ ((𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩ → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩))
87		df-ov 5895	. . . . . . . . . 10 ⊢ ((𝐺‘𝑣)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘𝑣))) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)
88	25	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → 𝐶 ∈ ℤ)
89	88, 32, 78	frec2uzuzd 10428	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → (𝐺‘𝑣) ∈ (ℤ≥‘𝐶))
90	25, 32, 26, 52, 23	frecuzrdgrrn 10434	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → (𝑅‘𝑣) ∈ ((ℤ≥‘𝐶) × 𝑆))
91		xp2nd 6186	. . . . . . . . . . . 12 ⊢ ((𝑅‘𝑣) ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘(𝑅‘𝑣)) ∈ 𝑆)
92	90, 91	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → (2nd ‘(𝑅‘𝑣)) ∈ 𝑆)
93		peano2uz 9608	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑣) ∈ (ℤ≥‘𝐶) → ((𝐺‘𝑣) + 1) ∈ (ℤ≥‘𝐶))
94	89, 93	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → ((𝐺‘𝑣) + 1) ∈ (ℤ≥‘𝐶))
95	52	caovclg 6045	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (ℤ≥‘𝐶) ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐹𝑤) ∈ 𝑆)
96	95	adantlr 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ (𝑧 ∈ (ℤ≥‘𝐶) ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐹𝑤) ∈ 𝑆)
97	96, 89, 92	caovcld 6046	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣))) ∈ 𝑆)
98		opelxp 4671	. . . . . . . . . . . 12 ⊢ (⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ (((𝐺‘𝑣) + 1) ∈ (ℤ≥‘𝐶) ∧ ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣))) ∈ 𝑆))
99	94, 97, 98	sylanbrc 417	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
100		oveq1 5899	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐺‘𝑣) → (𝑤 + 1) = ((𝐺‘𝑣) + 1))
101		oveq1 5899	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐺‘𝑣) → (𝑤𝐹𝑧) = ((𝐺‘𝑣)𝐹𝑧))
102	100, 101	opeq12d 3801	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐺‘𝑣) → ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩ = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹𝑧)⟩)
103		oveq2 5900	. . . . . . . . . . . . 13 ⊢ (𝑧 = (2nd ‘(𝑅‘𝑣)) → ((𝐺‘𝑣)𝐹𝑧) = ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣))))
104	103	opeq2d 3800	. . . . . . . . . . . 12 ⊢ (𝑧 = (2nd ‘(𝑅‘𝑣)) → ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹𝑧)⟩ = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
105		oveq1 5899	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝑥 + 1) = (𝑤 + 1))
106		oveq1 5899	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝑥𝐹𝑦) = (𝑤𝐹𝑦))
107	105, 106	opeq12d 3801	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩)
108		oveq2 5900	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝑤𝐹𝑦) = (𝑤𝐹𝑧))
109	108	opeq2d 3800	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
110	107, 109	cbvmpov 5972	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑤 ∈ (ℤ≥‘𝐶), 𝑧 ∈ 𝑆 ↦ ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
111	102, 104, 110	ovmpog 6027	. . . . . . . . . . 11 ⊢ (((𝐺‘𝑣) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘(𝑅‘𝑣)) ∈ 𝑆 ∧ ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝐺‘𝑣)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘𝑣))) = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
112	89, 92, 99, 111	syl3anc 1249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → ((𝐺‘𝑣)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘𝑣))) = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
113	87, 112	eqtr3id 2236	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
114	86, 113	sylan9eqr 2244	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)) = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
115	85, 114	eqtrd 2222	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
116	88, 32, 78	frec2uzsucd 10427	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → (𝐺‘suc 𝑣) = ((𝐺‘𝑣) + 1))
117	116	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → (𝐺‘suc 𝑣) = ((𝐺‘𝑣) + 1))
118	115	fveq2d 5535	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = (2nd ‘⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩))
119	88, 32, 78	frec2uzzd 10426	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → (𝐺‘𝑣) ∈ ℤ)
120	119	peano2zd 9403	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → ((𝐺‘𝑣) + 1) ∈ ℤ)
121	120	adantr 276	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → ((𝐺‘𝑣) + 1) ∈ ℤ)
122	97	adantr 276	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣))) ∈ 𝑆)
123		op2ndg 6171	. . . . . . . . . 10 ⊢ ((((𝐺‘𝑣) + 1) ∈ ℤ ∧ ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣))) ∈ 𝑆) → (2nd ‘⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩) = ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣))))
124	121, 122, 123	syl2anc 411	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → (2nd ‘⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩) = ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣))))
125	118, 124	eqtrd 2222	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣))))
126	117, 125	opeq12d 3801	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩ = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
127	115, 126	eqtr4d 2225	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ ω) ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
128	127	ex 115	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → ((𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
129	128	expcom 116	. . . 4 ⊢ (𝑣 ∈ ω → (𝜑 → ((𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)))
130	12, 17, 22, 41, 129	finds2 4615	. . 3 ⊢ (𝑧 ∈ ω → (𝜑 → (𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩))
131	7, 130	vtoclga 2818	. 2 ⊢ (𝐵 ∈ ω → (𝜑 → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩))
132	1, 131	mpcom 36	1 ⊢ (𝜑 → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)

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  1^st c1st 6158  2^nd c2nd 6159  freccfrec 6410  1c1 7837   + caddc 7839  ℤcz 9278  ℤ_≥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