Theorem om2uzrdg 13957

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. Normally 𝐹 is a function on the partition, and 𝐴 is a member of the partition. See also comment in om2uz0i 13948. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

Hypotheses

Ref	Expression
om2uz.1	⊢ 𝐶 ∈ ℤ
om2uz.2	⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)
uzrdg.1	⊢ 𝐴 ∈ V
uzrdg.2	⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)

Assertion

Ref	Expression
om2uzrdg	⊢ (𝐵 ∈ ω → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐶   𝑦,𝐺   𝑥,𝐹,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐺(𝑥)

Proof of Theorem om2uzrdg

Dummy variables 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6896	. . 3 ⊢ (𝑧 = ∅ → (𝑅‘𝑧) = (𝑅‘∅))
2		fveq2 6896	. . . 4 ⊢ (𝑧 = ∅ → (𝐺‘𝑧) = (𝐺‘∅))
3		2fveq3 6901	. . . 4 ⊢ (𝑧 = ∅ → (2nd ‘(𝑅‘𝑧)) = (2nd ‘(𝑅‘∅)))
4	2, 3	opeq12d 4883	. . 3 ⊢ (𝑧 = ∅ → ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
5	1, 4	eqeq12d 2741	. 2 ⊢ (𝑧 = ∅ → ((𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ ↔ (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩))
6		fveq2 6896	. . 3 ⊢ (𝑧 = 𝑣 → (𝑅‘𝑧) = (𝑅‘𝑣))
7		fveq2 6896	. . . 4 ⊢ (𝑧 = 𝑣 → (𝐺‘𝑧) = (𝐺‘𝑣))
8		2fveq3 6901	. . . 4 ⊢ (𝑧 = 𝑣 → (2nd ‘(𝑅‘𝑧)) = (2nd ‘(𝑅‘𝑣)))
9	7, 8	opeq12d 4883	. . 3 ⊢ (𝑧 = 𝑣 → ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)
10	6, 9	eqeq12d 2741	. 2 ⊢ (𝑧 = 𝑣 → ((𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ ↔ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩))
11		fveq2 6896	. . 3 ⊢ (𝑧 = suc 𝑣 → (𝑅‘𝑧) = (𝑅‘suc 𝑣))
12		fveq2 6896	. . . 4 ⊢ (𝑧 = suc 𝑣 → (𝐺‘𝑧) = (𝐺‘suc 𝑣))
13		2fveq3 6901	. . . 4 ⊢ (𝑧 = suc 𝑣 → (2nd ‘(𝑅‘𝑧)) = (2nd ‘(𝑅‘suc 𝑣)))
14	12, 13	opeq12d 4883	. . 3 ⊢ (𝑧 = suc 𝑣 → ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
15	11, 14	eqeq12d 2741	. 2 ⊢ (𝑧 = suc 𝑣 → ((𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ ↔ (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
16		fveq2 6896	. . 3 ⊢ (𝑧 = 𝐵 → (𝑅‘𝑧) = (𝑅‘𝐵))
17		fveq2 6896	. . . 4 ⊢ (𝑧 = 𝐵 → (𝐺‘𝑧) = (𝐺‘𝐵))
18		2fveq3 6901	. . . 4 ⊢ (𝑧 = 𝐵 → (2nd ‘(𝑅‘𝑧)) = (2nd ‘(𝑅‘𝐵)))
19	17, 18	opeq12d 4883	. . 3 ⊢ (𝑧 = 𝐵 → ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)
20	16, 19	eqeq12d 2741	. 2 ⊢ (𝑧 = 𝐵 → ((𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ ↔ (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩))
21		uzrdg.2	. . . . 5 ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)
22	21	fveq1i 6897	. . . 4 ⊢ (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅)
23		opex 5466	. . . . 5 ⊢ ⟨𝐶, 𝐴⟩ ∈ V
24		fr0g 8457	. . . . 5 ⊢ (⟨𝐶, 𝐴⟩ ∈ V → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅) = ⟨𝐶, 𝐴⟩)
25	23, 24	ax-mp 5	. . . 4 ⊢ ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅) = ⟨𝐶, 𝐴⟩
26	22, 25	eqtri 2753	. . 3 ⊢ (𝑅‘∅) = ⟨𝐶, 𝐴⟩
27		om2uz.1	. . . . 5 ⊢ 𝐶 ∈ ℤ
28		om2uz.2	. . . . 5 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)
29	27, 28	om2uz0i 13948	. . . 4 ⊢ (𝐺‘∅) = 𝐶
30	26	fveq2i 6899	. . . . 5 ⊢ (2nd ‘(𝑅‘∅)) = (2nd ‘⟨𝐶, 𝐴⟩)
31	27	elexi 3482	. . . . . 6 ⊢ 𝐶 ∈ V
32		uzrdg.1	. . . . . 6 ⊢ 𝐴 ∈ V
33	31, 32	op2nd 8003	. . . . 5 ⊢ (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴
34	30, 33	eqtri 2753	. . . 4 ⊢ (2nd ‘(𝑅‘∅)) = 𝐴
35	29, 34	opeq12i 4880	. . 3 ⊢ ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩ = ⟨𝐶, 𝐴⟩
36	26, 35	eqtr4i 2756	. 2 ⊢ (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩
37		frsuc 8458	. . . . . 6 ⊢ (𝑣 ∈ ω → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc 𝑣) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣)))
38	21	fveq1i 6897	. . . . . 6 ⊢ (𝑅‘suc 𝑣) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc 𝑣)
39	21	fveq1i 6897	. . . . . . 7 ⊢ (𝑅‘𝑣) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣)
40	39	fveq2i 6899	. . . . . 6 ⊢ ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣))
41	37, 38, 40	3eqtr4g 2790	. . . . 5 ⊢ (𝑣 ∈ ω → (𝑅‘suc 𝑣) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)))
42		fveq2 6896	. . . . . 6 ⊢ ((𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩ → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩))
43		df-ov 7422	. . . . . . 7 ⊢ ((𝐺‘𝑣)(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘𝑣))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)
44		fvex 6909	. . . . . . . 8 ⊢ (𝐺‘𝑣) ∈ V
45		fvex 6909	. . . . . . . 8 ⊢ (2nd ‘(𝑅‘𝑣)) ∈ V
46		oveq1 7426	. . . . . . . . . 10 ⊢ (𝑤 = (𝐺‘𝑣) → (𝑤 + 1) = ((𝐺‘𝑣) + 1))
47		oveq1 7426	. . . . . . . . . 10 ⊢ (𝑤 = (𝐺‘𝑣) → (𝑤𝐹𝑧) = ((𝐺‘𝑣)𝐹𝑧))
48	46, 47	opeq12d 4883	. . . . . . . . 9 ⊢ (𝑤 = (𝐺‘𝑣) → ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩ = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹𝑧)⟩)
49		oveq2 7427	. . . . . . . . . 10 ⊢ (𝑧 = (2nd ‘(𝑅‘𝑣)) → ((𝐺‘𝑣)𝐹𝑧) = ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣))))
50	49	opeq2d 4882	. . . . . . . . 9 ⊢ (𝑧 = (2nd ‘(𝑅‘𝑣)) → ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹𝑧)⟩ = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
51		oveq1 7426	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑥 + 1) = (𝑤 + 1))
52		oveq1 7426	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑥𝐹𝑦) = (𝑤𝐹𝑦))
53	51, 52	opeq12d 4883	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩)
54		oveq2 7427	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑤𝐹𝑦) = (𝑤𝐹𝑧))
55	54	opeq2d 4882	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ⟨(𝑤 + 1), (𝑤𝐹𝑦)⟩ = ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
56	53, 55	cbvmpov 7515	. . . . . . . . 9 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑤 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑤 + 1), (𝑤𝐹𝑧)⟩)
57		opex 5466	. . . . . . . . 9 ⊢ ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩ ∈ V
58	48, 50, 56, 57	ovmpo 7581	. . . . . . . 8 ⊢ (((𝐺‘𝑣) ∈ V ∧ (2nd ‘(𝑅‘𝑣)) ∈ V) → ((𝐺‘𝑣)(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘𝑣))) = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
59	44, 45, 58	mp2an 690	. . . . . . 7 ⊢ ((𝐺‘𝑣)(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘𝑣))) = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩
60	43, 59	eqtr3i 2755	. . . . . 6 ⊢ ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩
61	42, 60	eqtrdi 2781	. . . . 5 ⊢ ((𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩ → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)) = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
62	41, 61	sylan9eq 2785	. . . 4 ⊢ ((𝑣 ∈ ω ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
63	27, 28	om2uzsuci 13949	. . . . . 6 ⊢ (𝑣 ∈ ω → (𝐺‘suc 𝑣) = ((𝐺‘𝑣) + 1))
64	63	adantr 479	. . . . 5 ⊢ ((𝑣 ∈ ω ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → (𝐺‘suc 𝑣) = ((𝐺‘𝑣) + 1))
65	62	fveq2d 6900	. . . . . 6 ⊢ ((𝑣 ∈ ω ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = (2nd ‘⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩))
66		ovex 7452	. . . . . . 7 ⊢ ((𝐺‘𝑣) + 1) ∈ V
67		ovex 7452	. . . . . . 7 ⊢ ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣))) ∈ V
68	66, 67	op2nd 8003	. . . . . 6 ⊢ (2nd ‘⟨((𝐺‘𝑣) + 1), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩) = ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))
69	65, 68	eqtrdi 2781	. . . . 5 ⊢ ((𝑣 ∈ ω ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → (2nd ‘(𝑅‘suc 𝑣)) = ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣))))
70	64, 69	opeq12d 4883	. . . 4 ⊢ ((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)))⟩)
71	62, 70	eqtr4d 2768	. . 3 ⊢ ((𝑣 ∈ ω ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
72	71	ex 411	. 2 ⊢ (𝑣 ∈ ω → ((𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
73	5, 10, 15, 20, 36, 72	finds 7904	1 ⊢ (𝐵 ∈ ω → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3461  ∅c0 4322  ⟨cop 4636   ↦ cmpt 5232   ↾ cres 5680  suc csuc 6373  ‘cfv 6549  (class class class)co 7419   ∈ cmpo 7421  ωcom 7871  2^nd c2nd 7993  reccrdg 8430  1c1 11141   + caddc 11143  ℤcz 12591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431

This theorem is referenced by:  uzrdglem  13958  uzrdgfni  13959  uzrdgsuci  13961