Theorem om2noseqrdg 28328

Description: A helper lemma for the value of a recursive definition generator on a surreal sequence with characteristic function 𝐹(𝑥, 𝑦) and initial value 𝐴. (Contributed by Scott Fenton, 18-Apr-2025.)

Hypotheses

Ref	Expression
om2noseq.1	⊢ (𝜑 → 𝐶 ∈ No )
om2noseq.2	⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
om2noseq.3	⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
noseqrdg.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
noseqrdg.2	⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))

Assertion

Ref	Expression
om2noseqrdg	⊢ ((𝜑 ∧ 𝐵 ∈ ω) → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem om2noseqrdg

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

Step	Hyp	Ref	Expression
1		fveq2 6920	. . . . 5 ⊢ (𝑧 = ∅ → (𝑅‘𝑧) = (𝑅‘∅))
2		fveq2 6920	. . . . . 6 ⊢ (𝑧 = ∅ → (𝐺‘𝑧) = (𝐺‘∅))
3		2fveq3 6925	. . . . . 6 ⊢ (𝑧 = ∅ → (2nd ‘(𝑅‘𝑧)) = (2nd ‘(𝑅‘∅)))
4	2, 3	opeq12d 4905	. . . . 5 ⊢ (𝑧 = ∅ → ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
5	1, 4	eqeq12d 2756	. . . 4 ⊢ (𝑧 = ∅ → ((𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ ↔ (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩))
6	5	imbi2d 340	. . 3 ⊢ (𝑧 = ∅ → ((𝜑 → (𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩) ↔ (𝜑 → (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)))
7		fveq2 6920	. . . . 5 ⊢ (𝑧 = 𝑣 → (𝑅‘𝑧) = (𝑅‘𝑣))
8		fveq2 6920	. . . . . 6 ⊢ (𝑧 = 𝑣 → (𝐺‘𝑧) = (𝐺‘𝑣))
9		2fveq3 6925	. . . . . 6 ⊢ (𝑧 = 𝑣 → (2nd ‘(𝑅‘𝑧)) = (2nd ‘(𝑅‘𝑣)))
10	8, 9	opeq12d 4905	. . . . 5 ⊢ (𝑧 = 𝑣 → ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)
11	7, 10	eqeq12d 2756	. . . 4 ⊢ (𝑧 = 𝑣 → ((𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ ↔ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩))
12	11	imbi2d 340	. . 3 ⊢ (𝑧 = 𝑣 → ((𝜑 → (𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩) ↔ (𝜑 → (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)))
13		fveq2 6920	. . . . 5 ⊢ (𝑧 = suc 𝑣 → (𝑅‘𝑧) = (𝑅‘suc 𝑣))
14		fveq2 6920	. . . . . 6 ⊢ (𝑧 = suc 𝑣 → (𝐺‘𝑧) = (𝐺‘suc 𝑣))
15		2fveq3 6925	. . . . . 6 ⊢ (𝑧 = suc 𝑣 → (2nd ‘(𝑅‘𝑧)) = (2nd ‘(𝑅‘suc 𝑣)))
16	14, 15	opeq12d 4905	. . . . 5 ⊢ (𝑧 = suc 𝑣 → ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
17	13, 16	eqeq12d 2756	. . . 4 ⊢ (𝑧 = suc 𝑣 → ((𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ ↔ (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩))
18	17	imbi2d 340	. . 3 ⊢ (𝑧 = suc 𝑣 → ((𝜑 → (𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩) ↔ (𝜑 → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)))
19		fveq2 6920	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑅‘𝑧) = (𝑅‘𝐵))
20		fveq2 6920	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝐺‘𝑧) = (𝐺‘𝐵))
21		2fveq3 6925	. . . . . 6 ⊢ (𝑧 = 𝐵 → (2nd ‘(𝑅‘𝑧)) = (2nd ‘(𝑅‘𝐵)))
22	20, 21	opeq12d 4905	. . . . 5 ⊢ (𝑧 = 𝐵 → ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)
23	19, 22	eqeq12d 2756	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩ ↔ (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩))
24	23	imbi2d 340	. . 3 ⊢ (𝑧 = 𝐵 → ((𝜑 → (𝑅‘𝑧) = ⟨(𝐺‘𝑧), (2nd ‘(𝑅‘𝑧))⟩) ↔ (𝜑 → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)))
25		noseqrdg.2	. . . . . 6 ⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
26	25	fveq1d 6922	. . . . 5 ⊢ (𝜑 → (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅))
27		opex 5484	. . . . . 6 ⊢ ⟨𝐶, 𝐴⟩ ∈ V
28		fr0g 8492	. . . . . 6 ⊢ (⟨𝐶, 𝐴⟩ ∈ V → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅) = ⟨𝐶, 𝐴⟩)
29	27, 28	ax-mp 5	. . . . 5 ⊢ ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅) = ⟨𝐶, 𝐴⟩
30	26, 29	eqtrdi 2796	. . . 4 ⊢ (𝜑 → (𝑅‘∅) = ⟨𝐶, 𝐴⟩)
31		om2noseq.1	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ No )
32		om2noseq.2	. . . . . 6 ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
33	31, 32	om2noseq0 28320	. . . . 5 ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
34	30	fveq2d 6924	. . . . . 6 ⊢ (𝜑 → (2nd ‘(𝑅‘∅)) = (2nd ‘⟨𝐶, 𝐴⟩))
35		noseqrdg.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
36		op2ndg 8043	. . . . . . 7 ⊢ ((𝐶 ∈ No ∧ 𝐴 ∈ 𝑉) → (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴)
37	31, 35, 36	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝐶, 𝐴⟩) = 𝐴)
38	34, 37	eqtrd 2780	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝑅‘∅)) = 𝐴)
39	33, 38	opeq12d 4905	. . . 4 ⊢ (𝜑 → ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩ = ⟨𝐶, 𝐴⟩)
40	30, 39	eqtr4d 2783	. . 3 ⊢ (𝜑 → (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
41		frsuc 8493	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ω → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc 𝑣) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣)))
42	41	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc 𝑣) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣)))
43	25	fveq1d 6922	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅‘suc 𝑣) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc 𝑣))
44	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → (𝑅‘suc 𝑣) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc 𝑣))
45	25	fveq1d 6922	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑅‘𝑣) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣))
46	45	fveq2d 6924	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣)))
47	46	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘𝑣)))
48	42, 44, 47	3eqtr4d 2790	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → (𝑅‘suc 𝑣) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)))
49	48	adantrr 716	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)) → (𝑅‘suc 𝑣) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)))
50		fveq2 6920	. . . . . . . . . 10 ⊢ ((𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩ → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩))
51		df-ov 7451	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑣)(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘𝑣))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)
52		fvex 6933	. . . . . . . . . . . 12 ⊢ (𝐺‘𝑣) ∈ V
53		fvex 6933	. . . . . . . . . . . 12 ⊢ (2nd ‘(𝑅‘𝑣)) ∈ V
54		oveq1 7455	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐺‘𝑣) → (𝑤 +s 1s ) = ((𝐺‘𝑣) +s 1s ))
55		oveq1 7455	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐺‘𝑣) → (𝑤𝐹𝑧) = ((𝐺‘𝑣)𝐹𝑧))
56	54, 55	opeq12d 4905	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐺‘𝑣) → ⟨(𝑤 +s 1s ), (𝑤𝐹𝑧)⟩ = ⟨((𝐺‘𝑣) +s 1s ), ((𝐺‘𝑣)𝐹𝑧)⟩)
57		oveq2 7456	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (2nd ‘(𝑅‘𝑣)) → ((𝐺‘𝑣)𝐹𝑧) = ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣))))
58	57	opeq2d 4904	. . . . . . . . . . . . 13 ⊢ (𝑧 = (2nd ‘(𝑅‘𝑣)) → ⟨((𝐺‘𝑣) +s 1s ), ((𝐺‘𝑣)𝐹𝑧)⟩ = ⟨((𝐺‘𝑣) +s 1s ), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
59		oveq1 7455	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝑥 +s 1s ) = (𝑤 +s 1s ))
60		oveq1 7455	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝑥𝐹𝑦) = (𝑤𝐹𝑦))
61	59, 60	opeq12d 4905	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩ = ⟨(𝑤 +s 1s ), (𝑤𝐹𝑦)⟩)
62		oveq2 7456	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝑤𝐹𝑦) = (𝑤𝐹𝑧))
63	62	opeq2d 4904	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → ⟨(𝑤 +s 1s ), (𝑤𝐹𝑦)⟩ = ⟨(𝑤 +s 1s ), (𝑤𝐹𝑧)⟩)
64	61, 63	cbvmpov 7545	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩) = (𝑤 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑤 +s 1s ), (𝑤𝐹𝑧)⟩)
65		opex 5484	. . . . . . . . . . . . 13 ⊢ ⟨((𝐺‘𝑣) +s 1s ), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩ ∈ V
66	56, 58, 64, 65	ovmpo 7610	. . . . . . . . . . . 12 ⊢ (((𝐺‘𝑣) ∈ V ∧ (2nd ‘(𝑅‘𝑣)) ∈ V) → ((𝐺‘𝑣)(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘𝑣))) = ⟨((𝐺‘𝑣) +s 1s ), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
67	52, 53, 66	mp2an 691	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑣)(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘𝑣))) = ⟨((𝐺‘𝑣) +s 1s ), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩
68	51, 67	eqtr3i 2770	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) = ⟨((𝐺‘𝑣) +s 1s ), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩
69	50, 68	eqtrdi 2796	. . . . . . . . 9 ⊢ ((𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩ → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)) = ⟨((𝐺‘𝑣) +s 1s ), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
70	69	ad2antll 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)) → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑣)) = ⟨((𝐺‘𝑣) +s 1s ), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
71	49, 70	eqtrd 2780	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)) → (𝑅‘suc 𝑣) = ⟨((𝐺‘𝑣) +s 1s ), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
72	31	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → 𝐶 ∈ No )
73	32	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
74		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → 𝑣 ∈ ω)
75	72, 73, 74	om2noseqsuc 28321	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → (𝐺‘suc 𝑣) = ((𝐺‘𝑣) +s 1s ))
76	75	adantrr 716	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)) → (𝐺‘suc 𝑣) = ((𝐺‘𝑣) +s 1s ))
77	71	fveq2d 6924	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)) → (2nd ‘(𝑅‘suc 𝑣)) = (2nd ‘⟨((𝐺‘𝑣) +s 1s ), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩))
78		ovex 7481	. . . . . . . . . 10 ⊢ ((𝐺‘𝑣) +s 1s ) ∈ V
79		ovex 7481	. . . . . . . . . 10 ⊢ ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣))) ∈ V
80	78, 79	op2nd 8039	. . . . . . . . 9 ⊢ (2nd ‘⟨((𝐺‘𝑣) +s 1s ), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩) = ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))
81	77, 80	eqtrdi 2796	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)) → (2nd ‘(𝑅‘suc 𝑣)) = ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣))))
82	76, 81	opeq12d 4905	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)) → ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩ = ⟨((𝐺‘𝑣) +s 1s ), ((𝐺‘𝑣)𝐹(2nd ‘(𝑅‘𝑣)))⟩)
83	71, 82	eqtr4d 2783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩)) → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)
84	83	exp32 420	. . . . 5 ⊢ (𝜑 → (𝑣 ∈ ω → ((𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)))
85	84	com12 32	. . . 4 ⊢ (𝑣 ∈ ω → (𝜑 → ((𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩ → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)))
86	85	a2d 29	. . 3 ⊢ (𝑣 ∈ ω → ((𝜑 → (𝑅‘𝑣) = ⟨(𝐺‘𝑣), (2nd ‘(𝑅‘𝑣))⟩) → (𝜑 → (𝑅‘suc 𝑣) = ⟨(𝐺‘suc 𝑣), (2nd ‘(𝑅‘suc 𝑣))⟩)))
87	6, 12, 18, 24, 40, 86	finds 7936	. 2 ⊢ (𝐵 ∈ ω → (𝜑 → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩))
88	87	impcom 407	1 ⊢ ((𝜑 ∧ 𝐵 ∈ ω) → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  ⟨cop 4654   ↦ cmpt 5249   ↾ cres 5702   “ cima 5703  suc csuc 6397  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  ωcom 7903  2^nd c2nd 8029  reccrdg 8465   No csur 27702   1_s c1s 27886   +_s cadds 28010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466

This theorem is referenced by:  noseqrdglem  28329  noseqrdgfn  28330  noseqrdgsuc  28332