Theorem noseqrdgsuc 28332

Description: Successor value of a recursive definition generator on surreal sequences. (Contributed by Scott Fenton, 19-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))
noseqrdg.3	⊢ (𝜑 → 𝑆 = ran 𝑅)

Assertion

Ref	Expression
noseqrdgsuc	⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑆‘(𝐵 +s 1s )) = (𝐵𝐹(𝑆‘𝐵)))

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

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

Proof of Theorem noseqrdgsuc

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

Step	Hyp	Ref	Expression
1		om2noseq.1	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ No )
2		om2noseq.2	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
3		om2noseq.3	. . . . . . 7 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
4		noseqrdg.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		noseqrdg.2	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
6		noseqrdg.3	. . . . . . 7 ⊢ (𝜑 → 𝑆 = ran 𝑅)
7	1, 2, 3, 4, 5, 6	noseqrdgfn 28330	. . . . . 6 ⊢ (𝜑 → 𝑆 Fn 𝑍)
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝑆 Fn 𝑍)
9	8	fnfund 6680	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → Fun 𝑆)
10	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
11	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝐶 ∈ No )
12		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝐵 ∈ 𝑍)
13	10, 11, 12	noseqp1 28315	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐵 +s 1s ) ∈ 𝑍)
14	1, 2, 3, 4, 5	noseqrdglem 28329	. . . . . 6 ⊢ ((𝜑 ∧ (𝐵 +s 1s ) ∈ 𝑍) → ⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 +s 1s ))))⟩ ∈ ran 𝑅)
15	13, 14	syldan 590	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 +s 1s ))))⟩ ∈ ran 𝑅)
16	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝑆 = ran 𝑅)
17	15, 16	eleqtrrd 2847	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 +s 1s ))))⟩ ∈ 𝑆)
18		funopfv 6972	. . . 4 ⊢ (Fun 𝑆 → (⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 +s 1s ))))⟩ ∈ 𝑆 → (𝑆‘(𝐵 +s 1s )) = (2nd ‘(𝑅‘(◡𝐺‘(𝐵 +s 1s ))))))
19	9, 17, 18	sylc 65	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑆‘(𝐵 +s 1s )) = (2nd ‘(𝑅‘(◡𝐺‘(𝐵 +s 1s )))))
20	1, 2, 3	om2noseqf1o 28325	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ω–1-1-onto→𝑍)
21	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝐺:ω–1-1-onto→𝑍)
22		f1ocnvdm 7321	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝐵 ∈ 𝑍) → (◡𝐺‘𝐵) ∈ ω)
23	20, 22	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (◡𝐺‘𝐵) ∈ ω)
24		peano2 7929	. . . . . . . 8 ⊢ ((◡𝐺‘𝐵) ∈ ω → suc (◡𝐺‘𝐵) ∈ ω)
25	23, 24	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → suc (◡𝐺‘𝐵) ∈ ω)
26	21, 25	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐺:ω–1-1-onto→𝑍 ∧ suc (◡𝐺‘𝐵) ∈ ω))
27	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
28	11, 27, 23	om2noseqsuc 28321	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐺‘suc (◡𝐺‘𝐵)) = ((𝐺‘(◡𝐺‘𝐵)) +s 1s ))
29		f1ocnvfv2 7313	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝐵 ∈ 𝑍) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
30	20, 29	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
31	30	oveq1d 7463	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ((𝐺‘(◡𝐺‘𝐵)) +s 1s ) = (𝐵 +s 1s ))
32	28, 31	eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 +s 1s ))
33		f1ocnvfv 7314	. . . . . 6 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ suc (◡𝐺‘𝐵) ∈ ω) → ((𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 +s 1s ) → (◡𝐺‘(𝐵 +s 1s )) = suc (◡𝐺‘𝐵)))
34	26, 32, 33	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (◡𝐺‘(𝐵 +s 1s )) = suc (◡𝐺‘𝐵))
35	34	fveq2d 6924	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑅‘(◡𝐺‘(𝐵 +s 1s ))) = (𝑅‘suc (◡𝐺‘𝐵)))
36	35	fveq2d 6924	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (2nd ‘(𝑅‘(◡𝐺‘(𝐵 +s 1s )))) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))))
37	19, 36	eqtrd 2780	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑆‘(𝐵 +s 1s )) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))))
38		frsuc 8493	. . . . . . . . 9 ⊢ ((◡𝐺‘𝐵) ∈ ω → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(◡𝐺‘𝐵))))
39	38	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(◡𝐺‘𝐵))))
40	5	fveq1d 6922	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘suc (◡𝐺‘𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (◡𝐺‘𝐵)))
41	40	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → (𝑅‘suc (◡𝐺‘𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (◡𝐺‘𝐵)))
42	5	fveq1d 6922	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅‘(◡𝐺‘𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(◡𝐺‘𝐵)))
43	42	fveq2d 6924	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(◡𝐺‘𝐵))))
44	43	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(◡𝐺‘𝐵))))
45	39, 41, 44	3eqtr4d 2790	. . . . . . 7 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → (𝑅‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))))
46	1, 2, 3, 4, 5	om2noseqrdg 28328	. . . . . . . . 9 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → (𝑅‘(◡𝐺‘𝐵)) = ⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
47	46	fveq2d 6924	. . . . . . . 8 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩))
48		df-ov 7451	. . . . . . . 8 ⊢ ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
49	47, 48	eqtr4di 2798	. . . . . . 7 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
50	45, 49	eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → (𝑅‘suc (◡𝐺‘𝐵)) = ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
51		fvex 6933	. . . . . . 7 ⊢ (𝐺‘(◡𝐺‘𝐵)) ∈ V
52		fvex 6933	. . . . . . 7 ⊢ (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ V
53		oveq1 7455	. . . . . . . . 9 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧 +s 1s ) = ((𝐺‘(◡𝐺‘𝐵)) +s 1s ))
54		oveq1 7455	. . . . . . . . 9 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤))
55	53, 54	opeq12d 4905	. . . . . . . 8 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩ = ⟨((𝐺‘(◡𝐺‘𝐵)) +s 1s ), ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤)⟩)
56		oveq2 7456	. . . . . . . . 9 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
57	56	opeq2d 4904	. . . . . . . 8 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → ⟨((𝐺‘(◡𝐺‘𝐵)) +s 1s ), ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤)⟩ = ⟨((𝐺‘(◡𝐺‘𝐵)) +s 1s ), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
58		oveq1 7455	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 +s 1s ) = (𝑧 +s 1s ))
59		oveq1 7455	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦))
60	58, 59	opeq12d 4905	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩ = ⟨(𝑧 +s 1s ), (𝑧𝐹𝑦)⟩)
61		oveq2 7456	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤))
62	61	opeq2d 4904	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⟨(𝑧 +s 1s ), (𝑧𝐹𝑦)⟩ = ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩)
63	60, 62	cbvmpov 7545	. . . . . . . 8 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩) = (𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩)
64		opex 5484	. . . . . . . 8 ⊢ ⟨((𝐺‘(◡𝐺‘𝐵)) +s 1s ), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ V
65	55, 57, 63, 64	ovmpo 7610	. . . . . . 7 ⊢ (((𝐺‘(◡𝐺‘𝐵)) ∈ V ∧ (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ V) → ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = ⟨((𝐺‘(◡𝐺‘𝐵)) +s 1s ), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
66	51, 52, 65	mp2an 691	. . . . . 6 ⊢ ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = ⟨((𝐺‘(◡𝐺‘𝐵)) +s 1s ), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩
67	50, 66	eqtrdi 2796	. . . . 5 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → (𝑅‘suc (◡𝐺‘𝐵)) = ⟨((𝐺‘(◡𝐺‘𝐵)) +s 1s ), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
68	67	fveq2d 6924	. . . 4 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = (2nd ‘⟨((𝐺‘(◡𝐺‘𝐵)) +s 1s ), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩))
69		ovex 7481	. . . . 5 ⊢ ((𝐺‘(◡𝐺‘𝐵)) +s 1s ) ∈ V
70		ovex 7481	. . . . 5 ⊢ ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) ∈ V
71	69, 70	op2nd 8039	. . . 4 ⊢ (2nd ‘⟨((𝐺‘(◡𝐺‘𝐵)) +s 1s ), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))
72	68, 71	eqtrdi 2796	. . 3 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
73	23, 72	syldan 590	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
74	1, 2, 3, 4, 5	noseqrdglem 28329	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)
75	74, 16	eleqtrrd 2847	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑆)
76		funopfv 6972	. . . . 5 ⊢ (Fun 𝑆 → (⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑆 → (𝑆‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
77	9, 75, 76	sylc 65	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑆‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵))))
78	77	eqcomd 2746	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) = (𝑆‘𝐵))
79	30, 78	oveq12d 7466	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = (𝐵𝐹(𝑆‘𝐵)))
80	37, 73, 79	3eqtrd 2784	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑆‘(𝐵 +s 1s )) = (𝐵𝐹(𝑆‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ⟨cop 4654   ↦ cmpt 5249  ^◡ccnv 5699  ran crn 5701   ↾ cres 5702   “ cima 5703  suc csuc 6397  Fun wfun 6567   Fn wfn 6568  –1-1-onto→wf1o 6572  ‘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-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  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-rmo 3388  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-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  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-se 5653  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-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec2 28000  df-adds 28011

This theorem is referenced by:  seqsp1  28335