Theorem noseqrdgsuc 28289

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 28287	. . . . . 6 ⊢ (𝜑 → 𝑆 Fn 𝑍)
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝑆 Fn 𝑍)
9	8	fnfund 6594	. . . 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 28272	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐵 +s 1s ) ∈ 𝑍)
14	1, 2, 3, 4, 5	noseqrdglem 28286	. . . . . 6 ⊢ ((𝜑 ∧ (𝐵 +s 1s ) ∈ 𝑍) → ⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 +s 1s ))))⟩ ∈ ran 𝑅)
15	13, 14	syldan 592	. . . . 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 2840	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 +s 1s ))))⟩ ∈ 𝑆)
18		funopfv 6884	. . . 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 28282	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ω–1-1-onto→𝑍)
21	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝐺:ω–1-1-onto→𝑍)
22		f1ocnvdm 7233	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝐵 ∈ 𝑍) → (◡𝐺‘𝐵) ∈ ω)
23	20, 22	sylan 581	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (◡𝐺‘𝐵) ∈ ω)
24		peano2 7834	. . . . . . . 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 28278	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐺‘suc (◡𝐺‘𝐵)) = ((𝐺‘(◡𝐺‘𝐵)) +s 1s ))
29		f1ocnvfv2 7225	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝐵 ∈ 𝑍) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
30	20, 29	sylan 581	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
31	30	oveq1d 7375	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ((𝐺‘(◡𝐺‘𝐵)) +s 1s ) = (𝐵 +s 1s ))
32	28, 31	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 +s 1s ))
33		f1ocnvfv 7226	. . . . . 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 6839	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑅‘(◡𝐺‘(𝐵 +s 1s ))) = (𝑅‘suc (◡𝐺‘𝐵)))
36	35	fveq2d 6839	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (2nd ‘(𝑅‘(◡𝐺‘(𝐵 +s 1s )))) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))))
37	19, 36	eqtrd 2772	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑆‘(𝐵 +s 1s )) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))))
38		frsuc 8370	. . . . . . . . 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 6837	. . . . . . . . 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 6837	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅‘(◡𝐺‘𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(◡𝐺‘𝐵)))
43	42	fveq2d 6839	. . . . . . . . 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 2782	. . . . . . 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 28285	. . . . . . . . 9 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → (𝑅‘(◡𝐺‘𝐵)) = ⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
47	46	fveq2d 6839	. . . . . . . 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 7363	. . . . . . . 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 2790	. . . . . . 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 2772	. . . . . 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 6848	. . . . . . 7 ⊢ (𝐺‘(◡𝐺‘𝐵)) ∈ V
52		fvex 6848	. . . . . . 7 ⊢ (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ V
53		oveq1 7367	. . . . . . . . 9 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧 +s 1s ) = ((𝐺‘(◡𝐺‘𝐵)) +s 1s ))
54		oveq1 7367	. . . . . . . . 9 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤))
55	53, 54	opeq12d 4838	. . . . . . . 8 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩ = ⟨((𝐺‘(◡𝐺‘𝐵)) +s 1s ), ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤)⟩)
56		oveq2 7368	. . . . . . . . 9 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
57	56	opeq2d 4837	. . . . . . . 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 7367	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 +s 1s ) = (𝑧 +s 1s ))
59		oveq1 7367	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦))
60	58, 59	opeq12d 4838	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩ = ⟨(𝑧 +s 1s ), (𝑧𝐹𝑦)⟩)
61		oveq2 7368	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤))
62	61	opeq2d 4837	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⟨(𝑧 +s 1s ), (𝑧𝐹𝑦)⟩ = ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩)
63	60, 62	cbvmpov 7455	. . . . . . . 8 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩) = (𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩)
64		opex 5413	. . . . . . . 8 ⊢ ⟨((𝐺‘(◡𝐺‘𝐵)) +s 1s ), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ V
65	55, 57, 63, 64	ovmpo 7520	. . . . . . 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 693	. . . . . 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 2788	. . . . 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 6839	. . . 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 7393	. . . . 5 ⊢ ((𝐺‘(◡𝐺‘𝐵)) +s 1s ) ∈ V
70		ovex 7393	. . . . 5 ⊢ ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) ∈ V
71	69, 70	op2nd 7944	. . . 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 2788	. . 3 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
73	23, 72	syldan 592	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
74	1, 2, 3, 4, 5	noseqrdglem 28286	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)
75	74, 16	eleqtrrd 2840	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑆)
76		funopfv 6884	. . . . 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 2743	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) = (𝑆‘𝐵))
79	30, 78	oveq12d 7378	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = (𝐵𝐹(𝑆‘𝐵)))
80	37, 73, 79	3eqtrd 2776	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑆‘(𝐵 +s 1s )) = (𝐵𝐹(𝑆‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3441  ⟨cop 4587   ↦ cmpt 5180  ^◡ccnv 5624  ran crn 5626   ↾ cres 5627   “ cima 5628  suc csuc 6320  Fun wfun 6487   Fn wfn 6488  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7360   ∈ cmpo 7362  ωcom 7810  2^nd c2nd 7934  reccrdg 8342   No csur 27611   1_s c1s 27804   +_s cadds 27941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-nadd 8596  df-no 27614  df-slt 27615  df-bday 27616  df-sle 27717  df-sslt 27758  df-scut 27760  df-0s 27805  df-1s 27806  df-made 27825  df-old 27826  df-left 27828  df-right 27829  df-norec2 27931  df-adds 27942

This theorem is referenced by:  seqsp1  28292