Theorem noseqrdgsuc 28209

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 28207	. . . . . 6 ⊢ (𝜑 → 𝑆 Fn 𝑍)
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝑆 Fn 𝑍)
9	8	fnfund 6622	. . . 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 28192	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐵 +s 1s ) ∈ 𝑍)
14	1, 2, 3, 4, 5	noseqrdglem 28206	. . . . . 6 ⊢ ((𝜑 ∧ (𝐵 +s 1s ) ∈ 𝑍) → ⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 +s 1s ))))⟩ ∈ ran 𝑅)
15	13, 14	syldan 591	. . . . 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 2832	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 +s 1s ))))⟩ ∈ 𝑆)
18		funopfv 6913	. . . 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 28202	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ω–1-1-onto→𝑍)
21	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝐺:ω–1-1-onto→𝑍)
22		f1ocnvdm 7263	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝐵 ∈ 𝑍) → (◡𝐺‘𝐵) ∈ ω)
23	20, 22	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (◡𝐺‘𝐵) ∈ ω)
24		peano2 7869	. . . . . . . 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 28198	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐺‘suc (◡𝐺‘𝐵)) = ((𝐺‘(◡𝐺‘𝐵)) +s 1s ))
29		f1ocnvfv2 7255	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝐵 ∈ 𝑍) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
30	20, 29	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
31	30	oveq1d 7405	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ((𝐺‘(◡𝐺‘𝐵)) +s 1s ) = (𝐵 +s 1s ))
32	28, 31	eqtrd 2765	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 +s 1s ))
33		f1ocnvfv 7256	. . . . . 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 6865	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑅‘(◡𝐺‘(𝐵 +s 1s ))) = (𝑅‘suc (◡𝐺‘𝐵)))
36	35	fveq2d 6865	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (2nd ‘(𝑅‘(◡𝐺‘(𝐵 +s 1s )))) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))))
37	19, 36	eqtrd 2765	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑆‘(𝐵 +s 1s )) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))))
38		frsuc 8408	. . . . . . . . 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 6863	. . . . . . . . 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 6863	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅‘(◡𝐺‘𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(◡𝐺‘𝐵)))
43	42	fveq2d 6865	. . . . . . . . 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 2775	. . . . . . 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 28205	. . . . . . . . 9 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → (𝑅‘(◡𝐺‘𝐵)) = ⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
47	46	fveq2d 6865	. . . . . . . 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 7393	. . . . . . . 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 2783	. . . . . . 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 2765	. . . . . 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 6874	. . . . . . 7 ⊢ (𝐺‘(◡𝐺‘𝐵)) ∈ V
52		fvex 6874	. . . . . . 7 ⊢ (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ V
53		oveq1 7397	. . . . . . . . 9 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧 +s 1s ) = ((𝐺‘(◡𝐺‘𝐵)) +s 1s ))
54		oveq1 7397	. . . . . . . . 9 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤))
55	53, 54	opeq12d 4848	. . . . . . . 8 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩ = ⟨((𝐺‘(◡𝐺‘𝐵)) +s 1s ), ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤)⟩)
56		oveq2 7398	. . . . . . . . 9 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
57	56	opeq2d 4847	. . . . . . . 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 7397	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 +s 1s ) = (𝑧 +s 1s ))
59		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦))
60	58, 59	opeq12d 4848	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩ = ⟨(𝑧 +s 1s ), (𝑧𝐹𝑦)⟩)
61		oveq2 7398	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤))
62	61	opeq2d 4847	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⟨(𝑧 +s 1s ), (𝑧𝐹𝑦)⟩ = ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩)
63	60, 62	cbvmpov 7487	. . . . . . . 8 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩) = (𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩)
64		opex 5427	. . . . . . . 8 ⊢ ⟨((𝐺‘(◡𝐺‘𝐵)) +s 1s ), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ V
65	55, 57, 63, 64	ovmpo 7552	. . . . . . 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 692	. . . . . 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 2781	. . . . 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 6865	. . . 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 7423	. . . . 5 ⊢ ((𝐺‘(◡𝐺‘𝐵)) +s 1s ) ∈ V
70		ovex 7423	. . . . 5 ⊢ ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) ∈ V
71	69, 70	op2nd 7980	. . . 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 2781	. . 3 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
73	23, 72	syldan 591	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
74	1, 2, 3, 4, 5	noseqrdglem 28206	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)
75	74, 16	eleqtrrd 2832	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑆)
76		funopfv 6913	. . . . 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 2736	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) = (𝑆‘𝐵))
79	30, 78	oveq12d 7408	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = (𝐵𝐹(𝑆‘𝐵)))
80	37, 73, 79	3eqtrd 2769	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑆‘(𝐵 +s 1s )) = (𝐵𝐹(𝑆‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⟨cop 4598   ↦ cmpt 5191  ^◡ccnv 5640  ran crn 5642   ↾ cres 5643   “ cima 5644  suc csuc 6337  Fun wfun 6508   Fn wfn 6509  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  ωcom 7845  2^nd c2nd 7970  reccrdg 8380   No csur 27558   1_s c1s 27742   +_s cadds 27873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-nadd 8633  df-no 27561  df-slt 27562  df-bday 27563  df-sle 27664  df-sslt 27700  df-scut 27702  df-0s 27743  df-1s 27744  df-made 27762  df-old 27763  df-left 27765  df-right 27766  df-norec2 27863  df-adds 27874

This theorem is referenced by:  seqsp1  28212