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Theorem frecuzrdgsuc 10199
Description: Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10184 for the description of 𝐺 as the mapping from ω to (ℤ𝐶). (Contributed by Jim Kingdon, 28-May-2020.)
Hypotheses
Ref Expression
frec2uz.1 (𝜑𝐶 ∈ ℤ)
frec2uz.2 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
frecuzrdgrrn.a (𝜑𝐴𝑆)
frecuzrdgrrn.f ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
frecuzrdgrrn.2 𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
frecuzrdgtcl.3 (𝜑𝑇 = ran 𝑅)
Assertion
Ref Expression
frecuzrdgsuc ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑇‘(𝐵 + 1)) = (𝐵𝐹(𝑇𝐵)))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐶,𝑦   𝑦,𝐺   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝑅(𝑥,𝑦)   𝑇(𝑥,𝑦)   𝐺(𝑥)

Proof of Theorem frecuzrdgsuc
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frec2uz.1 . . . . . . 7 (𝜑𝐶 ∈ ℤ)
21adantr 274 . . . . . 6 ((𝜑𝐵 ∈ (ℤ𝐶)) → 𝐶 ∈ ℤ)
3 frec2uz.2 . . . . . 6 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
4 frecuzrdgrrn.a . . . . . . 7 (𝜑𝐴𝑆)
54adantr 274 . . . . . 6 ((𝜑𝐵 ∈ (ℤ𝐶)) → 𝐴𝑆)
6 frecuzrdgrrn.f . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
76adantlr 468 . . . . . 6 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
8 frecuzrdgrrn.2 . . . . . 6 𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
9 peano2uz 9390 . . . . . . 7 (𝐵 ∈ (ℤ𝐶) → (𝐵 + 1) ∈ (ℤ𝐶))
109adantl 275 . . . . . 6 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝐵 + 1) ∈ (ℤ𝐶))
112, 3, 5, 7, 8, 10frecuzrdglem 10196 . . . . 5 ((𝜑𝐵 ∈ (ℤ𝐶)) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))⟩ ∈ ran 𝑅)
12 frecuzrdgtcl.3 . . . . . 6 (𝜑𝑇 = ran 𝑅)
1312adantr 274 . . . . 5 ((𝜑𝐵 ∈ (ℤ𝐶)) → 𝑇 = ran 𝑅)
1411, 13eleqtrrd 2219 . . . 4 ((𝜑𝐵 ∈ (ℤ𝐶)) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))⟩ ∈ 𝑇)
151, 3, 4, 6, 8, 12frecuzrdgtcl 10197 . . . . . . 7 (𝜑𝑇:(ℤ𝐶)⟶𝑆)
16 ffun 5275 . . . . . . 7 (𝑇:(ℤ𝐶)⟶𝑆 → Fun 𝑇)
1715, 16syl 14 . . . . . 6 (𝜑 → Fun 𝑇)
18 funopfv 5461 . . . . . 6 (Fun 𝑇 → (⟨(𝐵 + 1), (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))⟩ ∈ 𝑇 → (𝑇‘(𝐵 + 1)) = (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))))
1917, 18syl 14 . . . . 5 (𝜑 → (⟨(𝐵 + 1), (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))⟩ ∈ 𝑇 → (𝑇‘(𝐵 + 1)) = (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))))
2019adantr 274 . . . 4 ((𝜑𝐵 ∈ (ℤ𝐶)) → (⟨(𝐵 + 1), (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))⟩ ∈ 𝑇 → (𝑇‘(𝐵 + 1)) = (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1))))))
2114, 20mpd 13 . . 3 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑇‘(𝐵 + 1)) = (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1)))))
221, 3frec2uzf1od 10191 . . . . . . . . 9 (𝜑𝐺:ω–1-1-onto→(ℤ𝐶))
23 f1ocnvdm 5682 . . . . . . . . 9 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝐵 ∈ (ℤ𝐶)) → (𝐺𝐵) ∈ ω)
2422, 23sylan 281 . . . . . . . 8 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝐺𝐵) ∈ ω)
252, 3, 24frec2uzsucd 10186 . . . . . . 7 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝐺‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵)) + 1))
26 f1ocnvfv2 5679 . . . . . . . . 9 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝐵 ∈ (ℤ𝐶)) → (𝐺‘(𝐺𝐵)) = 𝐵)
2722, 26sylan 281 . . . . . . . 8 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝐺‘(𝐺𝐵)) = 𝐵)
2827oveq1d 5789 . . . . . . 7 ((𝜑𝐵 ∈ (ℤ𝐶)) → ((𝐺‘(𝐺𝐵)) + 1) = (𝐵 + 1))
2925, 28eqtrd 2172 . . . . . 6 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝐺‘suc (𝐺𝐵)) = (𝐵 + 1))
30 peano2 4509 . . . . . . . 8 ((𝐺𝐵) ∈ ω → suc (𝐺𝐵) ∈ ω)
3124, 30syl 14 . . . . . . 7 ((𝜑𝐵 ∈ (ℤ𝐶)) → suc (𝐺𝐵) ∈ ω)
32 f1ocnvfv 5680 . . . . . . 7 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ suc (𝐺𝐵) ∈ ω) → ((𝐺‘suc (𝐺𝐵)) = (𝐵 + 1) → (𝐺‘(𝐵 + 1)) = suc (𝐺𝐵)))
3322, 31, 32syl2an2r 584 . . . . . 6 ((𝜑𝐵 ∈ (ℤ𝐶)) → ((𝐺‘suc (𝐺𝐵)) = (𝐵 + 1) → (𝐺‘(𝐵 + 1)) = suc (𝐺𝐵)))
3429, 33mpd 13 . . . . 5 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝐺‘(𝐵 + 1)) = suc (𝐺𝐵))
3534fveq2d 5425 . . . 4 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑅‘(𝐺‘(𝐵 + 1))) = (𝑅‘suc (𝐺𝐵)))
3635fveq2d 5425 . . 3 ((𝜑𝐵 ∈ (ℤ𝐶)) → (2nd ‘(𝑅‘(𝐺‘(𝐵 + 1)))) = (2nd ‘(𝑅‘suc (𝐺𝐵))))
3721, 36eqtrd 2172 . 2 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑇‘(𝐵 + 1)) = (2nd ‘(𝑅‘suc (𝐺𝐵))))
38 1st2nd2 6073 . . . . . . . . . . 11 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
3938adantl 275 . . . . . . . . . 10 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
4039fveq2d 5425 . . . . . . . . 9 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩))
41 df-ov 5777 . . . . . . . . . . 11 ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩)
42 xp1st 6063 . . . . . . . . . . . . 13 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → (1st𝑧) ∈ (ℤ𝐶))
4342adantl 275 . . . . . . . . . . . 12 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (1st𝑧) ∈ (ℤ𝐶))
44 xp2nd 6064 . . . . . . . . . . . . 13 (𝑧 ∈ ((ℤ𝐶) × 𝑆) → (2nd𝑧) ∈ 𝑆)
4544adantl 275 . . . . . . . . . . . 12 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → (2nd𝑧) ∈ 𝑆)
46 peano2uz 9390 . . . . . . . . . . . . . 14 ((1st𝑧) ∈ (ℤ𝐶) → ((1st𝑧) + 1) ∈ (ℤ𝐶))
4743, 46syl 14 . . . . . . . . . . . . 13 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧) + 1) ∈ (ℤ𝐶))
48 oveq2 5782 . . . . . . . . . . . . . . 15 (𝑦 = (2nd𝑧) → ((1st𝑧)𝐹𝑦) = ((1st𝑧)𝐹(2nd𝑧)))
4948eleq1d 2208 . . . . . . . . . . . . . 14 (𝑦 = (2nd𝑧) → (((1st𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
50 oveq1 5781 . . . . . . . . . . . . . . . . 17 (𝑥 = (1st𝑧) → (𝑥𝐹𝑦) = ((1st𝑧)𝐹𝑦))
5150eleq1d 2208 . . . . . . . . . . . . . . . 16 (𝑥 = (1st𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st𝑧)𝐹𝑦) ∈ 𝑆))
5251ralbidv 2437 . . . . . . . . . . . . . . 15 (𝑥 = (1st𝑧) → (∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦𝑆 ((1st𝑧)𝐹𝑦) ∈ 𝑆))
536ralrimivva 2514 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
5453ad2antrr 479 . . . . . . . . . . . . . . 15 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ𝐶)∀𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
5552, 54, 43rspcdva 2794 . . . . . . . . . . . . . 14 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ∀𝑦𝑆 ((1st𝑧)𝐹𝑦) ∈ 𝑆)
5649, 55, 45rspcdva 2794 . . . . . . . . . . . . 13 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆)
57 opelxp 4569 . . . . . . . . . . . . 13 (⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (((1st𝑧) + 1) ∈ (ℤ𝐶) ∧ ((1st𝑧)𝐹(2nd𝑧)) ∈ 𝑆))
5847, 56, 57sylanbrc 413 . . . . . . . . . . . 12 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆))
59 oveq1 5781 . . . . . . . . . . . . . 14 (𝑥 = (1st𝑧) → (𝑥 + 1) = ((1st𝑧) + 1))
6059, 50opeq12d 3713 . . . . . . . . . . . . 13 (𝑥 = (1st𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹𝑦)⟩)
6148opeq2d 3712 . . . . . . . . . . . . 13 (𝑦 = (2nd𝑧) → ⟨((1st𝑧) + 1), ((1st𝑧)𝐹𝑦)⟩ = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
62 eqid 2139 . . . . . . . . . . . . 13 (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
6360, 61, 62ovmpog 5905 . . . . . . . . . . . 12 (((1st𝑧) ∈ (ℤ𝐶) ∧ (2nd𝑧) ∈ 𝑆 ∧ ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩ ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
6443, 45, 58, 63syl3anc 1216 . . . . . . . . . . 11 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((1st𝑧)(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd𝑧)) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
6541, 64syl5eqr 2186 . . . . . . . . . 10 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩) = ⟨((1st𝑧) + 1), ((1st𝑧)𝐹(2nd𝑧))⟩)
6665, 58eqeltrd 2216 . . . . . . . . 9 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st𝑧), (2nd𝑧)⟩) ∈ ((ℤ𝐶) × 𝑆))
6740, 66eqeltrd 2216 . . . . . . . 8 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ 𝑧 ∈ ((ℤ𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆))
6867ralrimiva 2505 . . . . . . 7 ((𝜑𝐵 ∈ (ℤ𝐶)) → ∀𝑧 ∈ ((ℤ𝐶) × 𝑆)((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆))
69 uzid 9352 . . . . . . . . 9 (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ𝐶))
702, 69syl 14 . . . . . . . 8 ((𝜑𝐵 ∈ (ℤ𝐶)) → 𝐶 ∈ (ℤ𝐶))
71 opelxp 4569 . . . . . . . 8 (⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (𝐶 ∈ (ℤ𝐶) ∧ 𝐴𝑆))
7270, 5, 71sylanbrc 413 . . . . . . 7 ((𝜑𝐵 ∈ (ℤ𝐶)) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆))
73 frecsuc 6304 . . . . . . 7 ((∀𝑧 ∈ ((ℤ𝐶) × 𝑆)((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ𝐶) × 𝑆) ∧ ⟨𝐶, 𝐴⟩ ∈ ((ℤ𝐶) × 𝑆) ∧ (𝐺𝐵) ∈ ω) → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc (𝐺𝐵)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘(𝐺𝐵))))
7468, 72, 24, 73syl3anc 1216 . . . . . 6 ((𝜑𝐵 ∈ (ℤ𝐶)) → (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc (𝐺𝐵)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘(𝐺𝐵))))
758fveq1i 5422 . . . . . 6 (𝑅‘suc (𝐺𝐵)) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc (𝐺𝐵))
768fveq1i 5422 . . . . . . 7 (𝑅‘(𝐺𝐵)) = (frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘(𝐺𝐵))
7776fveq2i 5424 . . . . . 6 ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘(𝐺𝐵)))
7874, 75, 773eqtr4g 2197 . . . . 5 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑅‘suc (𝐺𝐵)) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))))
792, 3, 5, 7, 8, 24frec2uzrdg 10194 . . . . . . 7 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑅‘(𝐺𝐵)) = ⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
8079fveq2d 5425 . . . . . 6 ((𝜑𝐵 ∈ (ℤ𝐶)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩))
81 df-ov 5777 . . . . . 6 ((𝐺‘(𝐺𝐵))(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
8280, 81eqtr4di 2190 . . . . 5 ((𝜑𝐵 ∈ (ℤ𝐶)) → ((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))) = ((𝐺‘(𝐺𝐵))(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))))
832, 3, 24frec2uzuzd 10187 . . . . . 6 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝐺‘(𝐺𝐵)) ∈ (ℤ𝐶))
842, 3, 5, 7, 8frecuzrdgrrn 10193 . . . . . . . 8 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ (𝐺𝐵) ∈ ω) → (𝑅‘(𝐺𝐵)) ∈ ((ℤ𝐶) × 𝑆))
8524, 84mpdan 417 . . . . . . 7 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑅‘(𝐺𝐵)) ∈ ((ℤ𝐶) × 𝑆))
86 xp2nd 6064 . . . . . . 7 ((𝑅‘(𝐺𝐵)) ∈ ((ℤ𝐶) × 𝑆) → (2nd ‘(𝑅‘(𝐺𝐵))) ∈ 𝑆)
8785, 86syl 14 . . . . . 6 ((𝜑𝐵 ∈ (ℤ𝐶)) → (2nd ‘(𝑅‘(𝐺𝐵))) ∈ 𝑆)
8828, 10eqeltrd 2216 . . . . . . 7 ((𝜑𝐵 ∈ (ℤ𝐶)) → ((𝐺‘(𝐺𝐵)) + 1) ∈ (ℤ𝐶))
897caovclg 5923 . . . . . . . 8 (((𝜑𝐵 ∈ (ℤ𝐶)) ∧ (𝑧 ∈ (ℤ𝐶) ∧ 𝑤𝑆)) → (𝑧𝐹𝑤) ∈ 𝑆)
9089, 83, 87caovcld 5924 . . . . . . 7 ((𝜑𝐵 ∈ (ℤ𝐶)) → ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) ∈ 𝑆)
91 opelxp 4569 . . . . . . 7 (⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩ ∈ ((ℤ𝐶) × 𝑆) ↔ (((𝐺‘(𝐺𝐵)) + 1) ∈ (ℤ𝐶) ∧ ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) ∈ 𝑆))
9288, 90, 91sylanbrc 413 . . . . . 6 ((𝜑𝐵 ∈ (ℤ𝐶)) → ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩ ∈ ((ℤ𝐶) × 𝑆))
93 oveq1 5781 . . . . . . . 8 (𝑧 = (𝐺‘(𝐺𝐵)) → (𝑧 + 1) = ((𝐺‘(𝐺𝐵)) + 1))
94 oveq1 5781 . . . . . . . 8 (𝑧 = (𝐺‘(𝐺𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(𝐺𝐵))𝐹𝑤))
9593, 94opeq12d 3713 . . . . . . 7 (𝑧 = (𝐺‘(𝐺𝐵)) → ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩ = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹𝑤)⟩)
96 oveq2 5782 . . . . . . . 8 (𝑤 = (2nd ‘(𝑅‘(𝐺𝐵))) → ((𝐺‘(𝐺𝐵))𝐹𝑤) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
9796opeq2d 3712 . . . . . . 7 (𝑤 = (2nd ‘(𝑅‘(𝐺𝐵))) → ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹𝑤)⟩ = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
98 oveq1 5781 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1))
99 oveq1 5781 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦))
10098, 99opeq12d 3713 . . . . . . . 8 (𝑥 = 𝑧 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩)
101 oveq2 5782 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤))
102101opeq2d 3712 . . . . . . . 8 (𝑦 = 𝑤 → ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
103100, 102cbvmpov 5851 . . . . . . 7 (𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑧 ∈ (ℤ𝐶), 𝑤𝑆 ↦ ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
10495, 97, 103ovmpog 5905 . . . . . 6 (((𝐺‘(𝐺𝐵)) ∈ (ℤ𝐶) ∧ (2nd ‘(𝑅‘(𝐺𝐵))) ∈ 𝑆 ∧ ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩ ∈ ((ℤ𝐶) × 𝑆)) → ((𝐺‘(𝐺𝐵))(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
10583, 87, 92, 104syl3anc 1216 . . . . 5 ((𝜑𝐵 ∈ (ℤ𝐶)) → ((𝐺‘(𝐺𝐵))(𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
10678, 82, 1053eqtrd 2176 . . . 4 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑅‘suc (𝐺𝐵)) = ⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
107106fveq2d 5425 . . 3 ((𝜑𝐵 ∈ (ℤ𝐶)) → (2nd ‘(𝑅‘suc (𝐺𝐵))) = (2nd ‘⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩))
108 op2ndg 6049 . . . 4 ((((𝐺‘(𝐺𝐵)) + 1) ∈ (ℤ𝐶) ∧ ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) ∈ 𝑆) → (2nd ‘⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
10988, 90, 108syl2anc 408 . . 3 ((𝜑𝐵 ∈ (ℤ𝐶)) → (2nd ‘⟨((𝐺‘(𝐺𝐵)) + 1), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
110107, 109eqtrd 2172 . 2 ((𝜑𝐵 ∈ (ℤ𝐶)) → (2nd ‘(𝑅‘suc (𝐺𝐵))) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
111 simpr 109 . . . . . . 7 ((𝜑𝐵 ∈ (ℤ𝐶)) → 𝐵 ∈ (ℤ𝐶))
1122, 3, 5, 7, 8, 111frecuzrdglem 10196 . . . . . 6 ((𝜑𝐵 ∈ (ℤ𝐶)) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)
113112, 13eleqtrrd 2219 . . . . 5 ((𝜑𝐵 ∈ (ℤ𝐶)) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑇)
114 funopfv 5461 . . . . . . 7 (Fun 𝑇 → (⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑇 → (𝑇𝐵) = (2nd ‘(𝑅‘(𝐺𝐵)))))
11517, 114syl 14 . . . . . 6 (𝜑 → (⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑇 → (𝑇𝐵) = (2nd ‘(𝑅‘(𝐺𝐵)))))
116115adantr 274 . . . . 5 ((𝜑𝐵 ∈ (ℤ𝐶)) → (⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑇 → (𝑇𝐵) = (2nd ‘(𝑅‘(𝐺𝐵)))))
117113, 116mpd 13 . . . 4 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑇𝐵) = (2nd ‘(𝑅‘(𝐺𝐵))))
118117eqcomd 2145 . . 3 ((𝜑𝐵 ∈ (ℤ𝐶)) → (2nd ‘(𝑅‘(𝐺𝐵))) = (𝑇𝐵))
11927, 118oveq12d 5792 . 2 ((𝜑𝐵 ∈ (ℤ𝐶)) → ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) = (𝐵𝐹(𝑇𝐵)))
12037, 110, 1193eqtrd 2176 1 ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑇‘(𝐵 + 1)) = (𝐵𝐹(𝑇𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  wral 2416  cop 3530  cmpt 3989  suc csuc 4287  ωcom 4504   × cxp 4537  ccnv 4538  ran crn 4540  Fun wfun 5117  wf 5119  1-1-ontowf1o 5122  cfv 5123  (class class class)co 5774  cmpo 5776  1st c1st 6036  2nd c2nd 6037  freccfrec 6287  1c1 7633   + caddc 7635  cz 9066  cuz 9338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339
This theorem is referenced by: (None)
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