ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  seqvalcd GIF version

Theorem seqvalcd 10262
Description: Value of the sequence builder function. Similar to seq3val 10261 but the classes 𝐷 (type of each term) and 𝐶 (type of the value we are accumulating) do not need to be the same. (Contributed by Jim Kingdon, 9-Jul-2023.)
Hypotheses
Ref Expression
seqvalcd.m (𝜑𝑀 ∈ ℤ)
seqvalcd.r 𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)
seqvalcd.f0 (𝜑 → (𝐹𝑀) ∈ 𝐶)
seqvalcd.pl ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)
seqvalcd.fp1 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝑥) ∈ 𝐷)
Assertion
Ref Expression
seqvalcd (𝜑 → seq𝑀( + , 𝐹) = ran 𝑅)
Distinct variable groups:   𝑥, + ,𝑦,𝑤,𝑧   𝑥,𝐶,𝑦,𝑤,𝑧   𝑥,𝐷,𝑦   𝑥,𝐹,𝑦,𝑤,𝑧   𝑥,𝑀,𝑦,𝑤,𝑧   𝑥,𝑅,𝑦,𝑤,𝑧   𝜑,𝑥,𝑦,𝑤,𝑧
Allowed substitution hints:   𝐷(𝑧,𝑤)

Proof of Theorem seqvalcd
Dummy variables 𝑎 𝑏 𝑐 𝑘 𝑛 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-seqfrec 10249 . 2 seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
2 seqvalcd.m . . . . . 6 (𝜑𝑀 ∈ ℤ)
3 seqvalcd.f0 . . . . . 6 (𝜑 → (𝐹𝑀) ∈ 𝐶)
4 ssv 3123 . . . . . . 7 𝐶 ⊆ V
54a1i 9 . . . . . 6 (𝜑𝐶 ⊆ V)
6 eqidd 2141 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) → (𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) = (𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1)))))
7 simprr 522 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) ∧ (𝑧 = 𝑥𝑤 = 𝑦)) → 𝑤 = 𝑦)
8 simprl 521 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) ∧ (𝑧 = 𝑥𝑤 = 𝑦)) → 𝑧 = 𝑥)
98fvoveq1d 5803 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) ∧ (𝑧 = 𝑥𝑤 = 𝑦)) → (𝐹‘(𝑧 + 1)) = (𝐹‘(𝑥 + 1)))
107, 9oveq12d 5799 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) ∧ (𝑧 = 𝑥𝑤 = 𝑦)) → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑦 + (𝐹‘(𝑥 + 1))))
11 simprl 521 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) → 𝑥 ∈ (ℤ𝑀))
12 simprr 522 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) → 𝑦𝐶)
13 seqvalcd.pl . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)
1413ralrimivva 2517 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝐶𝑦𝐷 (𝑥 + 𝑦) ∈ 𝐶)
15 oveq1 5788 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑥 + 𝑦) = (𝑎 + 𝑦))
1615eleq1d 2209 . . . . . . . . . . . 12 (𝑥 = 𝑎 → ((𝑥 + 𝑦) ∈ 𝐶 ↔ (𝑎 + 𝑦) ∈ 𝐶))
17 oveq2 5789 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → (𝑎 + 𝑦) = (𝑎 + 𝑏))
1817eleq1d 2209 . . . . . . . . . . . 12 (𝑦 = 𝑏 → ((𝑎 + 𝑦) ∈ 𝐶 ↔ (𝑎 + 𝑏) ∈ 𝐶))
1916, 18cbvral2v 2668 . . . . . . . . . . 11 (∀𝑥𝐶𝑦𝐷 (𝑥 + 𝑦) ∈ 𝐶 ↔ ∀𝑎𝐶𝑏𝐷 (𝑎 + 𝑏) ∈ 𝐶)
2014, 19sylib 121 . . . . . . . . . 10 (𝜑 → ∀𝑎𝐶𝑏𝐷 (𝑎 + 𝑏) ∈ 𝐶)
2120adantr 274 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) → ∀𝑎𝐶𝑏𝐷 (𝑎 + 𝑏) ∈ 𝐶)
22 fveq2 5428 . . . . . . . . . . . 12 (𝑎 = (𝑥 + 1) → (𝐹𝑎) = (𝐹‘(𝑥 + 1)))
2322eleq1d 2209 . . . . . . . . . . 11 (𝑎 = (𝑥 + 1) → ((𝐹𝑎) ∈ 𝐷 ↔ (𝐹‘(𝑥 + 1)) ∈ 𝐷))
24 seqvalcd.fp1 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝑥) ∈ 𝐷)
2524ralrimiva 2508 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥 ∈ (ℤ‘(𝑀 + 1))(𝐹𝑥) ∈ 𝐷)
26 fveq2 5428 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
2726eleq1d 2209 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → ((𝐹𝑥) ∈ 𝐷 ↔ (𝐹𝑎) ∈ 𝐷))
2827cbvralv 2657 . . . . . . . . . . . . 13 (∀𝑥 ∈ (ℤ‘(𝑀 + 1))(𝐹𝑥) ∈ 𝐷 ↔ ∀𝑎 ∈ (ℤ‘(𝑀 + 1))(𝐹𝑎) ∈ 𝐷)
2925, 28sylib 121 . . . . . . . . . . . 12 (𝜑 → ∀𝑎 ∈ (ℤ‘(𝑀 + 1))(𝐹𝑎) ∈ 𝐷)
3029adantr 274 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) → ∀𝑎 ∈ (ℤ‘(𝑀 + 1))(𝐹𝑎) ∈ 𝐷)
31 eluzp1p1 9374 . . . . . . . . . . . 12 (𝑥 ∈ (ℤ𝑀) → (𝑥 + 1) ∈ (ℤ‘(𝑀 + 1)))
3211, 31syl 14 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) → (𝑥 + 1) ∈ (ℤ‘(𝑀 + 1)))
3323, 30, 32rspcdva 2797 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) → (𝐹‘(𝑥 + 1)) ∈ 𝐷)
34 oveq12 5790 . . . . . . . . . . . 12 ((𝑎 = 𝑦𝑏 = (𝐹‘(𝑥 + 1))) → (𝑎 + 𝑏) = (𝑦 + (𝐹‘(𝑥 + 1))))
3534eleq1d 2209 . . . . . . . . . . 11 ((𝑎 = 𝑦𝑏 = (𝐹‘(𝑥 + 1))) → ((𝑎 + 𝑏) ∈ 𝐶 ↔ (𝑦 + (𝐹‘(𝑥 + 1))) ∈ 𝐶))
3635rspc2gv 2804 . . . . . . . . . 10 ((𝑦𝐶 ∧ (𝐹‘(𝑥 + 1)) ∈ 𝐷) → (∀𝑎𝐶𝑏𝐷 (𝑎 + 𝑏) ∈ 𝐶 → (𝑦 + (𝐹‘(𝑥 + 1))) ∈ 𝐶))
3712, 33, 36syl2anc 409 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) → (∀𝑎𝐶𝑏𝐷 (𝑎 + 𝑏) ∈ 𝐶 → (𝑦 + (𝐹‘(𝑥 + 1))) ∈ 𝐶))
3821, 37mpd 13 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) → (𝑦 + (𝐹‘(𝑥 + 1))) ∈ 𝐶)
396, 10, 11, 12, 38ovmpod 5905 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) → (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = (𝑦 + (𝐹‘(𝑥 + 1))))
4039, 38eqeltrd 2217 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) → (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝐶)
41 seqvalcd.r . . . . . 6 𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)
422, 3, 5, 40, 41frecuzrdgrclt 10218 . . . . 5 (𝜑𝑅:ω⟶((ℤ𝑀) × 𝐶))
4342ffnd 5280 . . . 4 (𝜑𝑅 Fn ω)
44 1st2nd2 6080 . . . . . . . . . . . 12 (𝑢 ∈ ((ℤ𝑀) × 𝐶) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
4544adantl 275 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
4645fveq2d 5432 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st𝑢), (2nd𝑢)⟩))
47 df-ov 5784 . . . . . . . . . 10 ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd𝑢)) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st𝑢), (2nd𝑢)⟩)
4846, 47eqtr4di 2191 . . . . . . . . 9 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) = ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd𝑢)))
49 xp1st 6070 . . . . . . . . . . 11 (𝑢 ∈ ((ℤ𝑀) × 𝐶) → (1st𝑢) ∈ (ℤ𝑀))
5049adantl 275 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → (1st𝑢) ∈ (ℤ𝑀))
51 xp2nd 6071 . . . . . . . . . . . 12 (𝑢 ∈ ((ℤ𝑀) × 𝐶) → (2nd𝑢) ∈ 𝐶)
5251adantl 275 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → (2nd𝑢) ∈ 𝐶)
5352elexd 2702 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → (2nd𝑢) ∈ V)
54 peano2uz 9404 . . . . . . . . . . . 12 ((1st𝑢) ∈ (ℤ𝑀) → ((1st𝑢) + 1) ∈ (ℤ𝑀))
5550, 54syl 14 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((1st𝑢) + 1) ∈ (ℤ𝑀))
5614adantr 274 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ∀𝑥𝐶𝑦𝐷 (𝑥 + 𝑦) ∈ 𝐶)
57 fveq2 5428 . . . . . . . . . . . . . . 15 (𝑥 = ((1st𝑢) + 1) → (𝐹𝑥) = (𝐹‘((1st𝑢) + 1)))
5857eleq1d 2209 . . . . . . . . . . . . . 14 (𝑥 = ((1st𝑢) + 1) → ((𝐹𝑥) ∈ 𝐷 ↔ (𝐹‘((1st𝑢) + 1)) ∈ 𝐷))
5925adantr 274 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ∀𝑥 ∈ (ℤ‘(𝑀 + 1))(𝐹𝑥) ∈ 𝐷)
60 eluzp1p1 9374 . . . . . . . . . . . . . . 15 ((1st𝑢) ∈ (ℤ𝑀) → ((1st𝑢) + 1) ∈ (ℤ‘(𝑀 + 1)))
6150, 60syl 14 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((1st𝑢) + 1) ∈ (ℤ‘(𝑀 + 1)))
6258, 59, 61rspcdva 2797 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → (𝐹‘((1st𝑢) + 1)) ∈ 𝐷)
63 oveq12 5790 . . . . . . . . . . . . . . 15 ((𝑥 = (2nd𝑢) ∧ 𝑦 = (𝐹‘((1st𝑢) + 1))) → (𝑥 + 𝑦) = ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))))
6463eleq1d 2209 . . . . . . . . . . . . . 14 ((𝑥 = (2nd𝑢) ∧ 𝑦 = (𝐹‘((1st𝑢) + 1))) → ((𝑥 + 𝑦) ∈ 𝐶 ↔ ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))) ∈ 𝐶))
6564rspc2gv 2804 . . . . . . . . . . . . 13 (((2nd𝑢) ∈ 𝐶 ∧ (𝐹‘((1st𝑢) + 1)) ∈ 𝐷) → (∀𝑥𝐶𝑦𝐷 (𝑥 + 𝑦) ∈ 𝐶 → ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))) ∈ 𝐶))
6652, 62, 65syl2anc 409 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → (∀𝑥𝐶𝑦𝐷 (𝑥 + 𝑦) ∈ 𝐶 → ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))) ∈ 𝐶))
6756, 66mpd 13 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))) ∈ 𝐶)
6855, 67opelxpd 4579 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩ ∈ ((ℤ𝑀) × 𝐶))
69 oveq1 5788 . . . . . . . . . . . 12 (𝑥 = (1st𝑢) → (𝑥 + 1) = ((1st𝑢) + 1))
70 fvoveq1 5804 . . . . . . . . . . . . 13 (𝑥 = (1st𝑢) → (𝐹‘(𝑥 + 1)) = (𝐹‘((1st𝑢) + 1)))
7170oveq2d 5797 . . . . . . . . . . . 12 (𝑥 = (1st𝑢) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘((1st𝑢) + 1))))
7269, 71opeq12d 3720 . . . . . . . . . . 11 (𝑥 = (1st𝑢) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨((1st𝑢) + 1), (𝑦 + (𝐹‘((1st𝑢) + 1)))⟩)
73 oveq1 5788 . . . . . . . . . . . 12 (𝑦 = (2nd𝑢) → (𝑦 + (𝐹‘((1st𝑢) + 1))) = ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))))
7473opeq2d 3719 . . . . . . . . . . 11 (𝑦 = (2nd𝑢) → ⟨((1st𝑢) + 1), (𝑦 + (𝐹‘((1st𝑢) + 1)))⟩ = ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩)
75 eqid 2140 . . . . . . . . . . 11 (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)
7672, 74, 75ovmpog 5912 . . . . . . . . . 10 (((1st𝑢) ∈ (ℤ𝑀) ∧ (2nd𝑢) ∈ V ∧ ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩ ∈ ((ℤ𝑀) × 𝐶)) → ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd𝑢)) = ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩)
7750, 53, 68, 76syl3anc 1217 . . . . . . . . 9 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd𝑢)) = ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩)
7848, 77eqtrd 2173 . . . . . . . 8 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) = ⟨((1st𝑢) + 1), ((2nd𝑢) + (𝐹‘((1st𝑢) + 1)))⟩)
7978, 68eqeltrd 2217 . . . . . . 7 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝐶))
8079ralrimiva 2508 . . . . . 6 (𝜑 → ∀𝑢 ∈ ((ℤ𝑀) × 𝐶)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝐶))
81 uzid 9363 . . . . . . . 8 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
822, 81syl 14 . . . . . . 7 (𝜑𝑀 ∈ (ℤ𝑀))
8382, 3opelxpd 4579 . . . . . 6 (𝜑 → ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝐶))
8480, 83jca 304 . . . . 5 (𝜑 → (∀𝑢 ∈ ((ℤ𝑀) × 𝐶)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝐶) ∧ ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝐶)))
85 frecfcl 6309 . . . . 5 ((∀𝑢 ∈ ((ℤ𝑀) × 𝐶)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝐶) ∧ ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝐶)) → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩):ω⟶((ℤ𝑀) × 𝐶))
86 ffn 5279 . . . . 5 (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩):ω⟶((ℤ𝑀) × 𝐶) → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) Fn ω)
8784, 85, 863syl 17 . . . 4 (𝜑 → frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) Fn ω)
88 fveq2 5428 . . . . . . . 8 (𝑐 = ∅ → (𝑅𝑐) = (𝑅‘∅))
89 fveq2 5428 . . . . . . . 8 (𝑐 = ∅ → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅))
9088, 89eqeq12d 2155 . . . . . . 7 (𝑐 = ∅ → ((𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) ↔ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅)))
9190imbi2d 229 . . . . . 6 (𝑐 = ∅ → ((𝜑 → (𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅))))
92 fveq2 5428 . . . . . . . 8 (𝑐 = 𝑘 → (𝑅𝑐) = (𝑅𝑘))
93 fveq2 5428 . . . . . . . 8 (𝑐 = 𝑘 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘))
9492, 93eqeq12d 2155 . . . . . . 7 (𝑐 = 𝑘 → ((𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) ↔ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
9594imbi2d 229 . . . . . 6 (𝑐 = 𝑘 → ((𝜑 → (𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘))))
96 fveq2 5428 . . . . . . . 8 (𝑐 = suc 𝑘 → (𝑅𝑐) = (𝑅‘suc 𝑘))
97 fveq2 5428 . . . . . . . 8 (𝑐 = suc 𝑘 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))
9896, 97eqeq12d 2155 . . . . . . 7 (𝑐 = suc 𝑘 → ((𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) ↔ (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘)))
9998imbi2d 229 . . . . . 6 (𝑐 = suc 𝑘 → ((𝜑 → (𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))))
100 fveq2 5428 . . . . . . . 8 (𝑐 = 𝑛 → (𝑅𝑐) = (𝑅𝑛))
101 fveq2 5428 . . . . . . . 8 (𝑐 = 𝑛 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛))
102100, 101eqeq12d 2155 . . . . . . 7 (𝑐 = 𝑛 → ((𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐) ↔ (𝑅𝑛) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛)))
103102imbi2d 229 . . . . . 6 (𝑐 = 𝑛 → ((𝜑 → (𝑅𝑐) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅𝑛) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛))))
10441fveq1i 5429 . . . . . . . 8 (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅)
105 frec0g 6301 . . . . . . . . 9 (⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝐶) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
10683, 105syl 14 . . . . . . . 8 (𝜑 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
107104, 106syl5eq 2185 . . . . . . 7 (𝜑 → (𝑅‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
108 frec0g 6301 . . . . . . . 8 (⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝐶) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
10983, 108syl 14 . . . . . . 7 (𝜑 → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅) = ⟨𝑀, (𝐹𝑀)⟩)
110107, 109eqtr4d 2176 . . . . . 6 (𝜑 → (𝑅‘∅) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘∅))
11142ad2antlr 481 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → 𝑅:ω⟶((ℤ𝑀) × 𝐶))
112 simpll 519 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → 𝑘 ∈ ω)
113111, 112ffvelrnd 5563 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅𝑘) ∈ ((ℤ𝑀) × 𝐶))
114 xp1st 6070 . . . . . . . . . . 11 ((𝑅𝑘) ∈ ((ℤ𝑀) × 𝐶) → (1st ‘(𝑅𝑘)) ∈ (ℤ𝑀))
115113, 114syl 14 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (1st ‘(𝑅𝑘)) ∈ (ℤ𝑀))
116 xp2nd 6071 . . . . . . . . . . . 12 ((𝑅𝑘) ∈ ((ℤ𝑀) × 𝐶) → (2nd ‘(𝑅𝑘)) ∈ 𝐶)
117113, 116syl 14 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (2nd ‘(𝑅𝑘)) ∈ 𝐶)
118117elexd 2702 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (2nd ‘(𝑅𝑘)) ∈ V)
119 peano2uz 9404 . . . . . . . . . . . 12 ((1st ‘(𝑅𝑘)) ∈ (ℤ𝑀) → ((1st ‘(𝑅𝑘)) + 1) ∈ (ℤ𝑀))
120115, 119syl 14 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘)) + 1) ∈ (ℤ𝑀))
12114ad2antlr 481 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ∀𝑥𝐶𝑦𝐷 (𝑥 + 𝑦) ∈ 𝐶)
122 fveq2 5428 . . . . . . . . . . . . . . 15 (𝑎 = ((1st ‘(𝑅𝑘)) + 1) → (𝐹𝑎) = (𝐹‘((1st ‘(𝑅𝑘)) + 1)))
123122eleq1d 2209 . . . . . . . . . . . . . 14 (𝑎 = ((1st ‘(𝑅𝑘)) + 1) → ((𝐹𝑎) ∈ 𝐷 ↔ (𝐹‘((1st ‘(𝑅𝑘)) + 1)) ∈ 𝐷))
12429ad2antlr 481 . . . . . . . . . . . . . 14 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ∀𝑎 ∈ (ℤ‘(𝑀 + 1))(𝐹𝑎) ∈ 𝐷)
125 eluzp1p1 9374 . . . . . . . . . . . . . . 15 ((1st ‘(𝑅𝑘)) ∈ (ℤ𝑀) → ((1st ‘(𝑅𝑘)) + 1) ∈ (ℤ‘(𝑀 + 1)))
126115, 125syl 14 . . . . . . . . . . . . . 14 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘)) + 1) ∈ (ℤ‘(𝑀 + 1)))
127123, 124, 126rspcdva 2797 . . . . . . . . . . . . 13 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝐹‘((1st ‘(𝑅𝑘)) + 1)) ∈ 𝐷)
128 oveq12 5790 . . . . . . . . . . . . . . 15 ((𝑥 = (2nd ‘(𝑅𝑘)) ∧ 𝑦 = (𝐹‘((1st ‘(𝑅𝑘)) + 1))) → (𝑥 + 𝑦) = ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
129128eleq1d 2209 . . . . . . . . . . . . . 14 ((𝑥 = (2nd ‘(𝑅𝑘)) ∧ 𝑦 = (𝐹‘((1st ‘(𝑅𝑘)) + 1))) → ((𝑥 + 𝑦) ∈ 𝐶 ↔ ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))) ∈ 𝐶))
130129rspc2gv 2804 . . . . . . . . . . . . 13 (((2nd ‘(𝑅𝑘)) ∈ 𝐶 ∧ (𝐹‘((1st ‘(𝑅𝑘)) + 1)) ∈ 𝐷) → (∀𝑥𝐶𝑦𝐷 (𝑥 + 𝑦) ∈ 𝐶 → ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))) ∈ 𝐶))
131117, 127, 130syl2anc 409 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (∀𝑥𝐶𝑦𝐷 (𝑥 + 𝑦) ∈ 𝐶 → ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))) ∈ 𝐶))
132121, 131mpd 13 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))) ∈ 𝐶)
133120, 132opelxpd 4579 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩ ∈ ((ℤ𝑀) × 𝐶))
134 oveq1 5788 . . . . . . . . . . . 12 (𝑥 = (1st ‘(𝑅𝑘)) → (𝑥 + 1) = ((1st ‘(𝑅𝑘)) + 1))
135 fvoveq1 5804 . . . . . . . . . . . . 13 (𝑥 = (1st ‘(𝑅𝑘)) → (𝐹‘(𝑥 + 1)) = (𝐹‘((1st ‘(𝑅𝑘)) + 1)))
136135oveq2d 5797 . . . . . . . . . . . 12 (𝑥 = (1st ‘(𝑅𝑘)) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
137134, 136opeq12d 3720 . . . . . . . . . . 11 (𝑥 = (1st ‘(𝑅𝑘)) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨((1st ‘(𝑅𝑘)) + 1), (𝑦 + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
138 oveq1 5788 . . . . . . . . . . . 12 (𝑦 = (2nd ‘(𝑅𝑘)) → (𝑦 + (𝐹‘((1st ‘(𝑅𝑘)) + 1))) = ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
139138opeq2d 3719 . . . . . . . . . . 11 (𝑦 = (2nd ‘(𝑅𝑘)) → ⟨((1st ‘(𝑅𝑘)) + 1), (𝑦 + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩ = ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
140137, 139, 75ovmpog 5912 . . . . . . . . . 10 (((1st ‘(𝑅𝑘)) ∈ (ℤ𝑀) ∧ (2nd ‘(𝑅𝑘)) ∈ V ∧ ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩ ∈ ((ℤ𝑀) × 𝐶)) → ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))) = ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
141115, 118, 133, 140syl3anc 1217 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))) = ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
14280ad2antlr 481 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ∀𝑢 ∈ ((ℤ𝑀) × 𝐶)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝐶))
14383ad2antlr 481 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝐶))
144 frecsuc 6311 . . . . . . . . . . 11 ((∀𝑢 ∈ ((ℤ𝑀) × 𝐶)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝐶) ∧ ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝐶) ∧ 𝑘 ∈ ω) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
145142, 143, 112, 144syl3anc 1217 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
146 simpr 109 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘))
147146fveq2d 5432 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅𝑘)) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
148 1st2nd2 6080 . . . . . . . . . . . . 13 ((𝑅𝑘) ∈ ((ℤ𝑀) × 𝐶) → (𝑅𝑘) = ⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩)
149113, 148syl 14 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅𝑘) = ⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩)
150149fveq2d 5432 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅𝑘)) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩))
151 df-ov 5784 . . . . . . . . . . 11 ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩)
152150, 151eqtr4di 2191 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅𝑘)) = ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))))
153145, 147, 1523eqtr2d 2179 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅𝑘))))
15445fveq2d 5432 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st𝑢), (2nd𝑢)⟩))
155 df-ov 5784 . . . . . . . . . . . . . . . . . . 19 ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd𝑢)) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st𝑢), (2nd𝑢)⟩)
156154, 155eqtr4di 2191 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) = ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd𝑢)))
157 fvoveq1 5804 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = (1st𝑢) → (𝐹‘(𝑧 + 1)) = (𝐹‘((1st𝑢) + 1)))
158157oveq2d 5797 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = (1st𝑢) → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘((1st𝑢) + 1))))
159 oveq1 5788 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (2nd𝑢) → (𝑤 + (𝐹‘((1st𝑢) + 1))) = ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))))
160 eqid 2140 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) = (𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))
161158, 159, 160ovmpog 5912 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑢) ∈ (ℤ𝑀) ∧ (2nd𝑢) ∈ 𝐶 ∧ ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))) ∈ 𝐶) → ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢)) = ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))))
16250, 52, 67, 161syl3anc 1217 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢)) = ((2nd𝑢) + (𝐹‘((1st𝑢) + 1))))
163162, 67eqeltrd 2217 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢)) ∈ 𝐶)
16455, 163opelxpd 4579 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩ ∈ ((ℤ𝑀) × 𝐶))
165 oveq1 5788 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = (1st𝑢) → (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦))
16669, 165opeq12d 3720 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (1st𝑢) → ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ = ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)
167 oveq2 5789 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (2nd𝑢) → ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢)))
168167opeq2d 3719 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (2nd𝑢) → ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ = ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩)
169 eqid 2140 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)
170166, 168, 169ovmpog 5912 . . . . . . . . . . . . . . . . . . 19 (((1st𝑢) ∈ (ℤ𝑀) ∧ (2nd𝑢) ∈ V ∧ ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩ ∈ ((ℤ𝑀) × 𝐶)) → ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd𝑢)) = ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩)
17150, 53, 164, 170syl3anc 1217 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((1st𝑢)(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd𝑢)) = ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩)
172156, 171eqtrd 2173 . . . . . . . . . . . . . . . . 17 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) = ⟨((1st𝑢) + 1), ((1st𝑢)(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd𝑢))⟩)
173172, 164eqeltrd 2217 . . . . . . . . . . . . . . . 16 ((𝜑𝑢 ∈ ((ℤ𝑀) × 𝐶)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝐶))
174173ralrimiva 2508 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑢 ∈ ((ℤ𝑀) × 𝐶)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝐶))
175174ad2antlr 481 . . . . . . . . . . . . . 14 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ∀𝑢 ∈ ((ℤ𝑀) × 𝐶)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝐶))
176 frecsuc 6311 . . . . . . . . . . . . . 14 ((∀𝑢 ∈ ((ℤ𝑀) × 𝐶)((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ𝑀) × 𝐶) ∧ ⟨𝑀, (𝐹𝑀)⟩ ∈ ((ℤ𝑀) × 𝐶) ∧ 𝑘 ∈ ω) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
177175, 143, 112, 176syl3anc 1217 . . . . . . . . . . . . 13 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)))
17841fveq1i 5429 . . . . . . . . . . . . 13 (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘)
17941fveq1i 5429 . . . . . . . . . . . . . 14 (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)
180179fveq2i 5431 . . . . . . . . . . . . 13 ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(𝑅𝑘)) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘))
181177, 178, 1803eqtr4g 2198 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(𝑅𝑘)))
182149fveq2d 5432 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(𝑅𝑘)) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩))
183181, 182eqtrd 2173 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩))
184 df-ov 5784 . . . . . . . . . . 11 ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘(𝑅𝑘))) = ((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st ‘(𝑅𝑘)), (2nd ‘(𝑅𝑘))⟩)
185183, 184eqtr4di 2191 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘(𝑅𝑘))))
186 fvoveq1 5804 . . . . . . . . . . . . . . . 16 (𝑧 = (1st ‘(𝑅𝑘)) → (𝐹‘(𝑧 + 1)) = (𝐹‘((1st ‘(𝑅𝑘)) + 1)))
187186oveq2d 5797 . . . . . . . . . . . . . . 15 (𝑧 = (1st ‘(𝑅𝑘)) → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
188 oveq1 5788 . . . . . . . . . . . . . . 15 (𝑤 = (2nd ‘(𝑅𝑘)) → (𝑤 + (𝐹‘((1st ‘(𝑅𝑘)) + 1))) = ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
189187, 188, 160ovmpog 5912 . . . . . . . . . . . . . 14 (((1st ‘(𝑅𝑘)) ∈ (ℤ𝑀) ∧ (2nd ‘(𝑅𝑘)) ∈ 𝐶 ∧ ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))) ∈ 𝐶) → ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘))) = ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
190115, 117, 132, 189syl3anc 1217 . . . . . . . . . . . . 13 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘))) = ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1))))
191190, 132eqeltrd 2217 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘))) ∈ 𝐶)
192120, 191opelxpd 4579 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩ ∈ ((ℤ𝑀) × 𝐶))
193 oveq1 5788 . . . . . . . . . . . . 13 (𝑥 = (1st ‘(𝑅𝑘)) → (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦))
194134, 193opeq12d 3720 . . . . . . . . . . . 12 (𝑥 = (1st ‘(𝑅𝑘)) → ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ = ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)
195 oveq2 5789 . . . . . . . . . . . . 13 (𝑦 = (2nd ‘(𝑅𝑘)) → ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘))))
196195opeq2d 3719 . . . . . . . . . . . 12 (𝑦 = (2nd ‘(𝑅𝑘)) → ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ = ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩)
197194, 196, 169ovmpog 5912 . . . . . . . . . . 11 (((1st ‘(𝑅𝑘)) ∈ (ℤ𝑀) ∧ (2nd ‘(𝑅𝑘)) ∈ V ∧ ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩ ∈ ((ℤ𝑀) × 𝐶)) → ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘(𝑅𝑘))) = ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩)
198115, 118, 192, 197syl3anc 1217 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅𝑘))(𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘(𝑅𝑘))) = ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩)
199190opeq2d 3719 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → ⟨((1st ‘(𝑅𝑘)) + 1), ((1st ‘(𝑅𝑘))(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅𝑘)))⟩ = ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
200185, 198, 1993eqtrd 2177 . . . . . . . . 9 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ⟨((1st ‘(𝑅𝑘)) + 1), ((2nd ‘(𝑅𝑘)) + (𝐹‘((1st ‘(𝑅𝑘)) + 1)))⟩)
201141, 153, 2003eqtr4rd 2184 . . . . . . . 8 (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))
202201exp31 362 . . . . . . 7 (𝑘 ∈ ω → (𝜑 → ((𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘) → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))))
203202a2d 26 . . . . . 6 (𝑘 ∈ ω → ((𝜑 → (𝑅𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑘)) → (𝜑 → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘suc 𝑘))))
20491, 95, 99, 103, 110, 203finds 4521 . . . . 5 (𝑛 ∈ ω → (𝜑 → (𝑅𝑛) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛)))
205204impcom 124 . . . 4 ((𝜑𝑛 ∈ ω) → (𝑅𝑛) = (frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)‘𝑛))
20643, 87, 205eqfnfvd 5528 . . 3 (𝜑𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩))
207206rneqd 4775 . 2 (𝜑 → ran 𝑅 = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩))
2081, 207eqtr4id 2192 1 (𝜑 → seq𝑀( + , 𝐹) = ran 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wcel 1481  wral 2417  Vcvv 2689  wss 3075  c0 3367  cop 3534  suc csuc 4294  ωcom 4511   × cxp 4544  ran crn 4547   Fn wfn 5125  wf 5126  cfv 5130  (class class class)co 5781  cmpo 5783  1st c1st 6043  2nd c2nd 6044  freccfrec 6294  1c1 7644   + caddc 7646  cz 9077  cuz 9349  seqcseq 10248
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-0id 7751  ax-rnegex 7752  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-ltadd 7759
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-inn 8744  df-n0 9001  df-z 9078  df-uz 9350  df-seqfrec 10249
This theorem is referenced by:  seqf2  10267  seq1cd  10268  seqp1cd  10269
  Copyright terms: Public domain W3C validator