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Theorem pitonn 7847
Description: Mapping from N to . (Contributed by Jim Kingdon, 22-Apr-2020.)
Assertion
Ref Expression
pitonn (𝑁N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
Distinct variable groups:   𝑁,𝑙,𝑢   𝑦,𝑙,𝑢   𝑥,𝑦
Allowed substitution hints:   𝑁(𝑥,𝑦)

Proof of Theorem pitonn
Dummy variables 𝑤 𝑧 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3779 . . . . . . . . . . . . . . 15 (𝑤 = 1o → ⟨𝑤, 1o⟩ = ⟨1o, 1o⟩)
21eceq1d 6571 . . . . . . . . . . . . . 14 (𝑤 = 1o → [⟨𝑤, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
32breq2d 4016 . . . . . . . . . . . . 13 (𝑤 = 1o → (𝑙 <Q [⟨𝑤, 1o⟩] ~Q𝑙 <Q [⟨1o, 1o⟩] ~Q ))
43abbidv 2295 . . . . . . . . . . . 12 (𝑤 = 1o → {𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q } = {𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q })
52breq1d 4014 . . . . . . . . . . . . 13 (𝑤 = 1o → ([⟨𝑤, 1o⟩] ~Q <Q 𝑢 ↔ [⟨1o, 1o⟩] ~Q <Q 𝑢))
65abbidv 2295 . . . . . . . . . . . 12 (𝑤 = 1o → {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢})
74, 6opeq12d 3787 . . . . . . . . . . 11 (𝑤 = 1o → ⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩)
87oveq1d 5890 . . . . . . . . . 10 (𝑤 = 1o → (⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) = (⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
98opeq1d 3785 . . . . . . . . 9 (𝑤 = 1o → ⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩)
109eceq1d 6571 . . . . . . . 8 (𝑤 = 1o → [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
1110opeq1d 3785 . . . . . . 7 (𝑤 = 1o → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
1211eleq1d 2246 . . . . . 6 (𝑤 = 1o → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
1312imbi2d 230 . . . . 5 (𝑤 = 1o → (((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧) ↔ ((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)))
14 opeq1 3779 . . . . . . . . . . . . . . 15 (𝑤 = 𝑘 → ⟨𝑤, 1o⟩ = ⟨𝑘, 1o⟩)
1514eceq1d 6571 . . . . . . . . . . . . . 14 (𝑤 = 𝑘 → [⟨𝑤, 1o⟩] ~Q = [⟨𝑘, 1o⟩] ~Q )
1615breq2d 4016 . . . . . . . . . . . . 13 (𝑤 = 𝑘 → (𝑙 <Q [⟨𝑤, 1o⟩] ~Q𝑙 <Q [⟨𝑘, 1o⟩] ~Q ))
1716abbidv 2295 . . . . . . . . . . . 12 (𝑤 = 𝑘 → {𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q })
1815breq1d 4014 . . . . . . . . . . . . 13 (𝑤 = 𝑘 → ([⟨𝑤, 1o⟩] ~Q <Q 𝑢 ↔ [⟨𝑘, 1o⟩] ~Q <Q 𝑢))
1918abbidv 2295 . . . . . . . . . . . 12 (𝑤 = 𝑘 → {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢})
2017, 19opeq12d 3787 . . . . . . . . . . 11 (𝑤 = 𝑘 → ⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩)
2120oveq1d 5890 . . . . . . . . . 10 (𝑤 = 𝑘 → (⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) = (⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
2221opeq1d 3785 . . . . . . . . 9 (𝑤 = 𝑘 → ⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩)
2322eceq1d 6571 . . . . . . . 8 (𝑤 = 𝑘 → [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
2423opeq1d 3785 . . . . . . 7 (𝑤 = 𝑘 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
2524eleq1d 2246 . . . . . 6 (𝑤 = 𝑘 → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
2625imbi2d 230 . . . . 5 (𝑤 = 𝑘 → (((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧) ↔ ((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)))
27 opeq1 3779 . . . . . . . . . . . . . . 15 (𝑤 = (𝑘 +N 1o) → ⟨𝑤, 1o⟩ = ⟨(𝑘 +N 1o), 1o⟩)
2827eceq1d 6571 . . . . . . . . . . . . . 14 (𝑤 = (𝑘 +N 1o) → [⟨𝑤, 1o⟩] ~Q = [⟨(𝑘 +N 1o), 1o⟩] ~Q )
2928breq2d 4016 . . . . . . . . . . . . 13 (𝑤 = (𝑘 +N 1o) → (𝑙 <Q [⟨𝑤, 1o⟩] ~Q𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q ))
3029abbidv 2295 . . . . . . . . . . . 12 (𝑤 = (𝑘 +N 1o) → {𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q } = {𝑙𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q })
3128breq1d 4014 . . . . . . . . . . . . 13 (𝑤 = (𝑘 +N 1o) → ([⟨𝑤, 1o⟩] ~Q <Q 𝑢 ↔ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢))
3231abbidv 2295 . . . . . . . . . . . 12 (𝑤 = (𝑘 +N 1o) → {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢})
3330, 32opeq12d 3787 . . . . . . . . . . 11 (𝑤 = (𝑘 +N 1o) → ⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩)
3433oveq1d 5890 . . . . . . . . . 10 (𝑤 = (𝑘 +N 1o) → (⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) = (⟨{𝑙𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
3534opeq1d 3785 . . . . . . . . 9 (𝑤 = (𝑘 +N 1o) → ⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩)
3635eceq1d 6571 . . . . . . . 8 (𝑤 = (𝑘 +N 1o) → [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
3736opeq1d 3785 . . . . . . 7 (𝑤 = (𝑘 +N 1o) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
3837eleq1d 2246 . . . . . 6 (𝑤 = (𝑘 +N 1o) → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
3938imbi2d 230 . . . . 5 (𝑤 = (𝑘 +N 1o) → (((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧) ↔ ((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)))
40 opeq1 3779 . . . . . . . . . . . . . . 15 (𝑤 = 𝑁 → ⟨𝑤, 1o⟩ = ⟨𝑁, 1o⟩)
4140eceq1d 6571 . . . . . . . . . . . . . 14 (𝑤 = 𝑁 → [⟨𝑤, 1o⟩] ~Q = [⟨𝑁, 1o⟩] ~Q )
4241breq2d 4016 . . . . . . . . . . . . 13 (𝑤 = 𝑁 → (𝑙 <Q [⟨𝑤, 1o⟩] ~Q𝑙 <Q [⟨𝑁, 1o⟩] ~Q ))
4342abbidv 2295 . . . . . . . . . . . 12 (𝑤 = 𝑁 → {𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q })
4441breq1d 4014 . . . . . . . . . . . . 13 (𝑤 = 𝑁 → ([⟨𝑤, 1o⟩] ~Q <Q 𝑢 ↔ [⟨𝑁, 1o⟩] ~Q <Q 𝑢))
4544abbidv 2295 . . . . . . . . . . . 12 (𝑤 = 𝑁 → {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢})
4643, 45opeq12d 3787 . . . . . . . . . . 11 (𝑤 = 𝑁 → ⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩)
4746oveq1d 5890 . . . . . . . . . 10 (𝑤 = 𝑁 → (⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) = (⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
4847opeq1d 3785 . . . . . . . . 9 (𝑤 = 𝑁 → ⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩)
4948eceq1d 6571 . . . . . . . 8 (𝑤 = 𝑁 → [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
5049opeq1d 3785 . . . . . . 7 (𝑤 = 𝑁 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
5150eleq1d 2246 . . . . . 6 (𝑤 = 𝑁 → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
5251imbi2d 230 . . . . 5 (𝑤 = 𝑁 → (((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧) ↔ ((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)))
53 pitonnlem1 7844 . . . . . . . 8 ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
5453eleq1i 2243 . . . . . . 7 (⟨[⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ 1 ∈ 𝑧)
5554biimpri 133 . . . . . 6 (1 ∈ 𝑧 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)
5655adantr 276 . . . . 5 ((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)
57 oveq1 5882 . . . . . . . . . . 11 (𝑦 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (𝑦 + 1) = (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1))
5857eleq1d 2246 . . . . . . . . . 10 (𝑦 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝑦 + 1) ∈ 𝑧 ↔ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) ∈ 𝑧))
5958rspccv 2839 . . . . . . . . 9 (∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧 → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) ∈ 𝑧))
6059ad2antll 491 . . . . . . . 8 ((𝑘N ∧ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)) → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) ∈ 𝑧))
61 pitonnlem2 7846 . . . . . . . . . 10 (𝑘N → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
6261eleq1d 2246 . . . . . . . . 9 (𝑘N → ((⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) ∈ 𝑧 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
6362adantr 276 . . . . . . . 8 ((𝑘N ∧ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)) → ((⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) ∈ 𝑧 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
6460, 63sylibd 149 . . . . . . 7 ((𝑘N ∧ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)) → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
6564ex 115 . . . . . 6 (𝑘N → ((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)))
6665a2d 26 . . . . 5 (𝑘N → (((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧) → ((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)))
6713, 26, 39, 52, 56, 66indpi 7341 . . . 4 (𝑁N → ((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
6867alrimiv 1874 . . 3 (𝑁N → ∀𝑧((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
69 eleq2 2241 . . . . 5 (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧))
70 eleq2 2241 . . . . . 6 (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧))
7170raleqbi1dv 2681 . . . . 5 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧))
7269, 71anbi12d 473 . . . 4 (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)))
7372ralab 2898 . . 3 (∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ ∀𝑧((1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
7468, 73sylibr 134 . 2 (𝑁N → ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)
75 nnprlu 7552 . . . . . . 7 (𝑁N → ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
76 1pr 7553 . . . . . . 7 1PP
77 addclpr 7536 . . . . . . 7 ((⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P ∧ 1PP) → (⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
7875, 76, 77sylancl 413 . . . . . 6 (𝑁N → (⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
79 opelxpi 4659 . . . . . 6 (((⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 1PP) → ⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P))
8078, 76, 79sylancl 413 . . . . 5 (𝑁N → ⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P))
81 enrex 7736 . . . . . 6 ~R ∈ V
8281ecelqsi 6589 . . . . 5 (⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P) → [⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ ((P × P) / ~R ))
8380, 82syl 14 . . . 4 (𝑁N → [⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ ((P × P) / ~R ))
84 0r 7749 . . . 4 0RR
85 opelxpi 4659 . . . 4 (([⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ ((P × P) / ~R ) ∧ 0RR) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ (((P × P) / ~R ) × R))
8683, 84, 85sylancl 413 . . 3 (𝑁N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ (((P × P) / ~R ) × R))
87 elintg 3853 . . 3 (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ (((P × P) / ~R ) × R) → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
8886, 87syl 14 . 2 (𝑁N → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
8974, 88mpbird 167 1 (𝑁N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351   = wceq 1353  wcel 2148  {cab 2163  wral 2455  cop 3596   cint 3845   class class class wbr 4004   × cxp 4625  (class class class)co 5875  1oc1o 6410  [cec 6533   / cqs 6534  Ncnpi 7271   +N cpli 7272   ~Q ceq 7278   <Q cltq 7284  Pcnp 7290  1Pc1p 7291   +P cpp 7292   ~R cer 7295  Rcnr 7296  0Rc0r 7297  1c1 7812   + caddc 7814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-i1p 7466  df-iplp 7467  df-enr 7725  df-nr 7726  df-plr 7727  df-0r 7730  df-1r 7731  df-c 7817  df-1 7819  df-add 7822
This theorem is referenced by:  axarch  7890  axcaucvglemcl  7894  axcaucvglemval  7896  axcaucvglemcau  7897  axcaucvglemres  7898
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