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Theorem pitonn 6981
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 3576 . . . . . . . . . . . . . . 15 (𝑤 = 1𝑜 → ⟨𝑤, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
21eceq1d 6172 . . . . . . . . . . . . . 14 (𝑤 = 1𝑜 → [⟨𝑤, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
32breq2d 3803 . . . . . . . . . . . . 13 (𝑤 = 1𝑜 → (𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q ))
43abbidv 2171 . . . . . . . . . . . 12 (𝑤 = 1𝑜 → {𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q })
52breq1d 3801 . . . . . . . . . . . . 13 (𝑤 = 1𝑜 → ([⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢))
65abbidv 2171 . . . . . . . . . . . 12 (𝑤 = 1𝑜 → {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢})
74, 6opeq12d 3584 . . . . . . . . . . 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 5554 . . . . . . . . . 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 3582 . . . . . . . . 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 6172 . . . . . . . 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 3582 . . . . . . 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 2122 . . . . . 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 223 . . . . 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 3576 . . . . . . . . . . . . . . 15 (𝑤 = 𝑘 → ⟨𝑤, 1𝑜⟩ = ⟨𝑘, 1𝑜⟩)
1514eceq1d 6172 . . . . . . . . . . . . . 14 (𝑤 = 𝑘 → [⟨𝑤, 1𝑜⟩] ~Q = [⟨𝑘, 1𝑜⟩] ~Q )
1615breq2d 3803 . . . . . . . . . . . . 13 (𝑤 = 𝑘 → (𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q ))
1716abbidv 2171 . . . . . . . . . . . 12 (𝑤 = 𝑘 → {𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q })
1815breq1d 3801 . . . . . . . . . . . . 13 (𝑤 = 𝑘 → ([⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢))
1918abbidv 2171 . . . . . . . . . . . 12 (𝑤 = 𝑘 → {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢})
2017, 19opeq12d 3584 . . . . . . . . . . 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 5554 . . . . . . . . . 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 3582 . . . . . . . . 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 6172 . . . . . . . 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 3582 . . . . . . 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 2122 . . . . . 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 223 . . . . 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 3576 . . . . . . . . . . . . . . 15 (𝑤 = (𝑘 +N 1𝑜) → ⟨𝑤, 1𝑜⟩ = ⟨(𝑘 +N 1𝑜), 1𝑜⟩)
2827eceq1d 6172 . . . . . . . . . . . . . 14 (𝑤 = (𝑘 +N 1𝑜) → [⟨𝑤, 1𝑜⟩] ~Q = [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q )
2928breq2d 3803 . . . . . . . . . . . . 13 (𝑤 = (𝑘 +N 1𝑜) → (𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q ))
3029abbidv 2171 . . . . . . . . . . . 12 (𝑤 = (𝑘 +N 1𝑜) → {𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q })
3128breq1d 3801 . . . . . . . . . . . . 13 (𝑤 = (𝑘 +N 1𝑜) → ([⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢))
3231abbidv 2171 . . . . . . . . . . . 12 (𝑤 = (𝑘 +N 1𝑜) → {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢})
3330, 32opeq12d 3584 . . . . . . . . . . 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 5554 . . . . . . . . . 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 3582 . . . . . . . . 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 6172 . . . . . . . 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 3582 . . . . . . 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 2122 . . . . . 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 223 . . . . 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 3576 . . . . . . . . . . . . . . 15 (𝑤 = 𝑁 → ⟨𝑤, 1𝑜⟩ = ⟨𝑁, 1𝑜⟩)
4140eceq1d 6172 . . . . . . . . . . . . . 14 (𝑤 = 𝑁 → [⟨𝑤, 1𝑜⟩] ~Q = [⟨𝑁, 1𝑜⟩] ~Q )
4241breq2d 3803 . . . . . . . . . . . . 13 (𝑤 = 𝑁 → (𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q ))
4342abbidv 2171 . . . . . . . . . . . 12 (𝑤 = 𝑁 → {𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q })
4441breq1d 3801 . . . . . . . . . . . . 13 (𝑤 = 𝑁 → ([⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢))
4544abbidv 2171 . . . . . . . . . . . 12 (𝑤 = 𝑁 → {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢})
4643, 45opeq12d 3584 . . . . . . . . . . 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 5554 . . . . . . . . . 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 3582 . . . . . . . . 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 6172 . . . . . . . 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 3582 . . . . . . 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 2122 . . . . . 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 223 . . . . 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 6978 . . . . . . . 8 ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
5453eleq1i 2119 . . . . . . 7 (⟨[⟨(⟨{𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ 1 ∈ 𝑧)
5554biimpri 128 . . . . . 6 (1 ∈ 𝑧 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)
5655adantr 265 . . . . 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 5546 . . . . . . . . . . 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 2122 . . . . . . . . . 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 2670 . . . . . . . . 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 468 . . . . . . . 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 6980 . . . . . . . . . 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 2122 . . . . . . . . 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 265 . . . . . . . 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 142 . . . . . . 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 112 . . . . . 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 6497 . . . 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 1770 . . 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 2117 . . . . 5 (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧))
70 eleq2 2117 . . . . . 6 (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧))
7170raleqbi1dv 2530 . . . . 5 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧))
7269, 71anbi12d 450 . . . 4 (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)))
7372ralab 2723 . . 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 141 . 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 6708 . . . . . . 7 (𝑁N → ⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
76 1pr 6709 . . . . . . 7 1PP
77 addclpr 6692 . . . . . . 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 398 . . . . . 6 (𝑁N → (⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
79 opelxpi 4403 . . . . . 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 398 . . . . 5 (𝑁N → ⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P))
81 enrex 6879 . . . . . 6 ~R ∈ V
8281ecelqsi 6190 . . . . 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 6892 . . . 4 0RR
85 opelxpi 4403 . . . 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 398 . . 3 (𝑁N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ (((P × P) / ~R ) × R))
87 elintg 3650 . . 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 160 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 101  wb 102  wal 1257   = wceq 1259  wcel 1409  {cab 2042  wral 2323  cop 3405   cint 3642   class class class wbr 3791   × cxp 4370  (class class class)co 5539  1𝑜c1o 6024  [cec 6134   / cqs 6135  Ncnpi 6427   +N cpli 6428   ~Q ceq 6434   <Q cltq 6440  Pcnp 6446  1Pc1p 6447   +P cpp 6448   ~R cer 6451  Rcnr 6452  0Rc0r 6453  1c1 6947   + caddc 6949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-i1p 6622  df-iplp 6623  df-enr 6868  df-nr 6869  df-plr 6870  df-0r 6873  df-1r 6874  df-c 6952  df-1 6954  df-add 6957
This theorem is referenced by:  axarch  7022  axcaucvglemcl  7026  axcaucvglemval  7028  axcaucvglemcau  7029  axcaucvglemres  7030
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