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Theorem pitonn 7130
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 3590 . . . . . . . . . . . . . . 15 (𝑤 = 1𝑜 → ⟨𝑤, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
21eceq1d 6229 . . . . . . . . . . . . . 14 (𝑤 = 1𝑜 → [⟨𝑤, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
32breq2d 3817 . . . . . . . . . . . . 13 (𝑤 = 1𝑜 → (𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q ))
43abbidv 2200 . . . . . . . . . . . 12 (𝑤 = 1𝑜 → {𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q })
52breq1d 3815 . . . . . . . . . . . . 13 (𝑤 = 1𝑜 → ([⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢))
65abbidv 2200 . . . . . . . . . . . 12 (𝑤 = 1𝑜 → {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢})
74, 6opeq12d 3598 . . . . . . . . . . 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 5578 . . . . . . . . . 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 3596 . . . . . . . . 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 6229 . . . . . . . 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 3596 . . . . . . 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 2151 . . . . . 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 228 . . . . 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 3590 . . . . . . . . . . . . . . 15 (𝑤 = 𝑘 → ⟨𝑤, 1𝑜⟩ = ⟨𝑘, 1𝑜⟩)
1514eceq1d 6229 . . . . . . . . . . . . . 14 (𝑤 = 𝑘 → [⟨𝑤, 1𝑜⟩] ~Q = [⟨𝑘, 1𝑜⟩] ~Q )
1615breq2d 3817 . . . . . . . . . . . . 13 (𝑤 = 𝑘 → (𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q ))
1716abbidv 2200 . . . . . . . . . . . 12 (𝑤 = 𝑘 → {𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q })
1815breq1d 3815 . . . . . . . . . . . . 13 (𝑤 = 𝑘 → ([⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢))
1918abbidv 2200 . . . . . . . . . . . 12 (𝑤 = 𝑘 → {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢})
2017, 19opeq12d 3598 . . . . . . . . . . 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 5578 . . . . . . . . . 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 3596 . . . . . . . . 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 6229 . . . . . . . 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 3596 . . . . . . 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 2151 . . . . . 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 228 . . . . 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 3590 . . . . . . . . . . . . . . 15 (𝑤 = (𝑘 +N 1𝑜) → ⟨𝑤, 1𝑜⟩ = ⟨(𝑘 +N 1𝑜), 1𝑜⟩)
2827eceq1d 6229 . . . . . . . . . . . . . 14 (𝑤 = (𝑘 +N 1𝑜) → [⟨𝑤, 1𝑜⟩] ~Q = [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q )
2928breq2d 3817 . . . . . . . . . . . . 13 (𝑤 = (𝑘 +N 1𝑜) → (𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q ))
3029abbidv 2200 . . . . . . . . . . . 12 (𝑤 = (𝑘 +N 1𝑜) → {𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q })
3128breq1d 3815 . . . . . . . . . . . . 13 (𝑤 = (𝑘 +N 1𝑜) → ([⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢))
3231abbidv 2200 . . . . . . . . . . . 12 (𝑤 = (𝑘 +N 1𝑜) → {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢})
3330, 32opeq12d 3598 . . . . . . . . . . 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 5578 . . . . . . . . . 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 3596 . . . . . . . . 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 6229 . . . . . . . 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 3596 . . . . . . 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 2151 . . . . . 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 228 . . . . 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 3590 . . . . . . . . . . . . . . 15 (𝑤 = 𝑁 → ⟨𝑤, 1𝑜⟩ = ⟨𝑁, 1𝑜⟩)
4140eceq1d 6229 . . . . . . . . . . . . . 14 (𝑤 = 𝑁 → [⟨𝑤, 1𝑜⟩] ~Q = [⟨𝑁, 1𝑜⟩] ~Q )
4241breq2d 3817 . . . . . . . . . . . . 13 (𝑤 = 𝑁 → (𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q ))
4342abbidv 2200 . . . . . . . . . . . 12 (𝑤 = 𝑁 → {𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q })
4441breq1d 3815 . . . . . . . . . . . . 13 (𝑤 = 𝑁 → ([⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢))
4544abbidv 2200 . . . . . . . . . . . 12 (𝑤 = 𝑁 → {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢})
4643, 45opeq12d 3598 . . . . . . . . . . 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 5578 . . . . . . . . . 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 3596 . . . . . . . . 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 6229 . . . . . . . 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 3596 . . . . . . 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 2151 . . . . . 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 228 . . . . 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 7127 . . . . . . . 8 ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
5453eleq1i 2148 . . . . . . 7 (⟨[⟨(⟨{𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ 1 ∈ 𝑧)
5554biimpri 131 . . . . . 6 (1 ∈ 𝑧 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)
5655adantr 270 . . . . 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 5570 . . . . . . . . . . 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 2151 . . . . . . . . . 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 2707 . . . . . . . . 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 475 . . . . . . . 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 7129 . . . . . . . . . 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 2151 . . . . . . . . 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 270 . . . . . . . 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 147 . . . . . . 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 113 . . . . . 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 6646 . . . 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 1797 . . 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 2146 . . . . 5 (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧))
70 eleq2 2146 . . . . . 6 (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧))
7170raleqbi1dv 2562 . . . . 5 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧))
7269, 71anbi12d 457 . . . 4 (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)))
7372ralab 2761 . . 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 132 . 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 6857 . . . . . . 7 (𝑁N → ⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
76 1pr 6858 . . . . . . 7 1PP
77 addclpr 6841 . . . . . . 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 404 . . . . . 6 (𝑁N → (⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
79 opelxpi 4422 . . . . . 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 404 . . . . 5 (𝑁N → ⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P))
81 enrex 7028 . . . . . 6 ~R ∈ V
8281ecelqsi 6247 . . . . 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 7041 . . . 4 0RR
85 opelxpi 4422 . . . 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 404 . . 3 (𝑁N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ (((P × P) / ~R ) × R))
87 elintg 3664 . . 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 165 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 102  wb 103  wal 1283   = wceq 1285  wcel 1434  {cab 2069  wral 2353  cop 3419   cint 3656   class class class wbr 3805   × cxp 4389  (class class class)co 5563  1𝑜c1o 6078  [cec 6191   / cqs 6192  Ncnpi 6576   +N cpli 6577   ~Q ceq 6583   <Q cltq 6589  Pcnp 6595  1Pc1p 6596   +P cpp 6597   ~R cer 6600  Rcnr 6601  0Rc0r 6602  1c1 7096   + caddc 7098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-1o 6085  df-2o 6086  df-oadd 6089  df-omul 6090  df-er 6193  df-ec 6195  df-qs 6199  df-ni 6608  df-pli 6609  df-mi 6610  df-lti 6611  df-plpq 6648  df-mpq 6649  df-enq 6651  df-nqqs 6652  df-plqqs 6653  df-mqqs 6654  df-1nqqs 6655  df-rq 6656  df-ltnqqs 6657  df-enq0 6728  df-nq0 6729  df-0nq0 6730  df-plq0 6731  df-mq0 6732  df-inp 6770  df-i1p 6771  df-iplp 6772  df-enr 7017  df-nr 7018  df-plr 7019  df-0r 7022  df-1r 7023  df-c 7101  df-1 7103  df-add 7106
This theorem is referenced by:  axarch  7171  axcaucvglemcl  7175  axcaucvglemval  7177  axcaucvglemcau  7178  axcaucvglemres  7179
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