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Theorem pitonn 6704
Description: Mapping from N to . (Contributed by Jim Kingdon, 22-Apr-2020.)
Assertion
Ref Expression
pitonn (𝑛 N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R {x ∣ (1 x y x (y + 1) x)})
Distinct variable group:   𝑛,𝑙,u,x,y

Proof of Theorem pitonn
Dummy variables 𝑘 w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3540 . . . . . . . . . . . . . . 15 (w = 1𝑜 → ⟨w, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
21eceq1d 6078 . . . . . . . . . . . . . 14 (w = 1𝑜 → [⟨w, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
32breq2d 3767 . . . . . . . . . . . . 13 (w = 1𝑜 → (𝑙 <Q [⟨w, 1𝑜⟩] ~Q𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q ))
43abbidv 2152 . . . . . . . . . . . 12 (w = 1𝑜 → {𝑙𝑙 <Q [⟨w, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q })
52breq1d 3765 . . . . . . . . . . . . 13 (w = 1𝑜 → ([⟨w, 1𝑜⟩] ~Q <Q u ↔ [⟨1𝑜, 1𝑜⟩] ~Q <Q u))
65abbidv 2152 . . . . . . . . . . . 12 (w = 1𝑜 → {u ∣ [⟨w, 1𝑜⟩] ~Q <Q u} = {u ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q u})
74, 6opeq12d 3548 . . . . . . . . . . 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 5470 . . . . . . . . . 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 3546 . . . . . . . . 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 6078 . . . . . . . 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 3546 . . . . . . 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 2103 . . . . . 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 219 . . . . 5 (w = 1𝑜 → (((1 z y z (y + 1) z) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨w, 1𝑜⟩] ~Q }, {u ∣ [⟨w, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z) ↔ ((1 z y z (y + 1) z) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {u ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z)))
14 opeq1 3540 . . . . . . . . . . . . . . 15 (w = 𝑘 → ⟨w, 1𝑜⟩ = ⟨𝑘, 1𝑜⟩)
1514eceq1d 6078 . . . . . . . . . . . . . 14 (w = 𝑘 → [⟨w, 1𝑜⟩] ~Q = [⟨𝑘, 1𝑜⟩] ~Q )
1615breq2d 3767 . . . . . . . . . . . . 13 (w = 𝑘 → (𝑙 <Q [⟨w, 1𝑜⟩] ~Q𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q ))
1716abbidv 2152 . . . . . . . . . . . 12 (w = 𝑘 → {𝑙𝑙 <Q [⟨w, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q })
1815breq1d 3765 . . . . . . . . . . . . 13 (w = 𝑘 → ([⟨w, 1𝑜⟩] ~Q <Q u ↔ [⟨𝑘, 1𝑜⟩] ~Q <Q u))
1918abbidv 2152 . . . . . . . . . . . 12 (w = 𝑘 → {u ∣ [⟨w, 1𝑜⟩] ~Q <Q u} = {u ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q u})
2017, 19opeq12d 3548 . . . . . . . . . . 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 5470 . . . . . . . . . 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 3546 . . . . . . . . 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 6078 . . . . . . . 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 3546 . . . . . . 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 2103 . . . . . 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 219 . . . . 5 (w = 𝑘 → (((1 z y z (y + 1) z) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨w, 1𝑜⟩] ~Q }, {u ∣ [⟨w, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z) ↔ ((1 z y z (y + 1) z) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z)))
27 opeq1 3540 . . . . . . . . . . . . . . 15 (w = (𝑘 +N 1𝑜) → ⟨w, 1𝑜⟩ = ⟨(𝑘 +N 1𝑜), 1𝑜⟩)
2827eceq1d 6078 . . . . . . . . . . . . . 14 (w = (𝑘 +N 1𝑜) → [⟨w, 1𝑜⟩] ~Q = [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q )
2928breq2d 3767 . . . . . . . . . . . . 13 (w = (𝑘 +N 1𝑜) → (𝑙 <Q [⟨w, 1𝑜⟩] ~Q𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q ))
3029abbidv 2152 . . . . . . . . . . . 12 (w = (𝑘 +N 1𝑜) → {𝑙𝑙 <Q [⟨w, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q })
3128breq1d 3765 . . . . . . . . . . . . 13 (w = (𝑘 +N 1𝑜) → ([⟨w, 1𝑜⟩] ~Q <Q u ↔ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q u))
3231abbidv 2152 . . . . . . . . . . . 12 (w = (𝑘 +N 1𝑜) → {u ∣ [⟨w, 1𝑜⟩] ~Q <Q u} = {u ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q u})
3330, 32opeq12d 3548 . . . . . . . . . . 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 5470 . . . . . . . . . 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 3546 . . . . . . . . 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 6078 . . . . . . . 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 3546 . . . . . . 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 2103 . . . . . 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 219 . . . . 5 (w = (𝑘 +N 1𝑜) → (((1 z y z (y + 1) z) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨w, 1𝑜⟩] ~Q }, {u ∣ [⟨w, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z) ↔ ((1 z y z (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 3540 . . . . . . . . . . . . . . 15 (w = 𝑛 → ⟨w, 1𝑜⟩ = ⟨𝑛, 1𝑜⟩)
4140eceq1d 6078 . . . . . . . . . . . . . 14 (w = 𝑛 → [⟨w, 1𝑜⟩] ~Q = [⟨𝑛, 1𝑜⟩] ~Q )
4241breq2d 3767 . . . . . . . . . . . . 13 (w = 𝑛 → (𝑙 <Q [⟨w, 1𝑜⟩] ~Q𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q ))
4342abbidv 2152 . . . . . . . . . . . 12 (w = 𝑛 → {𝑙𝑙 <Q [⟨w, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q })
4441breq1d 3765 . . . . . . . . . . . . 13 (w = 𝑛 → ([⟨w, 1𝑜⟩] ~Q <Q u ↔ [⟨𝑛, 1𝑜⟩] ~Q <Q u))
4544abbidv 2152 . . . . . . . . . . . 12 (w = 𝑛 → {u ∣ [⟨w, 1𝑜⟩] ~Q <Q u} = {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u})
4643, 45opeq12d 3548 . . . . . . . . . . 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 5470 . . . . . . . . . 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 3546 . . . . . . . . 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 6078 . . . . . . . 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 3546 . . . . . . 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 2103 . . . . . 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 219 . . . . 5 (w = 𝑛 → (((1 z y z (y + 1) z) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨w, 1𝑜⟩] ~Q }, {u ∣ [⟨w, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z) ↔ ((1 z y z (y + 1) z) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z)))
53 pitonnlem1 6701 . . . . . . . 8 ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {u ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
5453eleq1i 2100 . . . . . . 7 (⟨[⟨(⟨{𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {u ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z ↔ 1 z)
5554biimpri 124 . . . . . 6 (1 z → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {u ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z)
5655adantr 261 . . . . 5 ((1 z y z (y + 1) z) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {u ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z)
57 oveq1 5462 . . . . . . . . . . 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 2103 . . . . . . . . . 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 2647 . . . . . . . . 9 (y z (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 460 . . . . . . . 8 ((𝑘 N (1 z y z (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 6703 . . . . . . . . . 10 (𝑘 N → (⟨[⟨(⟨{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 2103 . . . . . . . . 9 (𝑘 N → ((⟨[⟨(⟨{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 261 . . . . . . . 8 ((𝑘 N (1 z y z (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 138 . . . . . . 7 ((𝑘 N (1 z y z (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 108 . . . . . 6 (𝑘 N → ((1 z y z (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 23 . . . . 5 (𝑘 N → (((1 z y z (y + 1) z) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z) → ((1 z y z (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 6326 . . . 4 (𝑛 N → ((1 z y z (y + 1) z) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z))
6867alrimiv 1751 . . 3 (𝑛 Nz((1 z y z (y + 1) z) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z))
69 eleq2 2098 . . . . 5 (x = z → (1 x ↔ 1 z))
70 eleq2 2098 . . . . . 6 (x = z → ((y + 1) x ↔ (y + 1) z))
7170raleqbi1dv 2507 . . . . 5 (x = z → (y x (y + 1) xy z (y + 1) z))
7269, 71anbi12d 442 . . . 4 (x = z → ((1 x y x (y + 1) x) ↔ (1 z y z (y + 1) z)))
7372ralab 2695 . . 3 (z {x ∣ (1 x y x (y + 1) x)}⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R zz((1 z y z (y + 1) z) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z))
7468, 73sylibr 137 . 2 (𝑛 Nz {x ∣ (1 x y x (y + 1) x)}⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z)
75 nnprlu 6533 . . . . . . 7 (𝑛 N → ⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ P)
76 1pr 6534 . . . . . . 7 1P P
77 addclpr 6520 . . . . . . 7 ((⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ P 1P P) → (⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) P)
7875, 76, 77sylancl 392 . . . . . 6 (𝑛 N → (⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) P)
79 opelxpi 4319 . . . . . 6 (((⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) P 1P P) → ⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P (P × P))
8078, 76, 79sylancl 392 . . . . 5 (𝑛 N → ⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P (P × P))
81 enrex 6625 . . . . . 6 ~R V
8281ecelqsi 6096 . . . . 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 (𝑛 N → [⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R ((P × P) / ~R ))
84 0r 6638 . . . 4 0R R
85 opelxpi 4319 . . . 4 (([⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R ((P × P) / ~R ) 0R R) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R (((P × P) / ~R ) × R))
8683, 84, 85sylancl 392 . . 3 (𝑛 N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R (((P × P) / ~R ) × R))
87 elintg 3614 . . 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 y x (y + 1) x)} ↔ z {x ∣ (1 x y x (y + 1) x)}⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z))
8886, 87syl 14 . 2 (𝑛 N → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R {x ∣ (1 x y x (y + 1) x)} ↔ z {x ∣ (1 x y x (y + 1) x)}⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R z))
8974, 88mpbird 156 1 (𝑛 N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {u ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R , 0R {x ∣ (1 x y x (y + 1) x)})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   wcel 1390  {cab 2023  wral 2300  cop 3370   cint 3606   class class class wbr 3755   × cxp 4286  (class class class)co 5455  1𝑜c1o 5933  [cec 6040   / cqs 6041  Ncnpi 6256   +N cpli 6257   ~Q ceq 6263   <Q cltq 6269  Pcnp 6275  1Pc1p 6276   +P cpp 6277   ~R cer 6280  Rcnr 6281  0Rc0r 6282  1c1 6672   + caddc 6674
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-i1p 6449  df-iplp 6450  df-enr 6614  df-nr 6615  df-plr 6616  df-0r 6619  df-1r 6620  df-c 6677  df-1 6679  df-add 6682
This theorem is referenced by:  axarch  6733
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