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Theorem pitonnlem2 7363
Description: Lemma for pitonn 7364. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.)
Assertion
Ref Expression
pitonnlem2 (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⟩)
Distinct variable group:   𝐾,𝑙,𝑢

Proof of Theorem pitonnlem2
StepHypRef Expression
1 df-1 7337 . . . 4 1 = ⟨1R, 0R
21oveq2i 5645 . . 3 (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + ⟨1R, 0R⟩)
3 nnprlu 7091 . . . . . . . 8 (𝐾N → ⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
4 1pr 7092 . . . . . . . 8 1PP
5 addclpr 7075 . . . . . . . 8 ((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P ∧ 1PP) → (⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
63, 4, 5sylancl 404 . . . . . . 7 (𝐾N → (⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
7 opelxpi 4459 . . . . . . 7 (((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 1PP) → ⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P))
86, 4, 7sylancl 404 . . . . . 6 (𝐾N → ⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P))
9 enrex 7262 . . . . . . 7 ~R ∈ V
109ecelqsi 6326 . . . . . 6 (⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P) → [⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ ((P × P) / ~R ))
118, 10syl 14 . . . . 5 (𝐾N → [⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ ((P × P) / ~R ))
12 df-nr 7252 . . . . 5 R = ((P × P) / ~R )
1311, 12syl6eleqr 2181 . . . 4 (𝐾N → [⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR)
14 1sr 7276 . . . 4 1RR
15 addresr 7353 . . . 4 (([⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR ∧ 1RR) → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + ⟨1R, 0R⟩) = ⟨([⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R), 0R⟩)
1613, 14, 15sylancl 404 . . 3 (𝐾N → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + ⟨1R, 0R⟩) = ⟨([⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R), 0R⟩)
172, 16syl5eq 2132 . 2 (𝐾N → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = ⟨([⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R), 0R⟩)
18 pitonnlem1p1 7362 . . . . 5 ((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P → [⟨((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P), 1P⟩] ~R )
196, 18syl 14 . . . 4 (𝐾N → [⟨((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P), 1P⟩] ~R )
20 df-1r 7257 . . . . . 6 1R = [⟨(1P +P 1P), 1P⟩] ~R
2120oveq2i 5645 . . . . 5 ([⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R) = ([⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
22 addclpr 7075 . . . . . . . 8 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
234, 4, 22mp2an 417 . . . . . . 7 (1P +P 1P) ∈ P
24 addsrpr 7270 . . . . . . . 8 ((((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 1PP) ∧ ((1P +P 1P) ∈ P ∧ 1PP)) → ([⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
254, 24mpanl2 426 . . . . . . 7 (((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ ((1P +P 1P) ∈ P ∧ 1PP)) → ([⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
2623, 4, 25mpanr12 430 . . . . . 6 ((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P → ([⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
276, 26syl 14 . . . . 5 (𝐾N → ([⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
2821, 27syl5eq 2132 . . . 4 (𝐾N → ([⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R) = [⟨((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
29 addpinq1 7002 . . . . . . . . . . 11 (𝐾N → [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q = ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q))
3029breq2d 3849 . . . . . . . . . 10 (𝐾N → (𝑙 <Q [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q𝑙 <Q ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q)))
3130abbidv 2205 . . . . . . . . 9 (𝐾N → {𝑙𝑙 <Q [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q } = {𝑙𝑙 <Q ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q)})
3229breq1d 3847 . . . . . . . . . 10 (𝐾N → ([⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢 ↔ ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q) <Q 𝑢))
3332abbidv 2205 . . . . . . . . 9 (𝐾N → {𝑢 ∣ [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q) <Q 𝑢})
3431, 33opeq12d 3625 . . . . . . . 8 (𝐾N → ⟨{𝑙𝑙 <Q [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q)}, {𝑢 ∣ ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q) <Q 𝑢}⟩)
35 nnnq 6960 . . . . . . . . 9 (𝐾N → [⟨𝐾, 1𝑜⟩] ~QQ)
36 addnqpr1 7100 . . . . . . . . 9 ([⟨𝐾, 1𝑜⟩] ~QQ → ⟨{𝑙𝑙 <Q ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q)}, {𝑢 ∣ ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P))
3735, 36syl 14 . . . . . . . 8 (𝐾N → ⟨{𝑙𝑙 <Q ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q)}, {𝑢 ∣ ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P))
3834, 37eqtrd 2120 . . . . . . 7 (𝐾N → ⟨{𝑙𝑙 <Q [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P))
3938oveq1d 5649 . . . . . 6 (𝐾N → (⟨{𝑙𝑙 <Q [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) = ((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P))
4039opeq1d 3623 . . . . 5 (𝐾N → ⟨(⟨{𝑙𝑙 <Q [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P), 1P⟩)
4140eceq1d 6308 . . . 4 (𝐾N → [⟨(⟨{𝑙𝑙 <Q [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P), 1P⟩] ~R )
4219, 28, 413eqtr4d 2130 . . 3 (𝐾N → ([⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R) = [⟨(⟨{𝑙𝑙 <Q [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
4342opeq1d 3623 . 2 (𝐾N → ⟨([⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R), 0R⟩ = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
4417, 43eqtrd 2120 1 (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⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  {cab 2074  cop 3444   class class class wbr 3837   × cxp 4426  (class class class)co 5634  1𝑜c1o 6156  [cec 6270   / cqs 6271  Ncnpi 6810   +N cpli 6811   ~Q ceq 6817  Qcnq 6818  1Qc1q 6819   +Q cplq 6820   <Q cltq 6823  Pcnp 6829  1Pc1p 6830   +P cpp 6831   ~R cer 6834  Rcnr 6835  0Rc0r 6836  1Rc1r 6837   +R cplr 6839  1c1 7330   + caddc 7332
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-i1p 7005  df-iplp 7006  df-enr 7251  df-nr 7252  df-plr 7253  df-0r 7256  df-1r 7257  df-c 7335  df-1 7337  df-add 7340
This theorem is referenced by:  pitonn  7364  nntopi  7408
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