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Theorem pitonnlem2 6703
Description: Lemma for pitonn 6704. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.)
Assertion
Ref Expression
pitonnlem2 (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⟩)
Distinct variable group:   𝐾,𝑙,u

Proof of Theorem pitonnlem2
StepHypRef Expression
1 df-1 6679 . . . 4 1 = ⟨1R, 0R
21oveq2i 5466 . . 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 6533 . . . . . . . 8 (𝐾 N → ⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {u ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q u}⟩ P)
4 1pr 6534 . . . . . . . 8 1P P
5 addclpr 6520 . . . . . . . 8 ((⟨{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)
63, 4, 5sylancl 392 . . . . . . 7 (𝐾 N → (⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {u ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) P)
7 opelxpi 4319 . . . . . . 7 (((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {u ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) P 1P P) → ⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {u ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P (P × P))
86, 4, 7sylancl 392 . . . . . 6 (𝐾 N → ⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {u ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P (P × P))
9 enrex 6625 . . . . . . 7 ~R V
109ecelqsi 6096 . . . . . 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 (𝐾 N → [⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {u ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R ((P × P) / ~R ))
12 df-nr 6615 . . . . 5 R = ((P × P) / ~R )
1311, 12syl6eleqr 2128 . . . 4 (𝐾 N → [⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {u ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R R)
14 1sr 6639 . . . 4 1R R
15 addresr 6694 . . . 4 (([⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {u ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R R 1R R) → (⟨[⟨(⟨{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 392 . . 3 (𝐾 N → (⟨[⟨(⟨{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 2081 . 2 (𝐾 N → (⟨[⟨(⟨{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 6702 . . . . 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 (𝐾 N → [⟨((⟨{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 6620 . . . . . 6 1R = [⟨(1P +P 1P), 1P⟩] ~R
2120oveq2i 5466 . . . . 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 6520 . . . . . . . 8 ((1P P 1P P) → (1P +P 1P) P)
234, 4, 22mp2an 402 . . . . . . 7 (1P +P 1P) P
24 addsrpr 6633 . . . . . . . 8 ((((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {u ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) P 1P P) ((1P +P 1P) 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 )
254, 24mpanl2 411 . . . . . . 7 (((⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {u ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q u}⟩ +P 1P) P ((1P +P 1P) 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 )
2623, 4, 25mpanr12 415 . . . . . 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 (𝐾 N → ([⟨(⟨{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 2081 . . . 4 (𝐾 N → ([⟨(⟨{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 6446 . . . . . . . . . . 11 (𝐾 N → [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q = ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q))
3029breq2d 3767 . . . . . . . . . 10 (𝐾 N → (𝑙 <Q [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q𝑙 <Q ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q)))
3130abbidv 2152 . . . . . . . . 9 (𝐾 N → {𝑙𝑙 <Q [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q } = {𝑙𝑙 <Q ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q)})
3229breq1d 3765 . . . . . . . . . 10 (𝐾 N → ([⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q <Q u ↔ ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q) <Q u))
3332abbidv 2152 . . . . . . . . 9 (𝐾 N → {u ∣ [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q <Q u} = {u ∣ ([⟨𝐾, 1𝑜⟩] ~Q +Q 1Q) <Q u})
3431, 33opeq12d 3548 . . . . . . . 8 (𝐾 N → ⟨{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 6405 . . . . . . . . 9 (𝐾 N → [⟨𝐾, 1𝑜⟩] ~Q Q)
36 addnqpr1 6541 . . . . . . . . 9 ([⟨𝐾, 1𝑜⟩] ~Q Q → ⟨{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 (𝐾 N → ⟨{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 2069 . . . . . . 7 (𝐾 N → ⟨{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 5470 . . . . . 6 (𝐾 N → (⟨{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 3546 . . . . 5 (𝐾 N → ⟨(⟨{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 6078 . . . 4 (𝐾 N → [⟨(⟨{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 2079 . . 3 (𝐾 N → ([⟨(⟨{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 3546 . 2 (𝐾 N → ⟨([⟨(⟨{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 2069 1 (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⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  {cab 2023  cop 3370   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  Qcnq 6264  1Qc1q 6265   +Q cplq 6266   <Q cltq 6269  Pcnp 6275  1Pc1p 6276   +P cpp 6277   ~R cer 6280  Rcnr 6281  0Rc0r 6282  1Rc1r 6283   +R cplr 6285  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:  pitonn  6704
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