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Theorem prarloclem5 6598
 Description: A substitution of zero for 𝑦 and 𝑁 minus two for 𝑥. Lemma for prarloc 6601. (Contributed by Jim Kingdon, 4-Nov-2019.)
Assertion
Ref Expression
prarloclem5 (((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 1𝑜 <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐿,𝑦   𝑥,𝑁   𝑥,𝑃,𝑦   𝑥,𝑈,𝑦
Allowed substitution hint:   𝑁(𝑦)

Proof of Theorem prarloclem5
StepHypRef Expression
1 prarloclemn 6597 . . . 4 ((𝑁N ∧ 1𝑜 <N 𝑁) → ∃𝑥 ∈ ω (2𝑜 +𝑜 𝑥) = 𝑁)
213adant2 923 . . 3 ((𝑁N𝑃Q ∧ 1𝑜 <N 𝑁) → ∃𝑥 ∈ ω (2𝑜 +𝑜 𝑥) = 𝑁)
323ad2ant2 926 . 2 (((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 1𝑜 <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑥 ∈ ω (2𝑜 +𝑜 𝑥) = 𝑁)
4 elprnql 6579 . . . . . . 7 ((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) → 𝐴Q)
543ad2ant1 925 . . . . . 6 (((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 1𝑜 <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → 𝐴Q)
6 simp22 938 . . . . . 6 (((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 1𝑜 <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → 𝑃Q)
7 nqnq0 6539 . . . . . . . . 9 QQ0
87sseli 2941 . . . . . . . 8 (𝐴Q𝐴Q0)
9 nq0a0 6555 . . . . . . . 8 (𝐴Q0 → (𝐴 +Q0 0Q0) = 𝐴)
108, 9syl 14 . . . . . . 7 (𝐴Q → (𝐴 +Q0 0Q0) = 𝐴)
11 df-0nq0 6524 . . . . . . . . . 10 0Q0 = [⟨∅, 1𝑜⟩] ~Q0
1211oveq1i 5522 . . . . . . . . 9 (0Q0 ·Q0 𝑃) = ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)
137sseli 2941 . . . . . . . . . 10 (𝑃Q𝑃Q0)
14 nq0m0r 6554 . . . . . . . . . 10 (𝑃Q0 → (0Q0 ·Q0 𝑃) = 0Q0)
1513, 14syl 14 . . . . . . . . 9 (𝑃Q → (0Q0 ·Q0 𝑃) = 0Q0)
1612, 15syl5reqr 2087 . . . . . . . 8 (𝑃Q → 0Q0 = ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃))
1716oveq2d 5528 . . . . . . 7 (𝑃Q → (𝐴 +Q0 0Q0) = (𝐴 +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)))
1810, 17sylan9req 2093 . . . . . 6 ((𝐴Q𝑃Q) → 𝐴 = (𝐴 +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)))
195, 6, 18syl2anc 391 . . . . 5 (((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 1𝑜 <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → 𝐴 = (𝐴 +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)))
20 simp1r 929 . . . . 5 (((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 1𝑜 <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → 𝐴𝐿)
2119, 20eqeltrrd 2115 . . . 4 (((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 1𝑜 <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → (𝐴 +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿)
22 2onn 6094 . . . . . . . . . . . . . . 15 2𝑜 ∈ ω
23 nna0r 6057 . . . . . . . . . . . . . . 15 (2𝑜 ∈ ω → (∅ +𝑜 2𝑜) = 2𝑜)
2422, 23ax-mp 7 . . . . . . . . . . . . . 14 (∅ +𝑜 2𝑜) = 2𝑜
2524oveq1i 5522 . . . . . . . . . . . . 13 ((∅ +𝑜 2𝑜) +𝑜 𝑥) = (2𝑜 +𝑜 𝑥)
2625eqeq1i 2047 . . . . . . . . . . . 12 (((∅ +𝑜 2𝑜) +𝑜 𝑥) = 𝑁 ↔ (2𝑜 +𝑜 𝑥) = 𝑁)
2726biimpri 124 . . . . . . . . . . 11 ((2𝑜 +𝑜 𝑥) = 𝑁 → ((∅ +𝑜 2𝑜) +𝑜 𝑥) = 𝑁)
2827opeq1d 3555 . . . . . . . . . 10 ((2𝑜 +𝑜 𝑥) = 𝑁 → ⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩ = ⟨𝑁, 1𝑜⟩)
2928eceq1d 6142 . . . . . . . . 9 ((2𝑜 +𝑜 𝑥) = 𝑁 → [⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q = [⟨𝑁, 1𝑜⟩] ~Q )
3029oveq1d 5527 . . . . . . . 8 ((2𝑜 +𝑜 𝑥) = 𝑁 → ([⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃) = ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃))
3130oveq2d 5528 . . . . . . 7 ((2𝑜 +𝑜 𝑥) = 𝑁 → (𝐴 +Q ([⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) = (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)))
3231eleq1d 2106 . . . . . 6 ((2𝑜 +𝑜 𝑥) = 𝑁 → ((𝐴 +Q ([⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈 ↔ (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))
3332biimprcd 149 . . . . 5 ((𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈 → ((2𝑜 +𝑜 𝑥) = 𝑁 → (𝐴 +Q ([⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))
34333ad2ant3 927 . . . 4 (((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 1𝑜 <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ((2𝑜 +𝑜 𝑥) = 𝑁 → (𝐴 +Q ([⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))
35 peano1 4317 . . . . 5 ∅ ∈ ω
36 opeq1 3549 . . . . . . . . . . 11 (𝑦 = ∅ → ⟨𝑦, 1𝑜⟩ = ⟨∅, 1𝑜⟩)
3736eceq1d 6142 . . . . . . . . . 10 (𝑦 = ∅ → [⟨𝑦, 1𝑜⟩] ~Q0 = [⟨∅, 1𝑜⟩] ~Q0 )
3837oveq1d 5527 . . . . . . . . 9 (𝑦 = ∅ → ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃) = ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃))
3938oveq2d 5528 . . . . . . . 8 (𝑦 = ∅ → (𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) = (𝐴 +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)))
4039eleq1d 2106 . . . . . . 7 (𝑦 = ∅ → ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ↔ (𝐴 +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿))
41 oveq1 5519 . . . . . . . . . . . . 13 (𝑦 = ∅ → (𝑦 +𝑜 2𝑜) = (∅ +𝑜 2𝑜))
4241oveq1d 5527 . . . . . . . . . . . 12 (𝑦 = ∅ → ((𝑦 +𝑜 2𝑜) +𝑜 𝑥) = ((∅ +𝑜 2𝑜) +𝑜 𝑥))
4342opeq1d 3555 . . . . . . . . . . 11 (𝑦 = ∅ → ⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩ = ⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩)
4443eceq1d 6142 . . . . . . . . . 10 (𝑦 = ∅ → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q = [⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q )
4544oveq1d 5527 . . . . . . . . 9 (𝑦 = ∅ → ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃) = ([⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃))
4645oveq2d 5528 . . . . . . . 8 (𝑦 = ∅ → (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) = (𝐴 +Q ([⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)))
4746eleq1d 2106 . . . . . . 7 (𝑦 = ∅ → ((𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈 ↔ (𝐴 +Q ([⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))
4840, 47anbi12d 442 . . . . . 6 (𝑦 = ∅ → (((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) ↔ ((𝐴 +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈)))
4948rspcev 2656 . . . . 5 ((∅ ∈ ω ∧ ((𝐴 +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈)) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))
5035, 49mpan 400 . . . 4 (((𝐴 +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((∅ +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))
5121, 34, 50syl6an 1323 . . 3 (((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 1𝑜 <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ((2𝑜 +𝑜 𝑥) = 𝑁 → ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈)))
5251reximdv 2420 . 2 (((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 1𝑜 <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → (∃𝑥 ∈ ω (2𝑜 +𝑜 𝑥) = 𝑁 → ∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈)))
533, 52mpd 13 1 (((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 1𝑜 <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∃wrex 2307  ∅c0 3224  ⟨cop 3378   class class class wbr 3764  ωcom 4313  (class class class)co 5512  1𝑜c1o 5994  2𝑜c2o 5995   +𝑜 coa 5998  [cec 6104  Ncnpi 6370
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