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Theorem prarloclem5 6354
 Description: A substitution of zero for y and 𝑁 minus two for x. Lemma for prarloc 6357. (Contributed by Jim Kingdon, 4-Nov-2019.)
Assertion
Ref Expression
prarloclem5 (((⟨𝐿, 𝑈 P A 𝐿) (𝑁 N 𝑃 Q 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 6353 . . . 4 ((𝑁 N 1𝑜 <N 𝑁) → x 𝜔 (2𝑜 +𝑜 x) = 𝑁)
213adant2 911 . . 3 ((𝑁 N 𝑃 Q 1𝑜 <N 𝑁) → x 𝜔 (2𝑜 +𝑜 x) = 𝑁)
323ad2ant2 914 . 2 (((⟨𝐿, 𝑈 P A 𝐿) (𝑁 N 𝑃 Q 1𝑜 <N 𝑁) (A +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) 𝑈) → x 𝜔 (2𝑜 +𝑜 x) = 𝑁)
4 elprnql 6335 . . . . . . 7 ((⟨𝐿, 𝑈 P A 𝐿) → A Q)
543ad2ant1 913 . . . . . 6 (((⟨𝐿, 𝑈 P A 𝐿) (𝑁 N 𝑃 Q 1𝑜 <N 𝑁) (A +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) 𝑈) → A Q)
6 simp22 926 . . . . . 6 (((⟨𝐿, 𝑈 P A 𝐿) (𝑁 N 𝑃 Q 1𝑜 <N 𝑁) (A +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) 𝑈) → 𝑃 Q)
7 nqnq0 6296 . . . . . . . . 9 QQ0
87sseli 2918 . . . . . . . 8 (A QA Q0)
9 nq0a0 6312 . . . . . . . 8 (A Q0 → (A +Q0 0Q0) = A)
108, 9syl 14 . . . . . . 7 (A Q → (A +Q0 0Q0) = A)
11 df-0nq0 6281 . . . . . . . . . 10 0Q0 = [⟨∅, 1𝑜⟩] ~Q0
1211oveq1i 5446 . . . . . . . . 9 (0Q0 ·Q0 𝑃) = ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)
137sseli 2918 . . . . . . . . . 10 (𝑃 Q𝑃 Q0)
14 nq0m0r 6311 . . . . . . . . . 10 (𝑃 Q0 → (0Q0 ·Q0 𝑃) = 0Q0)
1513, 14syl 14 . . . . . . . . 9 (𝑃 Q → (0Q0 ·Q0 𝑃) = 0Q0)
1612, 15syl5reqr 2069 . . . . . . . 8 (𝑃 Q → 0Q0 = ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃))
1716oveq2d 5452 . . . . . . 7 (𝑃 Q → (A +Q0 0Q0) = (A +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)))
1810, 17sylan9req 2075 . . . . . 6 ((A Q 𝑃 Q) → A = (A +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)))
195, 6, 18syl2anc 393 . . . . 5 (((⟨𝐿, 𝑈 P A 𝐿) (𝑁 N 𝑃 Q 1𝑜 <N 𝑁) (A +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) 𝑈) → A = (A +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)))
20 simp1r 917 . . . . 5 (((⟨𝐿, 𝑈 P A 𝐿) (𝑁 N 𝑃 Q 1𝑜 <N 𝑁) (A +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) 𝑈) → A 𝐿)
2119, 20eqeltrrd 2097 . . . 4 (((⟨𝐿, 𝑈 P A 𝐿) (𝑁 N 𝑃 Q 1𝑜 <N 𝑁) (A +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) 𝑈) → (A +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)) 𝐿)
22 2onn 6005 . . . . . . . . . . . . . . 15 2𝑜 𝜔
23 nna0r 5972 . . . . . . . . . . . . . . 15 (2𝑜 𝜔 → (∅ +𝑜 2𝑜) = 2𝑜)
2422, 23ax-mp 7 . . . . . . . . . . . . . 14 (∅ +𝑜 2𝑜) = 2𝑜
2524oveq1i 5446 . . . . . . . . . . . . 13 ((∅ +𝑜 2𝑜) +𝑜 x) = (2𝑜 +𝑜 x)
2625eqeq1i 2029 . . . . . . . . . . . 12 (((∅ +𝑜 2𝑜) +𝑜 x) = 𝑁 ↔ (2𝑜 +𝑜 x) = 𝑁)
2726biimpri 124 . . . . . . . . . . 11 ((2𝑜 +𝑜 x) = 𝑁 → ((∅ +𝑜 2𝑜) +𝑜 x) = 𝑁)
2827opeq1d 3529 . . . . . . . . . 10 ((2𝑜 +𝑜 x) = 𝑁 → ⟨((∅ +𝑜 2𝑜) +𝑜 x), 1𝑜⟩ = ⟨𝑁, 1𝑜⟩)
2928eceq1d 6053 . . . . . . . . 9 ((2𝑜 +𝑜 x) = 𝑁 → [⟨((∅ +𝑜 2𝑜) +𝑜 x), 1𝑜⟩] ~Q = [⟨𝑁, 1𝑜⟩] ~Q )
3029oveq1d 5451 . . . . . . . 8 ((2𝑜 +𝑜 x) = 𝑁 → ([⟨((∅ +𝑜 2𝑜) +𝑜 x), 1𝑜⟩] ~Q ·Q 𝑃) = ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃))
3130oveq2d 5452 . . . . . . 7 ((2𝑜 +𝑜 x) = 𝑁 → (A +Q ([⟨((∅ +𝑜 2𝑜) +𝑜 x), 1𝑜⟩] ~Q ·Q 𝑃)) = (A +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)))
3231eleq1d 2088 . . . . . 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 915 . . . 4 (((⟨𝐿, 𝑈 P A 𝐿) (𝑁 N 𝑃 Q 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 4244 . . . . 5 𝜔
36 opeq1 3523 . . . . . . . . . . 11 (y = ∅ → ⟨y, 1𝑜⟩ = ⟨∅, 1𝑜⟩)
3736eceq1d 6053 . . . . . . . . . 10 (y = ∅ → [⟨y, 1𝑜⟩] ~Q0 = [⟨∅, 1𝑜⟩] ~Q0 )
3837oveq1d 5451 . . . . . . . . 9 (y = ∅ → ([⟨y, 1𝑜⟩] ~Q0 ·Q0 𝑃) = ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃))
3938oveq2d 5452 . . . . . . . 8 (y = ∅ → (A +Q0 ([⟨y, 1𝑜⟩] ~Q0 ·Q0 𝑃)) = (A +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)))
4039eleq1d 2088 . . . . . . 7 (y = ∅ → ((A +Q0 ([⟨y, 1𝑜⟩] ~Q0 ·Q0 𝑃)) 𝐿 ↔ (A +Q0 ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 𝑃)) 𝐿))
41 oveq1 5443 . . . . . . . . . . . . 13 (y = ∅ → (y +𝑜 2𝑜) = (∅ +𝑜 2𝑜))
4241oveq1d 5451 . . . . . . . . . . . 12 (y = ∅ → ((y +𝑜 2𝑜) +𝑜 x) = ((∅ +𝑜 2𝑜) +𝑜 x))
4342opeq1d 3529 . . . . . . . . . . 11 (y = ∅ → ⟨((y +𝑜 2𝑜) +𝑜 x), 1𝑜⟩ = ⟨((∅ +𝑜 2𝑜) +𝑜 x), 1𝑜⟩)
4443eceq1d 6053 . . . . . . . . . 10 (y = ∅ → [⟨((y +𝑜 2𝑜) +𝑜 x), 1𝑜⟩] ~Q = [⟨((∅ +𝑜 2𝑜) +𝑜 x), 1𝑜⟩] ~Q )
4544oveq1d 5451 . . . . . . . . 9 (y = ∅ → ([⟨((y +𝑜 2𝑜) +𝑜 x), 1𝑜⟩] ~Q ·Q 𝑃) = ([⟨((∅ +𝑜 2𝑜) +𝑜 x), 1𝑜⟩] ~Q ·Q 𝑃))
4645oveq2d 5452 . . . . . . . 8 (y = ∅ → (A +Q ([⟨((y +𝑜 2𝑜) +𝑜 x), 1𝑜⟩] ~Q ·Q 𝑃)) = (A +Q ([⟨((∅ +𝑜 2𝑜) +𝑜 x), 1𝑜⟩] ~Q ·Q 𝑃)))
4746eleq1d 2088 . . . . . . 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 445 . . . . . 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 2633 . . . . 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 402 . . . 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, 50ee12an 1302 . . 3 (((⟨𝐿, 𝑈 P A 𝐿) (𝑁 N 𝑃 Q 1𝑜 <N 𝑁) (A +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) 𝑈) → ((2𝑜 +𝑜 x) = 𝑁y 𝜔 ((A +Q0 ([⟨y, 1𝑜⟩] ~Q0 ·Q0 𝑃)) 𝐿 (A +Q ([⟨((y +𝑜 2𝑜) +𝑜 x), 1𝑜⟩] ~Q ·Q 𝑃)) 𝑈)))
5251reximdv 2398 . 2 (((⟨𝐿, 𝑈 P A 𝐿) (𝑁 N 𝑃 Q 1𝑜 <N 𝑁) (A +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) 𝑈) → (x 𝜔 (2𝑜 +𝑜 x) = 𝑁x 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 P A 𝐿) (𝑁 N 𝑃 Q 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 873   = wceq 1228   ∈ wcel 1374  ∃wrex 2285  ∅c0 3201  ⟨cop 3353   class class class wbr 3738  𝜔com 4240  (class class class)co 5436  1𝑜c1o 5909  2𝑜c2o 5910   +𝑜 coa 5913  [cec 6015  Ncnpi 6130
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