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Theorem prarloclemn 6482
 Description: Subtracting two from a positive integer. Lemma for prarloc 6486. (Contributed by Jim Kingdon, 5-Nov-2019.)
Assertion
Ref Expression
prarloclemn ((𝑁 N 1𝑜 <N 𝑁) → x 𝜔 (2𝑜 +𝑜 x) = 𝑁)
Distinct variable group:   x,𝑁

Proof of Theorem prarloclemn
StepHypRef Expression
1 simpl 102 . . 3 ((𝑁 N 1𝑜 <N 𝑁) → 𝑁 N)
2 1pi 6299 . . . . 5 1𝑜 N
3 ltpiord 6303 . . . . 5 ((1𝑜 N 𝑁 N) → (1𝑜 <N 𝑁 ↔ 1𝑜 𝑁))
42, 3mpan 400 . . . 4 (𝑁 N → (1𝑜 <N 𝑁 ↔ 1𝑜 𝑁))
54biimpa 280 . . 3 ((𝑁 N 1𝑜 <N 𝑁) → 1𝑜 𝑁)
6 piord 6295 . . . 4 (𝑁 N → Ord 𝑁)
7 ordsucss 4196 . . . 4 (Ord 𝑁 → (1𝑜 𝑁 → suc 1𝑜𝑁))
86, 7syl 14 . . 3 (𝑁 N → (1𝑜 𝑁 → suc 1𝑜𝑁))
91, 5, 8sylc 56 . 2 ((𝑁 N 1𝑜 <N 𝑁) → suc 1𝑜𝑁)
10 df-2o 5941 . . . 4 2𝑜 = suc 1𝑜
1110sseq1i 2963 . . 3 (2𝑜𝑁 ↔ suc 1𝑜𝑁)
12 pinn 6293 . . . . 5 (𝑁 N𝑁 𝜔)
13 2onn 6030 . . . . . 6 2𝑜 𝜔
14 nnawordex 6037 . . . . . 6 ((2𝑜 𝜔 𝑁 𝜔) → (2𝑜𝑁x 𝜔 (2𝑜 +𝑜 x) = 𝑁))
1513, 14mpan 400 . . . . 5 (𝑁 𝜔 → (2𝑜𝑁x 𝜔 (2𝑜 +𝑜 x) = 𝑁))
1612, 15syl 14 . . . 4 (𝑁 N → (2𝑜𝑁x 𝜔 (2𝑜 +𝑜 x) = 𝑁))
1716adantr 261 . . 3 ((𝑁 N 1𝑜 <N 𝑁) → (2𝑜𝑁x 𝜔 (2𝑜 +𝑜 x) = 𝑁))
1811, 17syl5bbr 183 . 2 ((𝑁 N 1𝑜 <N 𝑁) → (suc 1𝑜𝑁x 𝜔 (2𝑜 +𝑜 x) = 𝑁))
199, 18mpbid 135 1 ((𝑁 N 1𝑜 <N 𝑁) → x 𝜔 (2𝑜 +𝑜 x) = 𝑁)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∃wrex 2301   ⊆ wss 2911   class class class wbr 3755  Ord word 4065  suc csuc 4068  𝜔com 4256  (class class class)co 5455  1𝑜c1o 5933  2𝑜c2o 5934   +𝑜 coa 5937  Ncnpi 6256
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