Theorem lringnz 20594

Description: A local ring is a nonzero ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)

Hypotheses

Ref	Expression
lringnz.1	⊢ 1 = (1r‘𝑅)
lringnz.2	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
lringnz	⊢ (𝑅 ∈ LRing → 1 ≠ 0 )

Proof of Theorem lringnz

Step	Hyp	Ref	Expression
1		lringnzr 20592	. 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
2		lringnz.1	. . 3 ⊢ 1 = (1r‘𝑅)
3		lringnz.2	. . 3 ⊢ 0 = (0g‘𝑅)
4	2, 3	nzrnz 20566	. 2 ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 )
5	1, 4	syl 17	1 ⊢ (𝑅 ∈ LRing → 1 ≠ 0 )

Syntax hints:   → wi 4   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ‘cfv 6522  0_gc0g 17469  1_rcur 20232  NzRingcnzr 20563  LRingclring 20589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-iota 6478  df-fv 6530  df-nzr 20564  df-lring 20590

This theorem is referenced by: (None)