Theorem lringnz 20796

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 20794	. 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
2		lringnz.1	. . 3 ⊢ 1 = (1r‘𝑅)
3		lringnz.2	. . 3 ⊢ 0 = (0g‘𝑅)
4	2, 3	nzrnz 20766	. 2 ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 )
5	1, 4	syl 18	1 ⊢ (𝑅 ∈ LRing → 1 ≠ 0 )

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ‘cfv 6538  0_gc0g 17611  1_rcur 20408  NzRingcnzr 20763  LRingclring 20791

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-nzr 20764  df-lring 20792

This theorem is used by: (None)