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Metamath Proof Explorer

Theorem 01eq0ring

Description: If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019) (Proof shortened by SN, 23-Feb-2025)

Ref Expression
Hypotheses 0ring.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
0ring.0 ⊢ 0 = ( 0g ‘ 𝑅 )
0ring01eq.1 ⊢ 1 = ( 1r ‘ 𝑅 )
Assertion 01eq0ring ( ( 𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 } )

Proof

Step Hyp Ref Expression
1 0ring.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 0ring.0 ⊢ 0 = ( 0g ‘ 𝑅 )
3 0ring01eq.1 ⊢ 1 = ( 1r ‘ 𝑅 )
4 eqcom ⊢ ( 0 = 1 ↔ 1 = 0 )
5 1 2 ring0cl ⊢ ( 𝑅 ∈ Ring → 0 ∈ 𝐵 )
6 5 ne0d ⊢ ( 𝑅 ∈ Ring → 𝐵 ≠ ∅ )
7 5 adantr ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ) → 0 ∈ 𝐵 )
8 1 3 2 ring1eq0 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → ( 1 = 0 → 𝑥 = 0 ) )
9 7 8 mpd3an3 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ) → ( 1 = 0 → 𝑥 = 0 ) )
10 9 impancom ⊢ ( ( 𝑅 ∈ Ring ∧ 1 = 0 ) → ( 𝑥 ∈ 𝐵 → 𝑥 = 0 ) )
11 10 ralrimiv ⊢ ( ( 𝑅 ∈ Ring ∧ 1 = 0 ) → ∀ 𝑥 ∈ 𝐵 𝑥 = 0 )
12 eqsn ⊢ ( 𝐵 ≠ ∅ → ( 𝐵 = { 0 } ↔ ∀ 𝑥 ∈ 𝐵 𝑥 = 0 ) )
13 12 biimpar ⊢ ( ( 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 𝑥 = 0 ) → 𝐵 = { 0 } )
14 6 11 13 syl2an2r ⊢ ( ( 𝑅 ∈ Ring ∧ 1 = 0 ) → 𝐵 = { 0 } )
15 4 14 sylan2b ⊢ ( ( 𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 } )