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Metamath Proof Explorer

Theorem drnginvrn0d

Description: A multiplicative inverse in a division ring is nonzero. ( recne0d analog). (Contributed by SN, 14-Aug-2024)

Ref Expression
Hypotheses drnginvrn0d.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
drnginvrn0d.0 ⊢ 0 = ( 0g ‘ 𝑅 )
drnginvrn0d.i ⊢ 𝐼 = ( invr ‘ 𝑅 )
drnginvrn0d.r ⊢ ( 𝜑 → 𝑅 ∈ DivRing )
drnginvrn0d.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
drnginvrn0d.1 ⊢ ( 𝜑 → 𝑋 ≠ 0 )
Assertion drnginvrn0d ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ≠ 0 )

Proof

Step Hyp Ref Expression
1 drnginvrn0d.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 drnginvrn0d.0 ⊢ 0 = ( 0g ‘ 𝑅 )
3 drnginvrn0d.i ⊢ 𝐼 = ( invr ‘ 𝑅 )
4 drnginvrn0d.r ⊢ ( 𝜑 → 𝑅 ∈ DivRing )
5 drnginvrn0d.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 drnginvrn0d.1 ⊢ ( 𝜑 → 𝑋 ≠ 0 )
7 1 2 3 drnginvrn0 ⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 𝐼 ‘ 𝑋 ) ≠ 0 )
8 4 5 6 7 syl3anc ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ≠ 0 )