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Metamath Proof Explorer

Theorem drnginvrn0

Description: The multiplicative inverse in a division ring is nonzero. ( recne0 analog). (Contributed by NM, 19-Apr-2014)

Ref Expression
Hypotheses drnginvrcl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
drnginvrcl.z ⊢ 0 = ( 0g ‘ 𝑅 )
drnginvrcl.i ⊢ 𝐼 = ( invr ‘ 𝑅 )
Assertion drnginvrn0 ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 𝐼 ‘ 𝑋 ) ≠ 0 )

Proof

Step Hyp Ref Expression
1 drnginvrcl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 drnginvrcl.z ⊢ 0 = ( 0g ‘ 𝑅 )
3 drnginvrcl.i ⊢ 𝐼 = ( invr ‘ 𝑅 )
4 drngring ⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring )
5 eqid ⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 )
6 5 3 unitinvcl ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ ( Unit ‘ 𝑅 ) )
7 6 ex ⊢ ( 𝑅 ∈ Ring → ( 𝑋 ∈ ( Unit ‘ 𝑅 ) → ( 𝐼 ‘ 𝑋 ) ∈ ( Unit ‘ 𝑅 ) ) )
8 4 7 syl ⊢ ( 𝑅 ∈ DivRing → ( 𝑋 ∈ ( Unit ‘ 𝑅 ) → ( 𝐼 ‘ 𝑋 ) ∈ ( Unit ‘ 𝑅 ) ) )
9 1 5 2 drngunit ⊢ ( 𝑅 ∈ DivRing → ( 𝑋 ∈ ( Unit ‘ 𝑅 ) ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ) )
10 1 5 2 drngunit ⊢ ( 𝑅 ∈ DivRing → ( ( 𝐼 ‘ 𝑋 ) ∈ ( Unit ‘ 𝑅 ) ↔ ( ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐼 ‘ 𝑋 ) ≠ 0 ) ) )
11 8 9 10 3imtr3d ⊢ ( 𝑅 ∈ DivRing → ( ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐼 ‘ 𝑋 ) ≠ 0 ) ) )
12 11 3impib ⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐼 ‘ 𝑋 ) ≠ 0 ) )
13 12 simprd ⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 𝐼 ‘ 𝑋 ) ≠ 0 )