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Metamath Proof Explorer

Theorem grpridd

Description: The identity element of a group is a right identity. Deduction associated with grprid . (Contributed by SN, 29-Jan-2025)

Ref Expression
Hypotheses grpbn0.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
grplid.p ⊢ + = ( +g ‘ 𝐺 )
grplid.o ⊢ 0 = ( 0g ‘ 𝐺 )
grplidd.g ⊢ ( 𝜑 → 𝐺 ∈ Grp )
grplidd.1 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion grpridd ( 𝜑 → ( 𝑋 + 0 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 grpbn0.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grplid.p ⊢ + = ( +g ‘ 𝐺 )
3 grplid.o ⊢ 0 = ( 0g ‘ 𝐺 )
4 grplidd.g ⊢ ( 𝜑 → 𝐺 ∈ Grp )
5 grplidd.1 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 1 2 3 grprid ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 + 0 ) = 𝑋 )
7 4 5 6 syl2anc ⊢ ( 𝜑 → ( 𝑋 + 0 ) = 𝑋 )