This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem mgmlrid

Description: The identity element of a magma, if it exists, is a left and right identity. (Contributed by Mario Carneiro, 27-Dec-2014)

Ref Expression
Hypotheses ismgmid.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
ismgmid.o ⊢ 0 = ( 0g ‘ 𝐺 )
ismgmid.p ⊢ + = ( +g ‘ 𝐺 )
mgmidcl.e ⊢ ( 𝜑 → ∃ 𝑒 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) )
Assertion mgmlrid ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( 0 + 𝑋 ) = 𝑋 ∧ ( 𝑋 + 0 ) = 𝑋 ) )

Proof

Step Hyp Ref Expression
1 ismgmid.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 ismgmid.o ⊢ 0 = ( 0g ‘ 𝐺 )
3 ismgmid.p ⊢ + = ( +g ‘ 𝐺 )
4 mgmidcl.e ⊢ ( 𝜑 → ∃ 𝑒 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑒 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑒 ) = 𝑥 ) )
5 eqid ⊢ 0 = 0
6 1 2 3 4 ismgmid ⊢ ( 𝜑 → ( ( 0 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 0 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 0 ) = 𝑥 ) ) ↔ 0 = 0 ) )
7 5 6 mpbiri ⊢ ( 𝜑 → ( 0 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 0 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 0 ) = 𝑥 ) ) )
8 7 simprd ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ( ( 0 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 0 ) = 𝑥 ) )
9 oveq2 ⊢ ( 𝑥 = 𝑋 → ( 0 + 𝑥 ) = ( 0 + 𝑋 ) )
10 id ⊢ ( 𝑥 = 𝑋 → 𝑥 = 𝑋 )
11 9 10 eqeq12d ⊢ ( 𝑥 = 𝑋 → ( ( 0 + 𝑥 ) = 𝑥 ↔ ( 0 + 𝑋 ) = 𝑋 ) )
12 oveq1 ⊢ ( 𝑥 = 𝑋 → ( 𝑥 + 0 ) = ( 𝑋 + 0 ) )
13 12 10 eqeq12d ⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 + 0 ) = 𝑥 ↔ ( 𝑋 + 0 ) = 𝑋 ) )
14 11 13 anbi12d ⊢ ( 𝑥 = 𝑋 → ( ( ( 0 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 0 ) = 𝑥 ) ↔ ( ( 0 + 𝑋 ) = 𝑋 ∧ ( 𝑋 + 0 ) = 𝑋 ) ) )
15 14 rspccva ⊢ ( ( ∀ 𝑥 ∈ 𝐵 ( ( 0 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 0 ) = 𝑥 ) ∧ 𝑋 ∈ 𝐵 ) → ( ( 0 + 𝑋 ) = 𝑋 ∧ ( 𝑋 + 0 ) = 𝑋 ) )
16 8 15 sylan ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( 0 + 𝑋 ) = 𝑋 ∧ ( 𝑋 + 0 ) = 𝑋 ) )