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

Metamath Proof Explorer

Theorem mndvrid

Description: Tuple-wise right identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015)

Ref Expression
Hypotheses mndvcl.b ⊢ 𝐵 = ( Base ‘ 𝑀 )
mndvcl.p ⊢ + = ( +g ‘ 𝑀 )
mndvlid.z ⊢ 0 = ( 0g ‘ 𝑀 )
Assertion mndvrid ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝑋 ∘f + ( 𝐼 × { 0 } ) ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 mndvcl.b ⊢ 𝐵 = ( Base ‘ 𝑀 )
2 mndvcl.p ⊢ + = ( +g ‘ 𝑀 )
3 mndvlid.z ⊢ 0 = ( 0g ‘ 𝑀 )
4 elmapex ⊢ ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) → ( 𝐵 ∈ V ∧ 𝐼 ∈ V ) )
5 4 simprd ⊢ ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) → 𝐼 ∈ V )
6 5 adantl ⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝐼 ∈ V )
7 elmapi ⊢ ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) → 𝑋 : 𝐼 ⟶ 𝐵 )
8 7 adantl ⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝑋 : 𝐼 ⟶ 𝐵 )
9 1 3 mndidcl ⊢ ( 𝑀 ∈ Mnd → 0 ∈ 𝐵 )
10 9 adantr ⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → 0 ∈ 𝐵 )
11 1 2 3 mndrid ⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 + 0 ) = 𝑥 )
12 11 adantlr ⊢ ( ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 + 0 ) = 𝑥 )
13 6 8 10 12 caofid0r ⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝑋 ∘f + ( 𝐼 × { 0 } ) ) = 𝑋 )