Theorem mndvlid

Description: Tuple-wise left 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	mndvlid	⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( ( 𝐼 × { 0 } ) ∘f + 𝑋 ) = 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	mndlid	⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ) → ( 0 + 𝑥 ) = 𝑥 )
12	11	adantlr	⊢ ( ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 0 + 𝑥 ) = 𝑥 )
13	6 8 10 12	caofid0l	⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( ( 𝐼 × { 0 } ) ∘f + 𝑋 ) = 𝑋 )