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Metamath Proof Explorer

Theorem mndvcl

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

Ref Expression
Hypotheses mndvcl.b ⊢ 𝐵 = ( Base ‘ 𝑀 )
mndvcl.p ⊢ + = ( +g ‘ 𝑀 )
Assertion mndvcl ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝑋 ∘f + 𝑌 ) ∈ ( 𝐵 ↑m 𝐼 ) )

Proof

Step Hyp Ref Expression
1 mndvcl.b ⊢ 𝐵 = ( Base ‘ 𝑀 )
2 mndvcl.p ⊢ + = ( +g ‘ 𝑀 )
3 1 2 mndcl ⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 )
4 3 3expb ⊢ ( ( 𝑀 ∈ Mnd ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 )
5 4 3ad2antl1 ⊢ ( ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 )
6 elmapi ⊢ ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) → 𝑋 : 𝐼 ⟶ 𝐵 )
7 6 3ad2ant2 ⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝑋 : 𝐼 ⟶ 𝐵 )
8 elmapi ⊢ ( 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) → 𝑌 : 𝐼 ⟶ 𝐵 )
9 8 3ad2ant3 ⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝑌 : 𝐼 ⟶ 𝐵 )
10 elmapex ⊢ ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) → ( 𝐵 ∈ V ∧ 𝐼 ∈ V ) )
11 10 simprd ⊢ ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) → 𝐼 ∈ V )
12 11 3ad2ant2 ⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝐼 ∈ V )
13 inidm ⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼
14 5 7 9 12 12 13 off ⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝑋 ∘f + 𝑌 ) : 𝐼 ⟶ 𝐵 )
15 1 fvexi ⊢ 𝐵 ∈ V
16 elmapg ⊢ ( ( 𝐵 ∈ V ∧ 𝐼 ∈ V ) → ( ( 𝑋 ∘f + 𝑌 ) ∈ ( 𝐵 ↑m 𝐼 ) ↔ ( 𝑋 ∘f + 𝑌 ) : 𝐼 ⟶ 𝐵 ) )
17 15 12 16 sylancr ⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( ( 𝑋 ∘f + 𝑌 ) ∈ ( 𝐵 ↑m 𝐼 ) ↔ ( 𝑋 ∘f + 𝑌 ) : 𝐼 ⟶ 𝐵 ) )
18 14 17 mpbird ⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝑋 ∘f + 𝑌 ) ∈ ( 𝐵 ↑m 𝐼 ) )