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Metamath Proof Explorer

Theorem mndvass

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

		Ref	Expression
	Hypotheses	mndvcl.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
		mndvcl.p	⊢ + = ( +_g ‘ 𝑀 )
	Assertion	mndvass	⊢ ( ( 𝑀 ∈ Mnd ∧ ( 𝑋 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑍 ∈ ( 𝐵 ↑_m 𝐼 ) ) ) → ( ( 𝑋 ∘_f + 𝑌 ) ∘_f + 𝑍 ) = ( 𝑋 ∘_f + ( 𝑌 ∘_f + 𝑍 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mndvcl.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
2		mndvcl.p	⊢ + = ( +_g ‘ 𝑀 )
3		elmapex	⊢ ( 𝑋 ∈ ( 𝐵 ↑_m 𝐼 ) → ( 𝐵 ∈ V ∧ 𝐼 ∈ V ) )
4	3	simprd	⊢ ( 𝑋 ∈ ( 𝐵 ↑_m 𝐼 ) → 𝐼 ∈ V )
5	4	3ad2ant1	⊢ ( ( 𝑋 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑍 ∈ ( 𝐵 ↑_m 𝐼 ) ) → 𝐼 ∈ V )
6	5	adantl	⊢ ( ( 𝑀 ∈ Mnd ∧ ( 𝑋 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑍 ∈ ( 𝐵 ↑_m 𝐼 ) ) ) → 𝐼 ∈ V )
7		elmapi	⊢ ( 𝑋 ∈ ( 𝐵 ↑_m 𝐼 ) → 𝑋 : 𝐼 ⟶ 𝐵 )
8	7	3ad2ant1	⊢ ( ( 𝑋 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑍 ∈ ( 𝐵 ↑_m 𝐼 ) ) → 𝑋 : 𝐼 ⟶ 𝐵 )
9	8	adantl	⊢ ( ( 𝑀 ∈ Mnd ∧ ( 𝑋 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑍 ∈ ( 𝐵 ↑_m 𝐼 ) ) ) → 𝑋 : 𝐼 ⟶ 𝐵 )
10		elmapi	⊢ ( 𝑌 ∈ ( 𝐵 ↑_m 𝐼 ) → 𝑌 : 𝐼 ⟶ 𝐵 )
11	10	3ad2ant2	⊢ ( ( 𝑋 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑍 ∈ ( 𝐵 ↑_m 𝐼 ) ) → 𝑌 : 𝐼 ⟶ 𝐵 )
12	11	adantl	⊢ ( ( 𝑀 ∈ Mnd ∧ ( 𝑋 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑍 ∈ ( 𝐵 ↑_m 𝐼 ) ) ) → 𝑌 : 𝐼 ⟶ 𝐵 )
13		elmapi	⊢ ( 𝑍 ∈ ( 𝐵 ↑_m 𝐼 ) → 𝑍 : 𝐼 ⟶ 𝐵 )
14	13	3ad2ant3	⊢ ( ( 𝑋 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑍 ∈ ( 𝐵 ↑_m 𝐼 ) ) → 𝑍 : 𝐼 ⟶ 𝐵 )
15	14	adantl	⊢ ( ( 𝑀 ∈ Mnd ∧ ( 𝑋 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑍 ∈ ( 𝐵 ↑_m 𝐼 ) ) ) → 𝑍 : 𝐼 ⟶ 𝐵 )
16	1 2	mndass	⊢ ( ( 𝑀 ∈ Mnd ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) )
17	16	adantlr	⊢ ( ( ( 𝑀 ∈ Mnd ∧ ( 𝑋 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑍 ∈ ( 𝐵 ↑_m 𝐼 ) ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) )
18	6 9 12 15 17	caofass	⊢ ( ( 𝑀 ∈ Mnd ∧ ( 𝑋 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑_m 𝐼 ) ∧ 𝑍 ∈ ( 𝐵 ↑_m 𝐼 ) ) ) → ( ( 𝑋 ∘_f + 𝑌 ) ∘_f + 𝑍 ) = ( 𝑋 ∘_f + ( 𝑌 ∘_f + 𝑍 ) ) )