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Metamath Proof Explorer

Theorem cmn4

Description: Commutative/associative law for commutative monoids. (Contributed by NM, 4-Feb-2014) (Revised by Mario Carneiro, 21-Apr-2016)

Ref Expression
Hypotheses ablcom.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
ablcom.p ⊢ + = ( +g ‘ 𝐺 )
Assertion cmn4 ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + ( 𝑍 + 𝑊 ) ) = ( ( 𝑋 + 𝑍 ) + ( 𝑌 + 𝑊 ) ) )

Proof

Step Hyp Ref Expression
1 ablcom.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 ablcom.p ⊢ + = ( +g ‘ 𝐺 )
3 simp1 ⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → 𝐺 ∈ CMnd )
4 cmnmnd ⊢ ( 𝐺 ∈ CMnd → 𝐺 ∈ Mnd )
5 3 4 syl ⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → 𝐺 ∈ Mnd )
6 simp2l ⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
7 simp2r ⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
8 simp3l ⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
9 simp3r ⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → 𝑊 ∈ 𝐵 )
10 1 2 cmncom ⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 + 𝑍 ) = ( 𝑍 + 𝑌 ) )
11 3 7 8 10 syl3anc ⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( 𝑌 + 𝑍 ) = ( 𝑍 + 𝑌 ) )
12 1 2 5 6 7 8 9 11 mnd4g ⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + ( 𝑍 + 𝑊 ) ) = ( ( 𝑋 + 𝑍 ) + ( 𝑌 + 𝑊 ) ) )