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Metamath Proof Explorer

Theorem grpasscan2

Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009) (Revised by AV, 30-Aug-2021)

Ref Expression
Hypotheses grplcan.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
grplcan.p ⊢ + = ( +g ‘ 𝐺 )
grpasscan1.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
Assertion grpasscan2 ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 grplcan.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grplcan.p ⊢ + = ( +g ‘ 𝐺 )
3 grpasscan1.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
4 simp1 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐺 ∈ Grp )
5 simp2 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
6 1 3 grpinvcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 )
7 6 3adant2 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 )
8 simp3 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 )
9 1 2 grpass ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = ( 𝑋 + ( ( 𝑁 ‘ 𝑌 ) + 𝑌 ) ) )
10 4 5 7 8 9 syl13anc ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = ( 𝑋 + ( ( 𝑁 ‘ 𝑌 ) + 𝑌 ) ) )
11 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
12 1 2 11 3 grplinv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑌 ) + 𝑌 ) = ( 0g ‘ 𝐺 ) )
13 12 3adant2 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑌 ) + 𝑌 ) = ( 0g ‘ 𝐺 ) )
14 13 oveq2d ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + ( ( 𝑁 ‘ 𝑌 ) + 𝑌 ) ) = ( 𝑋 + ( 0g ‘ 𝐺 ) ) )
15 1 2 11 grprid ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 + ( 0g ‘ 𝐺 ) ) = 𝑋 )
16 15 3adant3 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + ( 0g ‘ 𝐺 ) ) = 𝑋 )
17 10 14 16 3eqtrd ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = 𝑋 )