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 + = ( +g𝐺 )
grpasscan1.n 𝑁 = ( invg𝐺 )
Assertion grpasscan2 ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( ( 𝑋 + ( 𝑁𝑌 ) ) + 𝑌 ) = 𝑋 )

Proof

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