This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem grpasscan1

Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008) (Revised by AV, 30-Aug-2021)

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

Proof

Step Hyp Ref Expression
1 grplcan.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grplcan.p ⊢ + = ( +g ‘ 𝐺 )
3 grpasscan1.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
4 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
5 1 2 4 3 grprinv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 + ( 𝑁 ‘ 𝑋 ) ) = ( 0g ‘ 𝐺 ) )
6 5 3adant3 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + ( 𝑁 ‘ 𝑋 ) ) = ( 0g ‘ 𝐺 ) )
7 6 oveq1d ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + ( 𝑁 ‘ 𝑋 ) ) + 𝑌 ) = ( ( 0g ‘ 𝐺 ) + 𝑌 ) )
8 1 3 grpinvcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 )
9 1 2 grpass ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 + ( 𝑁 ‘ 𝑋 ) ) + 𝑌 ) = ( 𝑋 + ( ( 𝑁 ‘ 𝑋 ) + 𝑌 ) ) )
10 9 3exp2 ⊢ ( 𝐺 ∈ Grp → ( 𝑋 ∈ 𝐵 → ( ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 → ( 𝑌 ∈ 𝐵 → ( ( 𝑋 + ( 𝑁 ‘ 𝑋 ) ) + 𝑌 ) = ( 𝑋 + ( ( 𝑁 ‘ 𝑋 ) + 𝑌 ) ) ) ) ) )
11 10 imp ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 → ( 𝑌 ∈ 𝐵 → ( ( 𝑋 + ( 𝑁 ‘ 𝑋 ) ) + 𝑌 ) = ( 𝑋 + ( ( 𝑁 ‘ 𝑋 ) + 𝑌 ) ) ) ) )
12 8 11 mpd ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 ∈ 𝐵 → ( ( 𝑋 + ( 𝑁 ‘ 𝑋 ) ) + 𝑌 ) = ( 𝑋 + ( ( 𝑁 ‘ 𝑋 ) + 𝑌 ) ) ) )
13 12 3impia ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + ( 𝑁 ‘ 𝑋 ) ) + 𝑌 ) = ( 𝑋 + ( ( 𝑁 ‘ 𝑋 ) + 𝑌 ) ) )
14 1 2 4 grplid ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( ( 0g ‘ 𝐺 ) + 𝑌 ) = 𝑌 )
15 14 3adant2 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 0g ‘ 𝐺 ) + 𝑌 ) = 𝑌 )
16 7 13 15 3eqtr3d ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + ( ( 𝑁 ‘ 𝑋 ) + 𝑌 ) ) = 𝑌 )