Metamath Proof Explorer


Theorem grppncan

Description: Cancellation law for subtraction ( pncan analog). (Contributed by NM, 16-Apr-2014)

Ref Expression
Hypotheses grpsubadd.b 𝐵 = ( Base ‘ 𝐺 )
grpsubadd.p + = ( +g𝐺 )
grpsubadd.m = ( -g𝐺 )
Assertion grppncan ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( ( 𝑋 + 𝑌 ) 𝑌 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 grpsubadd.b 𝐵 = ( Base ‘ 𝐺 )
2 grpsubadd.p + = ( +g𝐺 )
3 grpsubadd.m = ( -g𝐺 )
4 simp1 ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → 𝐺 ∈ Grp )
5 simp2 ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → 𝑋𝐵 )
6 simp3 ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → 𝑌𝐵 )
7 1 2 3 grpaddsubass ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑌𝐵 ) ) → ( ( 𝑋 + 𝑌 ) 𝑌 ) = ( 𝑋 + ( 𝑌 𝑌 ) ) )
8 4 5 6 6 7 syl13anc ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( ( 𝑋 + 𝑌 ) 𝑌 ) = ( 𝑋 + ( 𝑌 𝑌 ) ) )
9 eqid ( 0g𝐺 ) = ( 0g𝐺 )
10 1 9 3 grpsubid ( ( 𝐺 ∈ Grp ∧ 𝑌𝐵 ) → ( 𝑌 𝑌 ) = ( 0g𝐺 ) )
11 10 oveq2d ( ( 𝐺 ∈ Grp ∧ 𝑌𝐵 ) → ( 𝑋 + ( 𝑌 𝑌 ) ) = ( 𝑋 + ( 0g𝐺 ) ) )
12 11 3adant2 ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 + ( 𝑌 𝑌 ) ) = ( 𝑋 + ( 0g𝐺 ) ) )
13 1 2 9 grprid ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵 ) → ( 𝑋 + ( 0g𝐺 ) ) = 𝑋 )
14 13 3adant3 ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 + ( 0g𝐺 ) ) = 𝑋 )
15 8 12 14 3eqtrd ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( ( 𝑋 + 𝑌 ) 𝑌 ) = 𝑋 )