Metamath Proof Explorer


Theorem grppnpcan2

Description: Cancellation law for mixed addition and subtraction. ( pnpcan2 analog.) (Contributed by NM, 15-Feb-2008) (Revised by Mario Carneiro, 2-Dec-2014)

Ref Expression
Hypotheses grpsubadd.b 𝐵 = ( Base ‘ 𝐺 )
grpsubadd.p + = ( +g𝐺 )
grpsubadd.m = ( -g𝐺 )
Assertion grppnpcan2 ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 + 𝑍 ) ( 𝑌 + 𝑍 ) ) = ( 𝑋 𝑌 ) )

Proof

Step Hyp Ref Expression
1 grpsubadd.b 𝐵 = ( Base ‘ 𝐺 )
2 grpsubadd.p + = ( +g𝐺 )
3 grpsubadd.m = ( -g𝐺 )
4 simpl ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝐺 ∈ Grp )
5 1 2 grpcl ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵 ) → ( 𝑋 + 𝑍 ) ∈ 𝐵 )
6 5 3adant3r2 ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( 𝑋 + 𝑍 ) ∈ 𝐵 )
7 simpr3 ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝑍𝐵 )
8 simpr2 ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝑌𝐵 )
9 1 2 3 grpsubsub4 ( ( 𝐺 ∈ Grp ∧ ( ( 𝑋 + 𝑍 ) ∈ 𝐵𝑍𝐵𝑌𝐵 ) ) → ( ( ( 𝑋 + 𝑍 ) 𝑍 ) 𝑌 ) = ( ( 𝑋 + 𝑍 ) ( 𝑌 + 𝑍 ) ) )
10 4 6 7 8 9 syl13anc ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( ( 𝑋 + 𝑍 ) 𝑍 ) 𝑌 ) = ( ( 𝑋 + 𝑍 ) ( 𝑌 + 𝑍 ) ) )
11 1 2 3 grppncan ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵 ) → ( ( 𝑋 + 𝑍 ) 𝑍 ) = 𝑋 )
12 11 3adant3r2 ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 + 𝑍 ) 𝑍 ) = 𝑋 )
13 12 oveq1d ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( ( 𝑋 + 𝑍 ) 𝑍 ) 𝑌 ) = ( 𝑋 𝑌 ) )
14 10 13 eqtr3d ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 + 𝑍 ) ( 𝑌 + 𝑍 ) ) = ( 𝑋 𝑌 ) )