# Metamath Proof Explorer

Description: Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014)

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

### 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 simpr1 ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝑋𝐵 )
6 simpr2 ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝑌𝐵 )
7 eqid ( invg𝐺 ) = ( invg𝐺 )
8 1 7 grpinvcl ( ( 𝐺 ∈ Grp ∧ 𝑍𝐵 ) → ( ( invg𝐺 ) ‘ 𝑍 ) ∈ 𝐵 )
9 8 3ad2antr3 ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( invg𝐺 ) ‘ 𝑍 ) ∈ 𝐵 )
10 1 2 grpass ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵 ∧ ( ( invg𝐺 ) ‘ 𝑍 ) ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + ( ( invg𝐺 ) ‘ 𝑍 ) ) = ( 𝑋 + ( 𝑌 + ( ( invg𝐺 ) ‘ 𝑍 ) ) ) )
11 4 5 6 9 10 syl13anc ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + ( ( invg𝐺 ) ‘ 𝑍 ) ) = ( 𝑋 + ( 𝑌 + ( ( invg𝐺 ) ‘ 𝑍 ) ) ) )
12 1 2 grpcl ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
13 12 3adant3r3 ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
14 simpr3 ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝑍𝐵 )
15 1 2 7 3 grpsubval ( ( ( 𝑋 + 𝑌 ) ∈ 𝐵𝑍𝐵 ) → ( ( 𝑋 + 𝑌 ) 𝑍 ) = ( ( 𝑋 + 𝑌 ) + ( ( invg𝐺 ) ‘ 𝑍 ) ) )
16 13 14 15 syl2anc ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 + 𝑌 ) 𝑍 ) = ( ( 𝑋 + 𝑌 ) + ( ( invg𝐺 ) ‘ 𝑍 ) ) )
17 1 2 7 3 grpsubval ( ( 𝑌𝐵𝑍𝐵 ) → ( 𝑌 𝑍 ) = ( 𝑌 + ( ( invg𝐺 ) ‘ 𝑍 ) ) )
18 6 14 17 syl2anc ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( 𝑌 𝑍 ) = ( 𝑌 + ( ( invg𝐺 ) ‘ 𝑍 ) ) )
19 18 oveq2d ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( 𝑋 + ( 𝑌 𝑍 ) ) = ( 𝑋 + ( 𝑌 + ( ( invg𝐺 ) ‘ 𝑍 ) ) ) )
20 11 16 19 3eqtr4d ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 + 𝑌 ) 𝑍 ) = ( 𝑋 + ( 𝑌 𝑍 ) ) )