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Metamath Proof Explorer

Theorem grpaddsubass

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 ⊢ + = ( +g ‘ 𝐺 )
grpsubadd.m ⊢ − = ( -g ‘ 𝐺 )
Assertion grpaddsubass ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) − 𝑍 ) = ( 𝑋 + ( 𝑌 − 𝑍 ) ) )

Proof

Step Hyp Ref Expression
1 grpsubadd.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grpsubadd.p ⊢ + = ( +g ‘ 𝐺 )
3 grpsubadd.m ⊢ − = ( -g ‘ 𝐺 )
4 simpl ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐺 ∈ Grp )
5 simpr1 ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
6 simpr2 ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
7 eqid ⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
8 1 7 grpinvcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ∈ 𝐵 )
9 8 3ad2antr3 ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ∈ 𝐵 )
10 1 2 grpass ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) = ( 𝑋 + ( 𝑌 + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) ) )
11 4 5 6 9 10 syl13anc ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) = ( 𝑋 + ( 𝑌 + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) ) )
12 1 2 grpcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
13 12 3adant3r3 ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
14 simpr3 ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
15 1 2 7 3 grpsubval ⊢ ( ( ( 𝑋 + 𝑌 ) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝑋 + 𝑌 ) − 𝑍 ) = ( ( 𝑋 + 𝑌 ) + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) )
16 13 14 15 syl2anc ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) − 𝑍 ) = ( ( 𝑋 + 𝑌 ) + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) )
17 1 2 7 3 grpsubval ⊢ ( ( 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 − 𝑍 ) = ( 𝑌 + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) )
18 6 14 17 syl2anc ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑌 − 𝑍 ) = ( 𝑌 + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) )
19 18 oveq2d ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 + ( 𝑌 − 𝑍 ) ) = ( 𝑋 + ( 𝑌 + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) ) )
20 11 16 19 3eqtr4d ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) − 𝑍 ) = ( 𝑋 + ( 𝑌 − 𝑍 ) ) )