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Metamath Proof Explorer

Theorem grpsubadd

Description: Relationship between group subtraction and addition. (Contributed by NM, 31-Mar-2014)

Ref Expression
Hypotheses grpsubadd.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
grpsubadd.p ⊢ + = ( +g ‘ 𝐺 )
grpsubadd.m ⊢ − = ( -g ‘ 𝐺 )
Assertion grpsubadd ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) = 𝑍 ↔ ( 𝑍 + 𝑌 ) = 𝑋 ) )

Proof

Step Hyp Ref Expression
1 grpsubadd.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grpsubadd.p ⊢ + = ( +g ‘ 𝐺 )
3 grpsubadd.m ⊢ − = ( -g ‘ 𝐺 )
4 eqid ⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
5 1 2 4 3 grpsubval ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) )
6 5 3adant3 ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) )
7 6 adantl ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 − 𝑌 ) = ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) )
8 7 eqeq1d ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) = 𝑍 ↔ ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 𝑍 ) )
9 simpl ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐺 ∈ Grp )
10 simpr1 ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
11 1 4 grpinvcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 )
12 11 3ad2antr2 ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 )
13 1 2 grpcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ∈ 𝐵 )
14 9 10 12 13 syl3anc ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ∈ 𝐵 )
15 simpr3 ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
16 simpr2 ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
17 1 2 grprcan ⊢ ( ( 𝐺 ∈ Grp ∧ ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) + 𝑌 ) = ( 𝑍 + 𝑌 ) ↔ ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 𝑍 ) )
18 9 14 15 16 17 syl13anc ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) + 𝑌 ) = ( 𝑍 + 𝑌 ) ↔ ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 𝑍 ) )
19 1 2 grpass ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) + 𝑌 ) = ( 𝑋 + ( ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) + 𝑌 ) ) )
20 9 10 12 16 19 syl13anc ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) + 𝑌 ) = ( 𝑋 + ( ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) + 𝑌 ) ) )
21 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
22 1 2 21 4 grplinv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) + 𝑌 ) = ( 0g ‘ 𝐺 ) )
23 22 3ad2antr2 ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) + 𝑌 ) = ( 0g ‘ 𝐺 ) )
24 23 oveq2d ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 + ( ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) + 𝑌 ) ) = ( 𝑋 + ( 0g ‘ 𝐺 ) ) )
25 1 2 21 grprid ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 + ( 0g ‘ 𝐺 ) ) = 𝑋 )
26 25 3ad2antr1 ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 + ( 0g ‘ 𝐺 ) ) = 𝑋 )
27 20 24 26 3eqtrd ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) + 𝑌 ) = 𝑋 )
28 27 eqeq1d ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) + 𝑌 ) = ( 𝑍 + 𝑌 ) ↔ 𝑋 = ( 𝑍 + 𝑌 ) ) )
29 8 18 28 3bitr2d ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) = 𝑍 ↔ 𝑋 = ( 𝑍 + 𝑌 ) ) )
30 eqcom ⊢ ( 𝑋 = ( 𝑍 + 𝑌 ) ↔ ( 𝑍 + 𝑌 ) = 𝑋 )
31 29 30 bitrdi ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) = 𝑍 ↔ ( 𝑍 + 𝑌 ) = 𝑋 ) )