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Metamath Proof Explorer

Theorem grpsubeq0

Description: If the difference between two group elements is zero, they are equal. ( subeq0 analog.) (Contributed by NM, 31-Mar-2014)

Ref Expression
Hypotheses grpsubid.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
grpsubid.o ⊢ 0 = ( 0g ‘ 𝐺 )
grpsubid.m ⊢ − = ( -g ‘ 𝐺 )
Assertion grpsubeq0 ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 − 𝑌 ) = 0 ↔ 𝑋 = 𝑌 ) )

Proof

Step Hyp Ref Expression
1 grpsubid.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grpsubid.o ⊢ 0 = ( 0g ‘ 𝐺 )
3 grpsubid.m ⊢ − = ( -g ‘ 𝐺 )
4 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
5 eqid ⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
6 1 4 5 3 grpsubval ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) )
7 6 3adant1 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) )
8 7 eqeq1d ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 − 𝑌 ) = 0 ↔ ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 0 ) )
9 simp1 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐺 ∈ Grp )
10 1 5 grpinvcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 )
11 10 3adant2 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 )
12 simp2 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
13 1 4 2 5 grpinvid2 ⊢ ( ( 𝐺 ∈ Grp ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( ( invg ‘ 𝐺 ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 𝑋 ↔ ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 0 ) )
14 9 11 12 13 syl3anc ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( invg ‘ 𝐺 ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 𝑋 ↔ ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 0 ) )
15 1 5 grpinvinv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 𝑌 )
16 15 3adant2 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 𝑌 )
17 16 eqeq1d ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( invg ‘ 𝐺 ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 𝑋 ↔ 𝑌 = 𝑋 ) )
18 eqcom ⊢ ( 𝑌 = 𝑋 ↔ 𝑋 = 𝑌 )
19 17 18 bitrdi ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( invg ‘ 𝐺 ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 𝑋 ↔ 𝑋 = 𝑌 ) )
20 8 14 19 3bitr2d ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 − 𝑌 ) = 0 ↔ 𝑋 = 𝑌 ) )