Metamath Proof Explorer


Theorem grpinvsub

Description: Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014)

Ref Expression
Hypotheses grpsubcl.b 𝐵 = ( Base ‘ 𝐺 )
grpsubcl.m = ( -g𝐺 )
grpinvsub.n 𝑁 = ( invg𝐺 )
Assertion grpinvsub ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑁 ‘ ( 𝑋 𝑌 ) ) = ( 𝑌 𝑋 ) )

Proof

Step Hyp Ref Expression
1 grpsubcl.b 𝐵 = ( Base ‘ 𝐺 )
2 grpsubcl.m = ( -g𝐺 )
3 grpinvsub.n 𝑁 = ( invg𝐺 )
4 1 3 grpinvcl ( ( 𝐺 ∈ Grp ∧ 𝑌𝐵 ) → ( 𝑁𝑌 ) ∈ 𝐵 )
5 4 3adant2 ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑁𝑌 ) ∈ 𝐵 )
6 eqid ( +g𝐺 ) = ( +g𝐺 )
7 1 6 3 grpinvadd ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ ( 𝑁𝑌 ) ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑋 ( +g𝐺 ) ( 𝑁𝑌 ) ) ) = ( ( 𝑁 ‘ ( 𝑁𝑌 ) ) ( +g𝐺 ) ( 𝑁𝑋 ) ) )
8 5 7 syld3an3 ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑁 ‘ ( 𝑋 ( +g𝐺 ) ( 𝑁𝑌 ) ) ) = ( ( 𝑁 ‘ ( 𝑁𝑌 ) ) ( +g𝐺 ) ( 𝑁𝑋 ) ) )
9 1 3 grpinvinv ( ( 𝐺 ∈ Grp ∧ 𝑌𝐵 ) → ( 𝑁 ‘ ( 𝑁𝑌 ) ) = 𝑌 )
10 9 3adant2 ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑁 ‘ ( 𝑁𝑌 ) ) = 𝑌 )
11 10 oveq1d ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( ( 𝑁 ‘ ( 𝑁𝑌 ) ) ( +g𝐺 ) ( 𝑁𝑋 ) ) = ( 𝑌 ( +g𝐺 ) ( 𝑁𝑋 ) ) )
12 8 11 eqtrd ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑁 ‘ ( 𝑋 ( +g𝐺 ) ( 𝑁𝑌 ) ) ) = ( 𝑌 ( +g𝐺 ) ( 𝑁𝑋 ) ) )
13 1 6 3 2 grpsubval ( ( 𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝑌 ) = ( 𝑋 ( +g𝐺 ) ( 𝑁𝑌 ) ) )
14 13 3adant1 ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝑌 ) = ( 𝑋 ( +g𝐺 ) ( 𝑁𝑌 ) ) )
15 14 fveq2d ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑁 ‘ ( 𝑋 𝑌 ) ) = ( 𝑁 ‘ ( 𝑋 ( +g𝐺 ) ( 𝑁𝑌 ) ) ) )
16 1 6 3 2 grpsubval ( ( 𝑌𝐵𝑋𝐵 ) → ( 𝑌 𝑋 ) = ( 𝑌 ( +g𝐺 ) ( 𝑁𝑋 ) ) )
17 16 ancoms ( ( 𝑋𝐵𝑌𝐵 ) → ( 𝑌 𝑋 ) = ( 𝑌 ( +g𝐺 ) ( 𝑁𝑋 ) ) )
18 17 3adant1 ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑌 𝑋 ) = ( 𝑌 ( +g𝐺 ) ( 𝑁𝑋 ) ) )
19 12 15 18 3eqtr4d ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑁 ‘ ( 𝑋 𝑌 ) ) = ( 𝑌 𝑋 ) )