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 4
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 ) |
6 |
5
|
3adant2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 ) |
7 |
1 2
|
grpcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ∈ 𝐵 ) |
8 |
6 7
|
syld3an3 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ∈ 𝐵 ) |
9 |
1 2 4 3
|
grpsubval |
⊢ ( ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ∈ 𝐵 ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 ) → ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) − ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) + ( ( invg ‘ 𝐺 ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ) ) |
10 |
8 6 9
|
syl2anc |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) − ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) + ( ( invg ‘ 𝐺 ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ) ) |
11 |
1 2 3
|
grppncan |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 ) → ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) − ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 𝑋 ) |
12 |
6 11
|
syld3an3 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) − ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 𝑋 ) |
13 |
1 2 4 3
|
grpsubval |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ) |
14 |
13
|
3adant1 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ) |
15 |
14
|
eqcomd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = ( 𝑋 − 𝑌 ) ) |
16 |
1 4
|
grpinvinv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 𝑌 ) |
17 |
16
|
3adant2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = 𝑌 ) |
18 |
15 17
|
oveq12d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) + ( ( invg ‘ 𝐺 ) ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ) = ( ( 𝑋 − 𝑌 ) + 𝑌 ) ) |
19 |
10 12 18
|
3eqtr3rd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 − 𝑌 ) + 𝑌 ) = 𝑋 ) |