Step |
Hyp |
Ref |
Expression |
1 |
|
grpsubadd.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpsubadd.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grpsubadd.m |
⊢ − = ( -g ‘ 𝐺 ) |
4 |
|
simp1 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐺 ∈ Grp ) |
5 |
|
simp2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
6 |
|
simp3 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 ) |
7 |
1 2 3
|
grpaddsubass |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) − 𝑌 ) = ( 𝑋 + ( 𝑌 − 𝑌 ) ) ) |
8 |
4 5 6 6 7
|
syl13anc |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + 𝑌 ) − 𝑌 ) = ( 𝑋 + ( 𝑌 − 𝑌 ) ) ) |
9 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
10 |
1 9 3
|
grpsubid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 − 𝑌 ) = ( 0g ‘ 𝐺 ) ) |
11 |
10
|
oveq2d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + ( 𝑌 − 𝑌 ) ) = ( 𝑋 + ( 0g ‘ 𝐺 ) ) ) |
12 |
11
|
3adant2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + ( 𝑌 − 𝑌 ) ) = ( 𝑋 + ( 0g ‘ 𝐺 ) ) ) |
13 |
1 2 9
|
grprid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 + ( 0g ‘ 𝐺 ) ) = 𝑋 ) |
14 |
13
|
3adant3 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + ( 0g ‘ 𝐺 ) ) = 𝑋 ) |
15 |
8 12 14
|
3eqtrd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + 𝑌 ) − 𝑌 ) = 𝑋 ) |