Step |
Hyp |
Ref |
Expression |
1 |
|
grpsubadd.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpsubadd.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grpsubadd.m |
⊢ − = ( -g ‘ 𝐺 ) |
4 |
|
grpnpncan0.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
5 |
|
simpl |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐺 ∈ Grp ) |
6 |
|
simprl |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 ) |
7 |
|
simprr |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 ) |
8 |
1 2 3
|
grpnpncan |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) + ( 𝑌 − 𝑋 ) ) = ( 𝑋 − 𝑋 ) ) |
9 |
5 6 7 6 8
|
syl13anc |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) + ( 𝑌 − 𝑋 ) ) = ( 𝑋 − 𝑋 ) ) |
10 |
1 4 3
|
grpsubid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 − 𝑋 ) = 0 ) |
11 |
10
|
adantrr |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 − 𝑋 ) = 0 ) |
12 |
9 11
|
eqtrd |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) + ( 𝑌 − 𝑋 ) ) = 0 ) |