Step |
Hyp |
Ref |
Expression |
1 |
|
grpsubadd.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpsubadd.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grpsubadd.m |
⊢ − = ( -g ‘ 𝐺 ) |
4 |
|
simpl |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐺 ∈ Grp ) |
5 |
1 2
|
grpcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑋 + 𝑍 ) ∈ 𝐵 ) |
6 |
5
|
3adant3r2 |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 + 𝑍 ) ∈ 𝐵 ) |
7 |
|
simpr3 |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 ) |
8 |
|
simpr2 |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 ) |
9 |
1 2 3
|
grpsubsub4 |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( 𝑋 + 𝑍 ) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝑋 + 𝑍 ) − 𝑍 ) − 𝑌 ) = ( ( 𝑋 + 𝑍 ) − ( 𝑌 + 𝑍 ) ) ) |
10 |
4 6 7 8 9
|
syl13anc |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( ( 𝑋 + 𝑍 ) − 𝑍 ) − 𝑌 ) = ( ( 𝑋 + 𝑍 ) − ( 𝑌 + 𝑍 ) ) ) |
11 |
1 2 3
|
grppncan |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝑋 + 𝑍 ) − 𝑍 ) = 𝑋 ) |
12 |
11
|
3adant3r2 |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑍 ) − 𝑍 ) = 𝑋 ) |
13 |
12
|
oveq1d |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( ( 𝑋 + 𝑍 ) − 𝑍 ) − 𝑌 ) = ( 𝑋 − 𝑌 ) ) |
14 |
10 13
|
eqtr3d |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑍 ) − ( 𝑌 + 𝑍 ) ) = ( 𝑋 − 𝑌 ) ) |