Step |
Hyp |
Ref |
Expression |
1 |
|
ablnncan.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
ablnncan.m |
⊢ − = ( -g ‘ 𝐺 ) |
3 |
|
ablnncan.g |
⊢ ( 𝜑 → 𝐺 ∈ Abel ) |
4 |
|
ablnncan.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
5 |
|
ablnncan.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
6 |
|
ablsub32.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝐵 ) |
7 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
8 |
1 7
|
ablcom |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 ( +g ‘ 𝐺 ) 𝑍 ) = ( 𝑍 ( +g ‘ 𝐺 ) 𝑌 ) ) |
9 |
3 5 6 8
|
syl3anc |
⊢ ( 𝜑 → ( 𝑌 ( +g ‘ 𝐺 ) 𝑍 ) = ( 𝑍 ( +g ‘ 𝐺 ) 𝑌 ) ) |
10 |
9
|
oveq2d |
⊢ ( 𝜑 → ( 𝑋 − ( 𝑌 ( +g ‘ 𝐺 ) 𝑍 ) ) = ( 𝑋 − ( 𝑍 ( +g ‘ 𝐺 ) 𝑌 ) ) ) |
11 |
1 7 2 3 4 5 6
|
ablsubsub4 |
⊢ ( 𝜑 → ( ( 𝑋 − 𝑌 ) − 𝑍 ) = ( 𝑋 − ( 𝑌 ( +g ‘ 𝐺 ) 𝑍 ) ) ) |
12 |
1 7 2 3 4 6 5
|
ablsubsub4 |
⊢ ( 𝜑 → ( ( 𝑋 − 𝑍 ) − 𝑌 ) = ( 𝑋 − ( 𝑍 ( +g ‘ 𝐺 ) 𝑌 ) ) ) |
13 |
10 11 12
|
3eqtr4d |
⊢ ( 𝜑 → ( ( 𝑋 − 𝑌 ) − 𝑍 ) = ( ( 𝑋 − 𝑍 ) − 𝑌 ) ) |