Step |
Hyp |
Ref |
Expression |
1 |
|
ablinvadd.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
ablinvadd.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
ablinvadd.n |
⊢ 𝑁 = ( invg ‘ 𝐺 ) |
4 |
|
ablgrp |
⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp ) |
5 |
1 2 3
|
grpinvadd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝑁 ‘ 𝑌 ) + ( 𝑁 ‘ 𝑋 ) ) ) |
6 |
4 5
|
syl3an1 |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝑁 ‘ 𝑌 ) + ( 𝑁 ‘ 𝑋 ) ) ) |
7 |
|
simp1 |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐺 ∈ Abel ) |
8 |
4
|
3ad2ant1 |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐺 ∈ Grp ) |
9 |
|
simp2 |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
10 |
1 3
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 ) |
11 |
8 9 10
|
syl2anc |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 ) |
12 |
|
simp3 |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 ) |
13 |
1 3
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ) |
14 |
8 12 13
|
syl2anc |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ) |
15 |
1 2
|
ablcom |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑋 ) + ( 𝑁 ‘ 𝑌 ) ) = ( ( 𝑁 ‘ 𝑌 ) + ( 𝑁 ‘ 𝑋 ) ) ) |
16 |
7 11 14 15
|
syl3anc |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑋 ) + ( 𝑁 ‘ 𝑌 ) ) = ( ( 𝑁 ‘ 𝑌 ) + ( 𝑁 ‘ 𝑋 ) ) ) |
17 |
6 16
|
eqtr4d |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝑁 ‘ 𝑋 ) + ( 𝑁 ‘ 𝑌 ) ) ) |