Step |
Hyp |
Ref |
Expression |
1 |
|
mulgsubdi.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
mulgsubdi.t |
⊢ · = ( .g ‘ 𝐺 ) |
3 |
|
mulgsubdi.d |
⊢ − = ( -g ‘ 𝐺 ) |
4 |
|
simpl |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐺 ∈ Abel ) |
5 |
|
simpr1 |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑀 ∈ ℤ ) |
6 |
|
simpr2 |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 ) |
7 |
|
ablgrp |
⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp ) |
8 |
7
|
adantr |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐺 ∈ Grp ) |
9 |
|
simpr3 |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 ) |
10 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
11 |
1 10
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 ) |
12 |
8 9 11
|
syl2anc |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 ) |
13 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
14 |
1 2 13
|
mulgdi |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ∈ 𝐵 ) ) → ( 𝑀 · ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ) = ( ( 𝑀 · 𝑋 ) ( +g ‘ 𝐺 ) ( 𝑀 · ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ) ) |
15 |
4 5 6 12 14
|
syl13anc |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑀 · ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ) = ( ( 𝑀 · 𝑋 ) ( +g ‘ 𝐺 ) ( 𝑀 · ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ) ) |
16 |
1 2 10
|
mulginvcom |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑀 ∈ ℤ ∧ 𝑌 ∈ 𝐵 ) → ( 𝑀 · ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = ( ( invg ‘ 𝐺 ) ‘ ( 𝑀 · 𝑌 ) ) ) |
17 |
8 5 9 16
|
syl3anc |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑀 · ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = ( ( invg ‘ 𝐺 ) ‘ ( 𝑀 · 𝑌 ) ) ) |
18 |
17
|
oveq2d |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑀 · 𝑋 ) ( +g ‘ 𝐺 ) ( 𝑀 · ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ) = ( ( 𝑀 · 𝑋 ) ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ ( 𝑀 · 𝑌 ) ) ) ) |
19 |
15 18
|
eqtrd |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑀 · ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ) = ( ( 𝑀 · 𝑋 ) ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ ( 𝑀 · 𝑌 ) ) ) ) |
20 |
1 13 10 3
|
grpsubval |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ) |
21 |
6 9 20
|
syl2anc |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 − 𝑌 ) = ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ) |
22 |
21
|
oveq2d |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑀 · ( 𝑋 − 𝑌 ) ) = ( 𝑀 · ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) ) ) |
23 |
1 2
|
mulgcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 · 𝑋 ) ∈ 𝐵 ) |
24 |
8 5 6 23
|
syl3anc |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑀 · 𝑋 ) ∈ 𝐵 ) |
25 |
1 2
|
mulgcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑀 ∈ ℤ ∧ 𝑌 ∈ 𝐵 ) → ( 𝑀 · 𝑌 ) ∈ 𝐵 ) |
26 |
8 5 9 25
|
syl3anc |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑀 · 𝑌 ) ∈ 𝐵 ) |
27 |
1 13 10 3
|
grpsubval |
⊢ ( ( ( 𝑀 · 𝑋 ) ∈ 𝐵 ∧ ( 𝑀 · 𝑌 ) ∈ 𝐵 ) → ( ( 𝑀 · 𝑋 ) − ( 𝑀 · 𝑌 ) ) = ( ( 𝑀 · 𝑋 ) ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ ( 𝑀 · 𝑌 ) ) ) ) |
28 |
24 26 27
|
syl2anc |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑀 · 𝑋 ) − ( 𝑀 · 𝑌 ) ) = ( ( 𝑀 · 𝑋 ) ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ ( 𝑀 · 𝑌 ) ) ) ) |
29 |
19 22 28
|
3eqtr4d |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑀 · ( 𝑋 − 𝑌 ) ) = ( ( 𝑀 · 𝑋 ) − ( 𝑀 · 𝑌 ) ) ) |