Step |
Hyp |
Ref |
Expression |
1 |
|
mulgnn0z.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
mulgnn0z.t |
⊢ · = ( .g ‘ 𝐺 ) |
3 |
|
mulgnn0z.o |
⊢ 0 = ( 0g ‘ 𝐺 ) |
4 |
|
grpmnd |
⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd ) |
5 |
4
|
adantr |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) → 𝐺 ∈ Mnd ) |
6 |
1 2 3
|
mulgnn0z |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 · 0 ) = 0 ) |
7 |
5 6
|
sylan |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 · 0 ) = 0 ) |
8 |
|
simpll |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) ∧ - 𝑁 ∈ ℕ0 ) → 𝐺 ∈ Grp ) |
9 |
|
nn0z |
⊢ ( - 𝑁 ∈ ℕ0 → - 𝑁 ∈ ℤ ) |
10 |
9
|
adantl |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) ∧ - 𝑁 ∈ ℕ0 ) → - 𝑁 ∈ ℤ ) |
11 |
1 3
|
grpidcl |
⊢ ( 𝐺 ∈ Grp → 0 ∈ 𝐵 ) |
12 |
11
|
ad2antrr |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) ∧ - 𝑁 ∈ ℕ0 ) → 0 ∈ 𝐵 ) |
13 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
14 |
1 2 13
|
mulgneg |
⊢ ( ( 𝐺 ∈ Grp ∧ - 𝑁 ∈ ℤ ∧ 0 ∈ 𝐵 ) → ( - - 𝑁 · 0 ) = ( ( invg ‘ 𝐺 ) ‘ ( - 𝑁 · 0 ) ) ) |
15 |
8 10 12 14
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) ∧ - 𝑁 ∈ ℕ0 ) → ( - - 𝑁 · 0 ) = ( ( invg ‘ 𝐺 ) ‘ ( - 𝑁 · 0 ) ) ) |
16 |
|
zcn |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℂ ) |
17 |
16
|
ad2antlr |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) ∧ - 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ℂ ) |
18 |
17
|
negnegd |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) ∧ - 𝑁 ∈ ℕ0 ) → - - 𝑁 = 𝑁 ) |
19 |
18
|
oveq1d |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) ∧ - 𝑁 ∈ ℕ0 ) → ( - - 𝑁 · 0 ) = ( 𝑁 · 0 ) ) |
20 |
1 2 3
|
mulgnn0z |
⊢ ( ( 𝐺 ∈ Mnd ∧ - 𝑁 ∈ ℕ0 ) → ( - 𝑁 · 0 ) = 0 ) |
21 |
5 20
|
sylan |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) ∧ - 𝑁 ∈ ℕ0 ) → ( - 𝑁 · 0 ) = 0 ) |
22 |
21
|
fveq2d |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) ∧ - 𝑁 ∈ ℕ0 ) → ( ( invg ‘ 𝐺 ) ‘ ( - 𝑁 · 0 ) ) = ( ( invg ‘ 𝐺 ) ‘ 0 ) ) |
23 |
3 13
|
grpinvid |
⊢ ( 𝐺 ∈ Grp → ( ( invg ‘ 𝐺 ) ‘ 0 ) = 0 ) |
24 |
23
|
ad2antrr |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) ∧ - 𝑁 ∈ ℕ0 ) → ( ( invg ‘ 𝐺 ) ‘ 0 ) = 0 ) |
25 |
22 24
|
eqtrd |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) ∧ - 𝑁 ∈ ℕ0 ) → ( ( invg ‘ 𝐺 ) ‘ ( - 𝑁 · 0 ) ) = 0 ) |
26 |
15 19 25
|
3eqtr3d |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) ∧ - 𝑁 ∈ ℕ0 ) → ( 𝑁 · 0 ) = 0 ) |
27 |
|
elznn0 |
⊢ ( 𝑁 ∈ ℤ ↔ ( 𝑁 ∈ ℝ ∧ ( 𝑁 ∈ ℕ0 ∨ - 𝑁 ∈ ℕ0 ) ) ) |
28 |
27
|
simprbi |
⊢ ( 𝑁 ∈ ℤ → ( 𝑁 ∈ ℕ0 ∨ - 𝑁 ∈ ℕ0 ) ) |
29 |
28
|
adantl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ∈ ℕ0 ∨ - 𝑁 ∈ ℕ0 ) ) |
30 |
7 26 29
|
mpjaodan |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) → ( 𝑁 · 0 ) = 0 ) |