Step |
Hyp |
Ref |
Expression |
1 |
|
grpvlinv.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpvlinv.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grpvlinv.n |
⊢ 𝑁 = ( invg ‘ 𝐺 ) |
4 |
|
grpvlinv.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
5 |
|
simpll |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑥 ∈ 𝐼 ) → 𝐺 ∈ Grp ) |
6 |
|
elmapi |
⊢ ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) → 𝑋 : 𝐼 ⟶ 𝐵 ) |
7 |
6
|
adantl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝑋 : 𝐼 ⟶ 𝐵 ) |
8 |
7
|
ffvelrnda |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑥 ) ∈ 𝐵 ) |
9 |
1 2 4 3
|
grprinv |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ‘ 𝑥 ) ∈ 𝐵 ) → ( ( 𝑋 ‘ 𝑥 ) + ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ) = 0 ) |
10 |
5 8 9
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝑋 ‘ 𝑥 ) + ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ) = 0 ) |
11 |
10
|
mpteq2dva |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑋 ‘ 𝑥 ) + ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ 𝐼 ↦ 0 ) ) |
12 |
|
elmapex |
⊢ ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) → ( 𝐵 ∈ V ∧ 𝐼 ∈ V ) ) |
13 |
12
|
simprd |
⊢ ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) → 𝐼 ∈ V ) |
14 |
13
|
adantl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝐼 ∈ V ) |
15 |
|
fvexd |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ∈ V ) |
16 |
7
|
feqmptd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝑋 = ( 𝑥 ∈ 𝐼 ↦ ( 𝑋 ‘ 𝑥 ) ) ) |
17 |
1 3
|
grpinvf |
⊢ ( 𝐺 ∈ Grp → 𝑁 : 𝐵 ⟶ 𝐵 ) |
18 |
|
fcompt |
⊢ ( ( 𝑁 : 𝐵 ⟶ 𝐵 ∧ 𝑋 : 𝐼 ⟶ 𝐵 ) → ( 𝑁 ∘ 𝑋 ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ) ) |
19 |
17 6 18
|
syl2an |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝑁 ∘ 𝑋 ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ) ) |
20 |
14 8 15 16 19
|
offval2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝑋 ∘f + ( 𝑁 ∘ 𝑋 ) ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑋 ‘ 𝑥 ) + ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ) ) ) |
21 |
|
fconstmpt |
⊢ ( 𝐼 × { 0 } ) = ( 𝑥 ∈ 𝐼 ↦ 0 ) |
22 |
21
|
a1i |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝐼 × { 0 } ) = ( 𝑥 ∈ 𝐼 ↦ 0 ) ) |
23 |
11 20 22
|
3eqtr4d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝑋 ∘f + ( 𝑁 ∘ 𝑋 ) ) = ( 𝐼 × { 0 } ) ) |