Step |
Hyp |
Ref |
Expression |
1 |
|
frgpup.b |
⊢ 𝐵 = ( Base ‘ 𝐻 ) |
2 |
|
frgpup.n |
⊢ 𝑁 = ( invg ‘ 𝐻 ) |
3 |
|
frgpup.t |
⊢ 𝑇 = ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( 𝑁 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
4 |
|
frgpup.h |
⊢ ( 𝜑 → 𝐻 ∈ Grp ) |
5 |
|
frgpup.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
6 |
|
frgpup.a |
⊢ ( 𝜑 → 𝐹 : 𝐼 ⟶ 𝐵 ) |
7 |
6
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ) |
8 |
7
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ) |
9 |
1 2
|
grpinvcl |
⊢ ( ( 𝐻 ∈ Grp ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝐹 ‘ 𝑦 ) ) ∈ 𝐵 ) |
10 |
4 8 9
|
syl2an2r |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o ) ) → ( 𝑁 ‘ ( 𝐹 ‘ 𝑦 ) ) ∈ 𝐵 ) |
11 |
8 10
|
ifcld |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o ) ) → if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( 𝑁 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ∈ 𝐵 ) |
12 |
11
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐼 ∀ 𝑧 ∈ 2o if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( 𝑁 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ∈ 𝐵 ) |
13 |
3
|
fmpo |
⊢ ( ∀ 𝑦 ∈ 𝐼 ∀ 𝑧 ∈ 2o if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( 𝑁 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ∈ 𝐵 ↔ 𝑇 : ( 𝐼 × 2o ) ⟶ 𝐵 ) |
14 |
12 13
|
sylib |
⊢ ( 𝜑 → 𝑇 : ( 𝐼 × 2o ) ⟶ 𝐵 ) |