Step |
Hyp |
Ref |
Expression |
1 |
|
frgpup3.g |
⊢ 𝐺 = ( freeGrp ‘ 𝐼 ) |
2 |
|
frgpup3.b |
⊢ 𝐵 = ( Base ‘ 𝐻 ) |
3 |
|
frgpup3.u |
⊢ 𝑈 = ( varFGrp ‘ 𝐼 ) |
4 |
|
eqid |
⊢ ( invg ‘ 𝐻 ) = ( invg ‘ 𝐻 ) |
5 |
|
eqid |
⊢ ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) = ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
6 |
|
simp1 |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) → 𝐻 ∈ Grp ) |
7 |
|
simp2 |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) → 𝐼 ∈ 𝑉 ) |
8 |
|
simp3 |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) → 𝐹 : 𝐼 ⟶ 𝐵 ) |
9 |
|
eqid |
⊢ ( I ‘ Word ( 𝐼 × 2o ) ) = ( I ‘ Word ( 𝐼 × 2o ) ) |
10 |
|
eqid |
⊢ ( ~FG ‘ 𝐼 ) = ( ~FG ‘ 𝐼 ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
12 |
|
eqid |
⊢ ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) = ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) |
13 |
2 4 5 6 7 8 9 10 1 11 12
|
frgpup1 |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) → ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ∈ ( 𝐺 GrpHom 𝐻 ) ) |
14 |
6
|
adantr |
⊢ ( ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) ∧ 𝑘 ∈ 𝐼 ) → 𝐻 ∈ Grp ) |
15 |
7
|
adantr |
⊢ ( ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) ∧ 𝑘 ∈ 𝐼 ) → 𝐼 ∈ 𝑉 ) |
16 |
8
|
adantr |
⊢ ( ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) ∧ 𝑘 ∈ 𝐼 ) → 𝐹 : 𝐼 ⟶ 𝐵 ) |
17 |
|
simpr |
⊢ ( ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) ∧ 𝑘 ∈ 𝐼 ) → 𝑘 ∈ 𝐼 ) |
18 |
2 4 5 14 15 16 9 10 1 11 12 3 17
|
frgpup2 |
⊢ ( ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) ∧ 𝑘 ∈ 𝐼 ) → ( ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ‘ ( 𝑈 ‘ 𝑘 ) ) = ( 𝐹 ‘ 𝑘 ) ) |
19 |
18
|
mpteq2dva |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) → ( 𝑘 ∈ 𝐼 ↦ ( ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ‘ ( 𝑈 ‘ 𝑘 ) ) ) = ( 𝑘 ∈ 𝐼 ↦ ( 𝐹 ‘ 𝑘 ) ) ) |
20 |
11 2
|
ghmf |
⊢ ( ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ∈ ( 𝐺 GrpHom 𝐻 ) → ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) : ( Base ‘ 𝐺 ) ⟶ 𝐵 ) |
21 |
13 20
|
syl |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) → ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) : ( Base ‘ 𝐺 ) ⟶ 𝐵 ) |
22 |
10 3 1 11
|
vrgpf |
⊢ ( 𝐼 ∈ 𝑉 → 𝑈 : 𝐼 ⟶ ( Base ‘ 𝐺 ) ) |
23 |
7 22
|
syl |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) → 𝑈 : 𝐼 ⟶ ( Base ‘ 𝐺 ) ) |
24 |
|
fcompt |
⊢ ( ( ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) : ( Base ‘ 𝐺 ) ⟶ 𝐵 ∧ 𝑈 : 𝐼 ⟶ ( Base ‘ 𝐺 ) ) → ( ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ∘ 𝑈 ) = ( 𝑘 ∈ 𝐼 ↦ ( ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ‘ ( 𝑈 ‘ 𝑘 ) ) ) ) |
25 |
21 23 24
|
syl2anc |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) → ( ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ∘ 𝑈 ) = ( 𝑘 ∈ 𝐼 ↦ ( ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ‘ ( 𝑈 ‘ 𝑘 ) ) ) ) |
26 |
8
|
feqmptd |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) → 𝐹 = ( 𝑘 ∈ 𝐼 ↦ ( 𝐹 ‘ 𝑘 ) ) ) |
27 |
19 25 26
|
3eqtr4d |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) → ( ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ∘ 𝑈 ) = 𝐹 ) |
28 |
6
|
adantr |
⊢ ( ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) ∧ ( 𝑚 ∈ ( 𝐺 GrpHom 𝐻 ) ∧ ( 𝑚 ∘ 𝑈 ) = 𝐹 ) ) → 𝐻 ∈ Grp ) |
29 |
7
|
adantr |
⊢ ( ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) ∧ ( 𝑚 ∈ ( 𝐺 GrpHom 𝐻 ) ∧ ( 𝑚 ∘ 𝑈 ) = 𝐹 ) ) → 𝐼 ∈ 𝑉 ) |
30 |
8
|
adantr |
⊢ ( ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) ∧ ( 𝑚 ∈ ( 𝐺 GrpHom 𝐻 ) ∧ ( 𝑚 ∘ 𝑈 ) = 𝐹 ) ) → 𝐹 : 𝐼 ⟶ 𝐵 ) |
31 |
|
simprl |
⊢ ( ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) ∧ ( 𝑚 ∈ ( 𝐺 GrpHom 𝐻 ) ∧ ( 𝑚 ∘ 𝑈 ) = 𝐹 ) ) → 𝑚 ∈ ( 𝐺 GrpHom 𝐻 ) ) |
32 |
|
simprr |
⊢ ( ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) ∧ ( 𝑚 ∈ ( 𝐺 GrpHom 𝐻 ) ∧ ( 𝑚 ∘ 𝑈 ) = 𝐹 ) ) → ( 𝑚 ∘ 𝑈 ) = 𝐹 ) |
33 |
2 4 5 28 29 30 9 10 1 11 12 3 31 32
|
frgpup3lem |
⊢ ( ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) ∧ ( 𝑚 ∈ ( 𝐺 GrpHom 𝐻 ) ∧ ( 𝑚 ∘ 𝑈 ) = 𝐹 ) ) → 𝑚 = ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ) |
34 |
33
|
expr |
⊢ ( ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) ∧ 𝑚 ∈ ( 𝐺 GrpHom 𝐻 ) ) → ( ( 𝑚 ∘ 𝑈 ) = 𝐹 → 𝑚 = ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ) ) |
35 |
34
|
ralrimiva |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) → ∀ 𝑚 ∈ ( 𝐺 GrpHom 𝐻 ) ( ( 𝑚 ∘ 𝑈 ) = 𝐹 → 𝑚 = ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ) ) |
36 |
|
coeq1 |
⊢ ( 𝑚 = ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) → ( 𝑚 ∘ 𝑈 ) = ( ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ∘ 𝑈 ) ) |
37 |
36
|
eqeq1d |
⊢ ( 𝑚 = ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) → ( ( 𝑚 ∘ 𝑈 ) = 𝐹 ↔ ( ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ∘ 𝑈 ) = 𝐹 ) ) |
38 |
37
|
eqreu |
⊢ ( ( ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ∈ ( 𝐺 GrpHom 𝐻 ) ∧ ( ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ∘ 𝑈 ) = 𝐹 ∧ ∀ 𝑚 ∈ ( 𝐺 GrpHom 𝐻 ) ( ( 𝑚 ∘ 𝑈 ) = 𝐹 → 𝑚 = ran ( 𝑔 ∈ ( I ‘ Word ( 𝐼 × 2o ) ) ↦ 〈 [ 𝑔 ] ( ~FG ‘ 𝐼 ) , ( 𝐻 Σg ( ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹 ‘ 𝑦 ) , ( ( invg ‘ 𝐻 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ∘ 𝑔 ) ) 〉 ) ) ) → ∃! 𝑚 ∈ ( 𝐺 GrpHom 𝐻 ) ( 𝑚 ∘ 𝑈 ) = 𝐹 ) |
39 |
13 27 35 38
|
syl3anc |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ 𝐵 ) → ∃! 𝑚 ∈ ( 𝐺 GrpHom 𝐻 ) ( 𝑚 ∘ 𝑈 ) = 𝐹 ) |