| Step | Hyp | Ref
| Expression |
| 1 | | frgpup3.b |
. . 3
⊢ 𝐵 = (Base‘𝐻) |
| 2 | | eqid 2737 |
. . 3
⊢
(invg‘𝐻) = (invg‘𝐻) |
| 3 | | eqid 2737 |
. . 3
⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), ((invg‘𝐻)‘(𝐹‘𝑦)))) = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), ((invg‘𝐻)‘(𝐹‘𝑦)))) |
| 4 | | simp1 1137 |
. . 3
⊢ ((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) → 𝐻 ∈ Grp) |
| 5 | | simp2 1138 |
. . 3
⊢ ((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) → 𝐼 ∈ 𝑉) |
| 6 | | simp3 1139 |
. . 3
⊢ ((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) → 𝐹:𝐼⟶𝐵) |
| 7 | | eqid 2737 |
. . 3
⊢ ( I
‘Word (𝐼 ×
2o)) = ( I ‘Word (𝐼 × 2o)) |
| 8 | | eqid 2737 |
. . 3
⊢ (
~FG ‘𝐼) = ( ~FG ‘𝐼) |
| 9 | | frgpup3.g |
. . 3
⊢ 𝐺 = (freeGrp‘𝐼) |
| 10 | | eqid 2737 |
. . 3
⊢
(Base‘𝐺) =
(Base‘𝐺) |
| 11 | | eqid 2737 |
. . 3
⊢ 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))〉) |
| 12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11 | frgpup1 19793 |
. 2
⊢ ((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)) |
| 13 | 4 | adantr 480 |
. . . . 5
⊢ (((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑘 ∈ 𝐼) → 𝐻 ∈ Grp) |
| 14 | 5 | adantr 480 |
. . . . 5
⊢ (((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑘 ∈ 𝐼) → 𝐼 ∈ 𝑉) |
| 15 | 6 | adantr 480 |
. . . . 5
⊢ (((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑘 ∈ 𝐼) → 𝐹:𝐼⟶𝐵) |
| 16 | | frgpup3.u |
. . . . 5
⊢ 𝑈 =
(varFGrp‘𝐼) |
| 17 | | simpr 484 |
. . . . 5
⊢ (((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼) |
| 18 | 1, 2, 3, 13, 14, 15, 7, 8, 9,
10, 11, 16, 17 | frgpup2 19794 |
. . . 4
⊢ (((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 5242 |
. . 3
⊢ ((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 | 10, 1 | ghmf 19238 |
. . . . 5
⊢ (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 | 12, 20 | syl 17 |
. . . 4
⊢ ((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 | 8, 16, 9, 10 | vrgpf 19786 |
. . . . 5
⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶(Base‘𝐺)) |
| 23 | 5, 22 | syl 17 |
. . . 4
⊢ ((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) → 𝑈:𝐼⟶(Base‘𝐺)) |
| 24 | | fcompt 7153 |
. . . 4
⊢ ((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 584 |
. . 3
⊢ ((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 | 6 | feqmptd 6977 |
. . 3
⊢ ((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) → 𝐹 = (𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘))) |
| 27 | 19, 25, 26 | 3eqtr4d 2787 |
. 2
⊢ ((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 | 4 | adantr 480 |
. . . . 5
⊢ (((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚 ∘ 𝑈) = 𝐹)) → 𝐻 ∈ Grp) |
| 29 | 5 | adantr 480 |
. . . . 5
⊢ (((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚 ∘ 𝑈) = 𝐹)) → 𝐼 ∈ 𝑉) |
| 30 | 6 | adantr 480 |
. . . . 5
⊢ (((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚 ∘ 𝑈) = 𝐹)) → 𝐹:𝐼⟶𝐵) |
| 31 | | simprl 771 |
. . . . 5
⊢ (((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚 ∘ 𝑈) = 𝐹)) → 𝑚 ∈ (𝐺 GrpHom 𝐻)) |
| 32 | | simprr 773 |
. . . . 5
⊢ (((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) ∧ (𝑚 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑚 ∘ 𝑈) = 𝐹)) → (𝑚 ∘ 𝑈) = 𝐹) |
| 33 | 1, 2, 3, 28, 29, 30, 7, 8, 9,
10, 11, 16, 31, 32 | frgpup3lem 19795 |
. . . 4
⊢ (((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 456 |
. . 3
⊢ (((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 3146 |
. 2
⊢ ((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 5868 |
. . . 4
⊢ (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 2739 |
. . 3
⊢ (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 3735 |
. 2
⊢ ((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 | 12, 27, 35, 38 | syl3anc 1373 |
1
⊢ ((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) → ∃!𝑚 ∈ (𝐺 GrpHom 𝐻)(𝑚 ∘ 𝑈) = 𝐹) |