| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | frgpnabl.a | . 2
⊢ (𝜑 → 𝐴 ∈ 𝐼) | 
| 2 |  | 0ex 5307 | . . 3
⊢ ∅
∈ V | 
| 3 | 2 | a1i 11 | . 2
⊢ (𝜑 → ∅ ∈
V) | 
| 4 |  | frgpnabl.d | . . . . . . . 8
⊢ 𝐷 = (𝑊 ∖ ∪
𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) | 
| 5 |  | difss 4136 | . . . . . . . 8
⊢ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) ⊆ 𝑊 | 
| 6 | 4, 5 | eqsstri 4030 | . . . . . . 7
⊢ 𝐷 ⊆ 𝑊 | 
| 7 |  | frgpnabl.g | . . . . . . . . 9
⊢ 𝐺 = (freeGrp‘𝐼) | 
| 8 |  | frgpnabl.w | . . . . . . . . 9
⊢ 𝑊 = ( I ‘Word (𝐼 ×
2o)) | 
| 9 |  | frgpnabl.r | . . . . . . . . 9
⊢  ∼ = (
~FG ‘𝐼) | 
| 10 |  | frgpnabl.p | . . . . . . . . 9
⊢  + =
(+g‘𝐺) | 
| 11 |  | frgpnabl.m | . . . . . . . . 9
⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | 
| 12 |  | frgpnabl.t | . . . . . . . . 9
⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) | 
| 13 |  | frgpnabl.u | . . . . . . . . 9
⊢ 𝑈 =
(varFGrp‘𝐼) | 
| 14 |  | frgpnabl.i | . . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈ 𝑉) | 
| 15 |  | frgpnabl.b | . . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ 𝐼) | 
| 16 | 7, 8, 9, 10, 11, 12, 4, 13, 14, 15, 1 | frgpnabllem1 19891 | . . . . . . . 8
⊢ (𝜑 → 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
∈ (𝐷 ∩ ((𝑈‘𝐵) + (𝑈‘𝐴)))) | 
| 17 | 16 | elin1d 4204 | . . . . . . 7
⊢ (𝜑 → 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
∈ 𝐷) | 
| 18 | 6, 17 | sselid 3981 | . . . . . 6
⊢ (𝜑 → 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
∈ 𝑊) | 
| 19 |  | eqid 2737 | . . . . . . 7
⊢ (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈
(1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈
(1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) | 
| 20 | 8, 9, 11, 12, 4, 19 | efgredeu 19770 | . . . . . 6
⊢
(〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉 ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) | 
| 21 |  | reurmo 3383 | . . . . . 6
⊢
(∃!𝑑 ∈
𝐷 𝑑 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉 → ∃*𝑑 ∈ 𝐷 𝑑 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) | 
| 22 | 18, 20, 21 | 3syl 18 | . . . . 5
⊢ (𝜑 → ∃*𝑑 ∈ 𝐷 𝑑 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) | 
| 23 | 7, 8, 9, 10, 11, 12, 4, 13, 14, 1, 15 | frgpnabllem1 19891 | . . . . . 6
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵)))) | 
| 24 | 23 | elin1d 4204 | . . . . 5
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ 𝐷) | 
| 25 | 8, 9 | efger 19736 | . . . . . . . . 9
⊢  ∼ Er
𝑊 | 
| 26 | 25 | a1i 11 | . . . . . . . 8
⊢ (𝜑 → ∼ Er 𝑊) | 
| 27 | 7 | frgpgrp 19780 | . . . . . . . . . . 11
⊢ (𝐼 ∈ 𝑉 → 𝐺 ∈ Grp) | 
| 28 | 14, 27 | syl 17 | . . . . . . . . . 10
⊢ (𝜑 → 𝐺 ∈ Grp) | 
| 29 |  | eqid 2737 | . . . . . . . . . . . . 13
⊢
(Base‘𝐺) =
(Base‘𝐺) | 
| 30 | 9, 13, 7, 29 | vrgpf 19786 | . . . . . . . . . . . 12
⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶(Base‘𝐺)) | 
| 31 | 14, 30 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑈:𝐼⟶(Base‘𝐺)) | 
| 32 | 31, 1 | ffvelcdmd 7105 | . . . . . . . . . 10
⊢ (𝜑 → (𝑈‘𝐴) ∈ (Base‘𝐺)) | 
| 33 | 31, 15 | ffvelcdmd 7105 | . . . . . . . . . 10
⊢ (𝜑 → (𝑈‘𝐵) ∈ (Base‘𝐺)) | 
| 34 | 29, 10 | grpcl 18959 | . . . . . . . . . 10
⊢ ((𝐺 ∈ Grp ∧ (𝑈‘𝐴) ∈ (Base‘𝐺) ∧ (𝑈‘𝐵) ∈ (Base‘𝐺)) → ((𝑈‘𝐴) + (𝑈‘𝐵)) ∈ (Base‘𝐺)) | 
| 35 | 28, 32, 33, 34 | syl3anc 1373 | . . . . . . . . 9
⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) ∈ (Base‘𝐺)) | 
| 36 |  | eqid 2737 | . . . . . . . . . . . 12
⊢
(freeMnd‘(𝐼
× 2o)) = (freeMnd‘(𝐼 × 2o)) | 
| 37 | 7, 36, 9 | frgpval 19776 | . . . . . . . . . . 11
⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o))
/s ∼ )) | 
| 38 | 14, 37 | syl 17 | . . . . . . . . . 10
⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2o))
/s ∼ )) | 
| 39 |  | 2on 8520 | . . . . . . . . . . . . . 14
⊢
2o ∈ On | 
| 40 |  | xpexg 7770 | . . . . . . . . . . . . . 14
⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) →
(𝐼 × 2o)
∈ V) | 
| 41 | 14, 39, 40 | sylancl 586 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝐼 × 2o) ∈
V) | 
| 42 |  | wrdexg 14562 | . . . . . . . . . . . . 13
⊢ ((𝐼 × 2o) ∈ V
→ Word (𝐼 ×
2o) ∈ V) | 
| 43 |  | fvi 6985 | . . . . . . . . . . . . 13
⊢ (Word
(𝐼 × 2o)
∈ V → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 ×
2o)) | 
| 44 | 41, 42, 43 | 3syl 18 | . . . . . . . . . . . 12
⊢ (𝜑 → ( I ‘Word (𝐼 × 2o)) = Word
(𝐼 ×
2o)) | 
| 45 | 8, 44 | eqtrid 2789 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o)) | 
| 46 |  | eqid 2737 | . . . . . . . . . . . . 13
⊢
(Base‘(freeMnd‘(𝐼 × 2o))) =
(Base‘(freeMnd‘(𝐼 × 2o))) | 
| 47 | 36, 46 | frmdbas 18865 | . . . . . . . . . . . 12
⊢ ((𝐼 × 2o) ∈ V
→ (Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 ×
2o)) | 
| 48 | 41, 47 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 →
(Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 ×
2o)) | 
| 49 | 45, 48 | eqtr4d 2780 | . . . . . . . . . 10
⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 ×
2o)))) | 
| 50 | 9 | fvexi 6920 | . . . . . . . . . . 11
⊢  ∼ ∈
V | 
| 51 | 50 | a1i 11 | . . . . . . . . . 10
⊢ (𝜑 → ∼ ∈
V) | 
| 52 |  | fvexd 6921 | . . . . . . . . . 10
⊢ (𝜑 → (freeMnd‘(𝐼 × 2o)) ∈
V) | 
| 53 | 38, 49, 51, 52 | qusbas 17590 | . . . . . . . . 9
⊢ (𝜑 → (𝑊 / ∼ ) =
(Base‘𝐺)) | 
| 54 | 35, 53 | eleqtrrd 2844 | . . . . . . . 8
⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) ∈ (𝑊 / ∼ )) | 
| 55 | 23 | elin2d 4205 | . . . . . . . 8
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ ((𝑈‘𝐴) + (𝑈‘𝐵))) | 
| 56 |  | qsel 8836 | . . . . . . . 8
⊢ (( ∼ Er
𝑊 ∧ ((𝑈‘𝐴) + (𝑈‘𝐵)) ∈ (𝑊 / ∼ ) ∧
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 ∈ ((𝑈‘𝐴) + (𝑈‘𝐵))) → ((𝑈‘𝐴) + (𝑈‘𝐵)) = [〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉] ∼
) | 
| 57 | 26, 54, 55, 56 | syl3anc 1373 | . . . . . . 7
⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = [〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉] ∼
) | 
| 58 | 16 | elin2d 4205 | . . . . . . . . 9
⊢ (𝜑 → 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
∈ ((𝑈‘𝐵) + (𝑈‘𝐴))) | 
| 59 |  | frgpnabl.n | . . . . . . . . 9
⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = ((𝑈‘𝐵) + (𝑈‘𝐴))) | 
| 60 | 58, 59 | eleqtrrd 2844 | . . . . . . . 8
⊢ (𝜑 → 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
∈ ((𝑈‘𝐴) + (𝑈‘𝐵))) | 
| 61 |  | qsel 8836 | . . . . . . . 8
⊢ (( ∼ Er
𝑊 ∧ ((𝑈‘𝐴) + (𝑈‘𝐵)) ∈ (𝑊 / ∼ ) ∧
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉 ∈ ((𝑈‘𝐴) + (𝑈‘𝐵))) → ((𝑈‘𝐴) + (𝑈‘𝐵)) = [〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉] ∼
) | 
| 62 | 26, 54, 60, 61 | syl3anc 1373 | . . . . . . 7
⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = [〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉] ∼
) | 
| 63 | 57, 62 | eqtr3d 2779 | . . . . . 6
⊢ (𝜑 → [〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉]
∼
= [〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉] ∼ ) | 
| 64 | 6, 24 | sselid 3981 | . . . . . . 7
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ 𝑊) | 
| 65 | 26, 64 | erth 8796 | . . . . . 6
⊢ (𝜑 → (〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉 ↔ [〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉] ∼ =
[〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉] ∼ )) | 
| 66 | 63, 65 | mpbird 257 | . . . . 5
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) | 
| 67 | 26, 18 | erref 8765 | . . . . 5
⊢ (𝜑 → 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) | 
| 68 |  | breq1 5146 | . . . . . 6
⊢ (𝑑 = 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
→ (𝑑 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉 ↔ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉)) | 
| 69 |  | breq1 5146 | . . . . . 6
⊢ (𝑑 = 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
→ (𝑑 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉 ↔ 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉)) | 
| 70 | 68, 69 | rmoi 3891 | . . . . 5
⊢
((∃*𝑑 ∈
𝐷 𝑑 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉 ∧ (〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ 𝐷 ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) ∧ (〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉 ∈ 𝐷 ∧ 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉)) → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 =
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) | 
| 71 | 22, 24, 66, 17, 67, 70 | syl122anc 1381 | . . . 4
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 =
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) | 
| 72 | 71 | fveq1d 6908 | . . 3
⊢ (𝜑 → (〈“〈𝐴, ∅〉〈𝐵,
∅〉”〉‘0) = (〈“〈𝐵, ∅〉〈𝐴,
∅〉”〉‘0)) | 
| 73 |  | opex 5469 | . . . 4
⊢
〈𝐴,
∅〉 ∈ V | 
| 74 |  | s2fv0 14926 | . . . 4
⊢
(〈𝐴,
∅〉 ∈ V → (〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉‘0) =
〈𝐴,
∅〉) | 
| 75 | 73, 74 | ax-mp 5 | . . 3
⊢
(〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉‘0) =
〈𝐴,
∅〉 | 
| 76 |  | opex 5469 | . . . 4
⊢
〈𝐵,
∅〉 ∈ V | 
| 77 |  | s2fv0 14926 | . . . 4
⊢
(〈𝐵,
∅〉 ∈ V → (〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉‘0) =
〈𝐵,
∅〉) | 
| 78 | 76, 77 | ax-mp 5 | . . 3
⊢
(〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉‘0) =
〈𝐵,
∅〉 | 
| 79 | 72, 75, 78 | 3eqtr3g 2800 | . 2
⊢ (𝜑 → 〈𝐴, ∅〉 = 〈𝐵, ∅〉) | 
| 80 |  | opthg 5482 | . . 3
⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ V) → (〈𝐴, ∅〉 = 〈𝐵, ∅〉 ↔ (𝐴 = 𝐵 ∧ ∅ = ∅))) | 
| 81 | 80 | simprbda 498 | . 2
⊢ (((𝐴 ∈ 𝐼 ∧ ∅ ∈ V) ∧ 〈𝐴, ∅〉 = 〈𝐵, ∅〉) → 𝐴 = 𝐵) | 
| 82 | 1, 3, 79, 81 | syl21anc 838 | 1
⊢ (𝜑 → 𝐴 = 𝐵) |