Step | Hyp | Ref
| Expression |
1 | | frgpnabl.a |
. 2
⊢ (𝜑 → 𝐴 ∈ 𝐼) |
2 | | 0ex 4718 |
. . 3
⊢ ∅
∈ V |
3 | 2 | a1i 11 |
. 2
⊢ (𝜑 → ∅ ∈
V) |
4 | | frgpnabl.d |
. . . . . . . 8
⊢ 𝐷 = (𝑊 ∖ ∪
𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) |
5 | | difss 3699 |
. . . . . . . 8
⊢ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) ⊆ 𝑊 |
6 | 4, 5 | eqsstri 3598 |
. . . . . . 7
⊢ 𝐷 ⊆ 𝑊 |
7 | | inss1 3795 |
. . . . . . . 8
⊢ (𝐷 ∩ ((𝑈‘𝐵) + (𝑈‘𝐴))) ⊆ 𝐷 |
8 | | frgpnabl.g |
. . . . . . . . 9
⊢ 𝐺 = (freeGrp‘𝐼) |
9 | | frgpnabl.w |
. . . . . . . . 9
⊢ 𝑊 = ( I ‘Word (𝐼 ×
2𝑜)) |
10 | | frgpnabl.r |
. . . . . . . . 9
⊢ ∼ = (
~FG ‘𝐼) |
11 | | frgpnabl.p |
. . . . . . . . 9
⊢ + =
(+g‘𝐺) |
12 | | frgpnabl.m |
. . . . . . . . 9
⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦
〈𝑦,
(1𝑜 ∖ 𝑧)〉) |
13 | | frgpnabl.t |
. . . . . . . . 9
⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦
(𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) |
14 | | frgpnabl.u |
. . . . . . . . 9
⊢ 𝑈 =
(varFGrp‘𝐼) |
15 | | frgpnabl.i |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈ V) |
16 | | frgpnabl.b |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ 𝐼) |
17 | 8, 9, 10, 11, 12, 13, 4, 14, 15, 16, 1 | frgpnabllem1 18099 |
. . . . . . . 8
⊢ (𝜑 → 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
∈ (𝐷 ∩ ((𝑈‘𝐵) + (𝑈‘𝐴)))) |
18 | 7, 17 | sseldi 3566 |
. . . . . . 7
⊢ (𝜑 → 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
∈ 𝐷) |
19 | 6, 18 | sseldi 3566 |
. . . . . 6
⊢ (𝜑 → 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
∈ 𝑊) |
20 | | eqid 2610 |
. . . . . . 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))) |
21 | 9, 10, 12, 13, 4, 20 | efgredeu 17988 |
. . . . . 6
⊢
(〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉 ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) |
22 | | reurmo 3138 |
. . . . . 6
⊢
(∃!𝑑 ∈
𝐷 𝑑 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉 → ∃*𝑑 ∈ 𝐷 𝑑 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) |
23 | 19, 21, 22 | 3syl 18 |
. . . . 5
⊢ (𝜑 → ∃*𝑑 ∈ 𝐷 𝑑 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) |
24 | | inss1 3795 |
. . . . . 6
⊢ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵))) ⊆ 𝐷 |
25 | 8, 9, 10, 11, 12, 13, 4, 14, 15, 1, 16 | frgpnabllem1 18099 |
. . . . . 6
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵)))) |
26 | 24, 25 | sseldi 3566 |
. . . . 5
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ 𝐷) |
27 | 9, 10 | efger 17954 |
. . . . . . . . 9
⊢ ∼ Er
𝑊 |
28 | 27 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ∼ Er 𝑊) |
29 | 8 | frgpgrp 17998 |
. . . . . . . . . . 11
⊢ (𝐼 ∈ V → 𝐺 ∈ Grp) |
30 | 15, 29 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 ∈ Grp) |
31 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(Base‘𝐺) =
(Base‘𝐺) |
32 | 10, 14, 8, 31 | vrgpf 18004 |
. . . . . . . . . . . 12
⊢ (𝐼 ∈ V → 𝑈:𝐼⟶(Base‘𝐺)) |
33 | 15, 32 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑈:𝐼⟶(Base‘𝐺)) |
34 | 33, 1 | ffvelrnd 6268 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑈‘𝐴) ∈ (Base‘𝐺)) |
35 | 33, 16 | ffvelrnd 6268 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑈‘𝐵) ∈ (Base‘𝐺)) |
36 | 31, 11 | grpcl 17253 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ Grp ∧ (𝑈‘𝐴) ∈ (Base‘𝐺) ∧ (𝑈‘𝐵) ∈ (Base‘𝐺)) → ((𝑈‘𝐴) + (𝑈‘𝐵)) ∈ (Base‘𝐺)) |
37 | 30, 34, 35, 36 | syl3anc 1318 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) ∈ (Base‘𝐺)) |
38 | | eqid 2610 |
. . . . . . . . . . . 12
⊢
(freeMnd‘(𝐼
× 2𝑜)) = (freeMnd‘(𝐼 ×
2𝑜)) |
39 | 8, 38, 10 | frgpval 17994 |
. . . . . . . . . . 11
⊢ (𝐼 ∈ V → 𝐺 = ((freeMnd‘(𝐼 × 2𝑜))
/s ∼ )) |
40 | 15, 39 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2𝑜))
/s ∼ )) |
41 | | 2on 7455 |
. . . . . . . . . . . . . 14
⊢
2𝑜 ∈ On |
42 | | xpexg 6858 |
. . . . . . . . . . . . . 14
⊢ ((𝐼 ∈ V ∧
2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈
V) |
43 | 15, 41, 42 | sylancl 693 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐼 × 2𝑜) ∈
V) |
44 | | wrdexg 13170 |
. . . . . . . . . . . . 13
⊢ ((𝐼 × 2𝑜)
∈ V → Word (𝐼
× 2𝑜) ∈ V) |
45 | | fvi 6165 |
. . . . . . . . . . . . 13
⊢ (Word
(𝐼 ×
2𝑜) ∈ V → ( I ‘Word (𝐼 × 2𝑜)) = Word
(𝐼 ×
2𝑜)) |
46 | 43, 44, 45 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → ( I ‘Word (𝐼 × 2𝑜))
= Word (𝐼 ×
2𝑜)) |
47 | 9, 46 | syl5eq 2656 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑊 = Word (𝐼 ×
2𝑜)) |
48 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(Base‘(freeMnd‘(𝐼 × 2𝑜))) =
(Base‘(freeMnd‘(𝐼 ×
2𝑜))) |
49 | 38, 48 | frmdbas 17212 |
. . . . . . . . . . . 12
⊢ ((𝐼 × 2𝑜)
∈ V → (Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word
(𝐼 ×
2𝑜)) |
50 | 43, 49 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 →
(Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word
(𝐼 ×
2𝑜)) |
51 | 47, 50 | eqtr4d 2647 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 ×
2𝑜)))) |
52 | | fvex 6113 |
. . . . . . . . . . . 12
⊢ (
~FG ‘𝐼) ∈ V |
53 | 10, 52 | eqeltri 2684 |
. . . . . . . . . . 11
⊢ ∼ ∈
V |
54 | 53 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ∼ ∈
V) |
55 | | fvex 6113 |
. . . . . . . . . . 11
⊢
(freeMnd‘(𝐼
× 2𝑜)) ∈ V |
56 | 55 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → (freeMnd‘(𝐼 × 2𝑜))
∈ V) |
57 | 40, 51, 54, 56 | qusbas 16028 |
. . . . . . . . 9
⊢ (𝜑 → (𝑊 / ∼ ) =
(Base‘𝐺)) |
58 | 37, 57 | eleqtrrd 2691 |
. . . . . . . 8
⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) ∈ (𝑊 / ∼ )) |
59 | | inss2 3796 |
. . . . . . . . 9
⊢ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵))) ⊆ ((𝑈‘𝐴) + (𝑈‘𝐵)) |
60 | 59, 25 | sseldi 3566 |
. . . . . . . 8
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ ((𝑈‘𝐴) + (𝑈‘𝐵))) |
61 | | qsel 7713 |
. . . . . . . 8
⊢ (( ∼ Er
𝑊 ∧ ((𝑈‘𝐴) + (𝑈‘𝐵)) ∈ (𝑊 / ∼ ) ∧
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 ∈ ((𝑈‘𝐴) + (𝑈‘𝐵))) → ((𝑈‘𝐴) + (𝑈‘𝐵)) = [〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉] ∼
) |
62 | 28, 58, 60, 61 | syl3anc 1318 |
. . . . . . 7
⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = [〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉] ∼
) |
63 | | inss2 3796 |
. . . . . . . . . 10
⊢ (𝐷 ∩ ((𝑈‘𝐵) + (𝑈‘𝐴))) ⊆ ((𝑈‘𝐵) + (𝑈‘𝐴)) |
64 | 63, 17 | sseldi 3566 |
. . . . . . . . 9
⊢ (𝜑 → 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
∈ ((𝑈‘𝐵) + (𝑈‘𝐴))) |
65 | | frgpnabl.n |
. . . . . . . . 9
⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = ((𝑈‘𝐵) + (𝑈‘𝐴))) |
66 | 64, 65 | eleqtrrd 2691 |
. . . . . . . 8
⊢ (𝜑 → 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
∈ ((𝑈‘𝐴) + (𝑈‘𝐵))) |
67 | | qsel 7713 |
. . . . . . . 8
⊢ (( ∼ Er
𝑊 ∧ ((𝑈‘𝐴) + (𝑈‘𝐵)) ∈ (𝑊 / ∼ ) ∧
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉 ∈ ((𝑈‘𝐴) + (𝑈‘𝐵))) → ((𝑈‘𝐴) + (𝑈‘𝐵)) = [〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉] ∼
) |
68 | 28, 58, 66, 67 | syl3anc 1318 |
. . . . . . 7
⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = [〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉] ∼
) |
69 | 62, 68 | eqtr3d 2646 |
. . . . . 6
⊢ (𝜑 → [〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉]
∼
= [〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉] ∼ ) |
70 | 6, 26 | sseldi 3566 |
. . . . . . 7
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ 𝑊) |
71 | 28, 70 | erth 7678 |
. . . . . 6
⊢ (𝜑 → (〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉 ↔ [〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉] ∼ =
[〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉] ∼ )) |
72 | 69, 71 | mpbird 246 |
. . . . 5
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) |
73 | 28, 19 | erref 7649 |
. . . . 5
⊢ (𝜑 → 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) |
74 | | breq1 4586 |
. . . . . 6
⊢ (𝑑 = 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
→ (𝑑 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉 ↔ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉)) |
75 | | breq1 4586 |
. . . . . 6
⊢ (𝑑 = 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
→ (𝑑 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉 ↔ 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉)) |
76 | 74, 75 | rmoi 3496 |
. . . . 5
⊢
((∃*𝑑 ∈
𝐷 𝑑 ∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉 ∧ (〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ 𝐷 ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) ∧ (〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉 ∈ 𝐷 ∧ 〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉
∼
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉)) → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 =
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) |
77 | 23, 26, 72, 18, 73, 76 | syl122anc 1327 |
. . . 4
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 =
〈“〈𝐵,
∅〉〈𝐴,
∅〉”〉) |
78 | 77 | fveq1d 6105 |
. . 3
⊢ (𝜑 → (〈“〈𝐴, ∅〉〈𝐵,
∅〉”〉‘0) = (〈“〈𝐵, ∅〉〈𝐴,
∅〉”〉‘0)) |
79 | | opex 4859 |
. . . 4
⊢
〈𝐴,
∅〉 ∈ V |
80 | | s2fv0 13482 |
. . . 4
⊢
(〈𝐴,
∅〉 ∈ V → (〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉‘0) =
〈𝐴,
∅〉) |
81 | 79, 80 | ax-mp 5 |
. . 3
⊢
(〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉‘0) =
〈𝐴,
∅〉 |
82 | | opex 4859 |
. . . 4
⊢
〈𝐵,
∅〉 ∈ V |
83 | | s2fv0 13482 |
. . . 4
⊢
(〈𝐵,
∅〉 ∈ V → (〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉‘0) =
〈𝐵,
∅〉) |
84 | 82, 83 | ax-mp 5 |
. . 3
⊢
(〈“〈𝐵, ∅〉〈𝐴, ∅〉”〉‘0) =
〈𝐵,
∅〉 |
85 | 78, 81, 84 | 3eqtr3g 2667 |
. 2
⊢ (𝜑 → 〈𝐴, ∅〉 = 〈𝐵, ∅〉) |
86 | | opthg 4872 |
. . 3
⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ V) → (〈𝐴, ∅〉 = 〈𝐵, ∅〉 ↔ (𝐴 = 𝐵 ∧ ∅ = ∅))) |
87 | 86 | simprbda 651 |
. 2
⊢ (((𝐴 ∈ 𝐼 ∧ ∅ ∈ V) ∧ 〈𝐴, ∅〉 = 〈𝐵, ∅〉) → 𝐴 = 𝐵) |
88 | 1, 3, 85, 87 | syl21anc 1317 |
1
⊢ (𝜑 → 𝐴 = 𝐵) |