Step | Hyp | Ref
| Expression |
1 | | frgpnabl.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝐼) |
2 | | 0ex 4790 |
. . . . . . . . 9
⊢ ∅
∈ V |
3 | 2 | prid1 4297 |
. . . . . . . 8
⊢ ∅
∈ {∅, 1𝑜} |
4 | | df2o3 7573 |
. . . . . . . 8
⊢
2𝑜 = {∅,
1𝑜} |
5 | 3, 4 | eleqtrri 2700 |
. . . . . . 7
⊢ ∅
∈ 2𝑜 |
6 | | opelxpi 5148 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2𝑜)
→ 〈𝐴,
∅〉 ∈ (𝐼
× 2𝑜)) |
7 | 1, 5, 6 | sylancl 694 |
. . . . . 6
⊢ (𝜑 → 〈𝐴, ∅〉 ∈ (𝐼 ×
2𝑜)) |
8 | | frgpnabl.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ 𝐼) |
9 | | opelxpi 5148 |
. . . . . . 7
⊢ ((𝐵 ∈ 𝐼 ∧ ∅ ∈ 2𝑜)
→ 〈𝐵,
∅〉 ∈ (𝐼
× 2𝑜)) |
10 | 8, 5, 9 | sylancl 694 |
. . . . . 6
⊢ (𝜑 → 〈𝐵, ∅〉 ∈ (𝐼 ×
2𝑜)) |
11 | 7, 10 | s2cld 13616 |
. . . . 5
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ Word (𝐼 ×
2𝑜)) |
12 | | frgpnabl.w |
. . . . . 6
⊢ 𝑊 = ( I ‘Word (𝐼 ×
2𝑜)) |
13 | | frgpnabl.i |
. . . . . . . 8
⊢ (𝜑 → 𝐼 ∈ V) |
14 | | 2on 7568 |
. . . . . . . 8
⊢
2𝑜 ∈ On |
15 | | xpexg 6960 |
. . . . . . . 8
⊢ ((𝐼 ∈ V ∧
2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈
V) |
16 | 13, 14, 15 | sylancl 694 |
. . . . . . 7
⊢ (𝜑 → (𝐼 × 2𝑜) ∈
V) |
17 | | wrdexg 13315 |
. . . . . . 7
⊢ ((𝐼 × 2𝑜)
∈ V → Word (𝐼
× 2𝑜) ∈ V) |
18 | | fvi 6255 |
. . . . . . 7
⊢ (Word
(𝐼 ×
2𝑜) ∈ V → ( I ‘Word (𝐼 × 2𝑜)) = Word
(𝐼 ×
2𝑜)) |
19 | 16, 17, 18 | 3syl 18 |
. . . . . 6
⊢ (𝜑 → ( I ‘Word (𝐼 × 2𝑜))
= Word (𝐼 ×
2𝑜)) |
20 | 12, 19 | syl5eq 2668 |
. . . . 5
⊢ (𝜑 → 𝑊 = Word (𝐼 ×
2𝑜)) |
21 | 11, 20 | eleqtrrd 2704 |
. . . 4
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ 𝑊) |
22 | | 1n0 7575 |
. . . . . . 7
⊢
1𝑜 ≠ ∅ |
23 | | 2cn 11091 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℂ |
24 | 23 | addid2i 10224 |
. . . . . . . . . . . . 13
⊢ (0 + 2) =
2 |
25 | | s2len 13634 |
. . . . . . . . . . . . 13
⊢
(#‘〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉) =
2 |
26 | 24, 25 | eqtr4i 2647 |
. . . . . . . . . . . 12
⊢ (0 + 2) =
(#‘〈“〈𝐴, ∅〉〈𝐵,
∅〉”〉) |
27 | | frgpnabl.r |
. . . . . . . . . . . . . 14
⊢ ∼ = (
~FG ‘𝐼) |
28 | | frgpnabl.m |
. . . . . . . . . . . . . 14
⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦
〈𝑦,
(1𝑜 ∖ 𝑧)〉) |
29 | | frgpnabl.t |
. . . . . . . . . . . . . 14
⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦
(𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) |
30 | 12, 27, 28, 29 | efgtlen 18139 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝑊 ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑇‘𝑥)) → (#‘〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉) =
((#‘𝑥) +
2)) |
31 | 30 | adantll 750 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑇‘𝑥)) → (#‘〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉) =
((#‘𝑥) +
2)) |
32 | 26, 31 | syl5eq 2668 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑇‘𝑥)) → (0 + 2) = ((#‘𝑥) + 2)) |
33 | 32 | ex 450 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ ran (𝑇‘𝑥) → (0 + 2) =
((#‘𝑥) +
2))) |
34 | | 0cnd 10033 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 0 ∈ ℂ) |
35 | | simpr 477 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑥 ∈ 𝑊) |
36 | 12 | efgrcl 18128 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 ×
2𝑜))) |
37 | 36 | simprd 479 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ 𝑊 → 𝑊 = Word (𝐼 ×
2𝑜)) |
38 | 37 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑊 = Word (𝐼 ×
2𝑜)) |
39 | 35, 38 | eleqtrd 2703 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑥 ∈ Word (𝐼 ×
2𝑜)) |
40 | | lencl 13324 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ Word (𝐼 × 2𝑜) →
(#‘𝑥) ∈
ℕ0) |
41 | 39, 40 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (#‘𝑥) ∈
ℕ0) |
42 | 41 | nn0cnd 11353 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (#‘𝑥) ∈ ℂ) |
43 | | 2cnd 11093 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 2 ∈ ℂ) |
44 | 34, 42, 43 | addcan2d 10240 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ((0 + 2) = ((#‘𝑥) + 2) ↔ 0 = (#‘𝑥))) |
45 | 33, 44 | sylibd 229 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ ran (𝑇‘𝑥) → 0 = (#‘𝑥))) |
46 | 12, 27, 28, 29 | efgtf 18135 |
. . . . . . . . . . . . . . . . . 18
⊢ (∅
∈ 𝑊 → ((𝑇‘∅) = (𝑎 ∈ (0...(#‘∅)),
𝑏 ∈ (𝐼 × 2𝑜) ↦
(∅ splice 〈𝑎,
𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉)) ∧ (𝑇‘∅):((0...(#‘∅))
× (𝐼 ×
2𝑜))⟶𝑊)) |
47 | 46 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → ((𝑇‘∅) = (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦
(∅ splice 〈𝑎,
𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉)) ∧ (𝑇‘∅):((0...(#‘∅))
× (𝐼 ×
2𝑜))⟶𝑊)) |
48 | 47 | simpld 475 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (𝑇‘∅) = (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦
(∅ splice 〈𝑎,
𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉))) |
49 | 48 | rneqd 5353 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → ran (𝑇‘∅) = ran (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦
(∅ splice 〈𝑎,
𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉))) |
50 | 49 | eleq2d 2687 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ∅ ∈ 𝑊) →
(〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 ∈ ran (𝑇‘∅) ↔
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 ∈ ran (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦
(∅ splice 〈𝑎,
𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉)))) |
51 | | eqid 2622 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 ∈ (0...(#‘∅)),
𝑏 ∈ (𝐼 × 2𝑜) ↦
(∅ splice 〈𝑎,
𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉)) = (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦
(∅ splice 〈𝑎,
𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉)) |
52 | | ovex 6678 |
. . . . . . . . . . . . . . . 16
⊢ (∅
splice 〈𝑎, 𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉) ∈
V |
53 | 51, 52 | elrnmpt2 6773 |
. . . . . . . . . . . . . . 15
⊢
(〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑎 ∈
(0...(#‘∅)), 𝑏
∈ (𝐼 ×
2𝑜) ↦ (∅ splice 〈𝑎, 𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉)) ↔ ∃𝑎 ∈
(0...(#‘∅))∃𝑏 ∈ (𝐼 ×
2𝑜)〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 = (∅
splice 〈𝑎, 𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉)) |
54 | | wrd0 13330 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ∅
∈ Word (𝐼 ×
2𝑜) |
55 | 54 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
∅ ∈ Word (𝐼
× 2𝑜)) |
56 | | simprr 796 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
𝑏 ∈ (𝐼 ×
2𝑜)) |
57 | 28 | efgmf 18126 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 𝑀:(𝐼 ×
2𝑜)⟶(𝐼 ×
2𝑜) |
58 | 57 | ffvelrni 6358 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑏 ∈ (𝐼 × 2𝑜) →
(𝑀‘𝑏) ∈ (𝐼 ×
2𝑜)) |
59 | 56, 58 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
(𝑀‘𝑏) ∈ (𝐼 ×
2𝑜)) |
60 | 56, 59 | s2cld 13616 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
〈“𝑏(𝑀‘𝑏)”〉 ∈ Word (𝐼 ×
2𝑜)) |
61 | | ccatlid 13369 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (∅
∈ Word (𝐼 ×
2𝑜) → (∅ ++ ∅) = ∅) |
62 | 54, 61 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (∅
++ ∅) = ∅ |
63 | 62 | oveq1i 6660 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((∅
++ ∅) ++ ∅) = (∅ ++ ∅) |
64 | 63, 62 | eqtr2i 2645 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ∅ =
((∅ ++ ∅) ++ ∅) |
65 | 64 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
∅ = ((∅ ++ ∅) ++ ∅)) |
66 | | simprl 794 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
𝑎 ∈
(0...(#‘∅))) |
67 | | hash0 13158 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(#‘∅) = 0 |
68 | 67 | oveq2i 6661 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(0...(#‘∅)) = (0...0) |
69 | 66, 68 | syl6eleq 2711 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
𝑎 ∈
(0...0)) |
70 | | elfz1eq 12352 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑎 ∈ (0...0) → 𝑎 = 0) |
71 | 69, 70 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
𝑎 = 0) |
72 | 71, 67 | syl6eqr 2674 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
𝑎 =
(#‘∅)) |
73 | 67 | oveq2i 6661 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎 + (#‘∅)) = (𝑎 + 0) |
74 | | 0cn 10032 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 0 ∈
ℂ |
75 | 71, 74 | syl6eqel 2709 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
𝑎 ∈
ℂ) |
76 | 75 | addid1d 10236 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
(𝑎 + 0) = 𝑎) |
77 | 73, 76 | syl5req 2669 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
𝑎 = (𝑎 + (#‘∅))) |
78 | 55, 55, 55, 60, 65, 72, 77 | splval2 13508 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
(∅ splice 〈𝑎,
𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉) = ((∅ ++
〈“𝑏(𝑀‘𝑏)”〉) ++ ∅)) |
79 | | ccatlid 13369 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(〈“𝑏(𝑀‘𝑏)”〉 ∈ Word (𝐼 × 2𝑜) →
(∅ ++ 〈“𝑏(𝑀‘𝑏)”〉) = 〈“𝑏(𝑀‘𝑏)”〉) |
80 | 79 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(〈“𝑏(𝑀‘𝑏)”〉 ∈ Word (𝐼 × 2𝑜) →
((∅ ++ 〈“𝑏(𝑀‘𝑏)”〉) ++ ∅) =
(〈“𝑏(𝑀‘𝑏)”〉 ++ ∅)) |
81 | | ccatrid 13370 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(〈“𝑏(𝑀‘𝑏)”〉 ∈ Word (𝐼 × 2𝑜) →
(〈“𝑏(𝑀‘𝑏)”〉 ++ ∅) =
〈“𝑏(𝑀‘𝑏)”〉) |
82 | 80, 81 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(〈“𝑏(𝑀‘𝑏)”〉 ∈ Word (𝐼 × 2𝑜) →
((∅ ++ 〈“𝑏(𝑀‘𝑏)”〉) ++ ∅) =
〈“𝑏(𝑀‘𝑏)”〉) |
83 | 60, 82 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
((∅ ++ 〈“𝑏(𝑀‘𝑏)”〉) ++ ∅) =
〈“𝑏(𝑀‘𝑏)”〉) |
84 | 78, 83 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
(∅ splice 〈𝑎,
𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉) = 〈“𝑏(𝑀‘𝑏)”〉) |
85 | 84 | eqeq2d 2632 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
(〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = (∅ splice 〈𝑎, 𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉) ↔
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉)) |
86 | 1 | ad3antrrr 766 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉) → 𝐴 ∈ 𝐼) |
87 | | 1on 7567 |
. . . . . . . . . . . . . . . . . . . 20
⊢
1𝑜 ∈ On |
88 | 87 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉) → 1𝑜
∈ On) |
89 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉) →
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉) |
90 | 89 | fveq1d 6193 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉) →
(〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉‘1) = (〈“𝑏(𝑀‘𝑏)”〉‘1)) |
91 | | opex 4932 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
〈𝐵,
∅〉 ∈ V |
92 | | s2fv1 13633 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(〈𝐵,
∅〉 ∈ V → (〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉‘1) =
〈𝐵,
∅〉) |
93 | 91, 92 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉‘1) =
〈𝐵,
∅〉 |
94 | | fvex 6201 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑀‘𝑏) ∈ V |
95 | | s2fv1 13633 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑀‘𝑏) ∈ V → (〈“𝑏(𝑀‘𝑏)”〉‘1) = (𝑀‘𝑏)) |
96 | 94, 95 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(〈“𝑏(𝑀‘𝑏)”〉‘1) = (𝑀‘𝑏) |
97 | 90, 93, 96 | 3eqtr3g 2679 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉) → 〈𝐵, ∅〉 = (𝑀‘𝑏)) |
98 | 89 | fveq1d 6193 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉) →
(〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉‘0) = (〈“𝑏(𝑀‘𝑏)”〉‘0)) |
99 | | opex 4932 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
〈𝐴,
∅〉 ∈ V |
100 | | s2fv0 13632 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(〈𝐴,
∅〉 ∈ V → (〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉‘0) =
〈𝐴,
∅〉) |
101 | 99, 100 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉‘0) =
〈𝐴,
∅〉 |
102 | | vex 3203 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 𝑏 ∈ V |
103 | | s2fv0 13632 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑏 ∈ V →
(〈“𝑏(𝑀‘𝑏)”〉‘0) = 𝑏) |
104 | 102, 103 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(〈“𝑏(𝑀‘𝑏)”〉‘0) = 𝑏 |
105 | 98, 101, 104 | 3eqtr3g 2679 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉) → 〈𝐴, ∅〉 = 𝑏) |
106 | 105 | fveq2d 6195 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉) → (𝑀‘〈𝐴, ∅〉) = (𝑀‘𝑏)) |
107 | 28 | efgmval 18125 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2𝑜)
→ (𝐴𝑀∅) = 〈𝐴, (1𝑜 ∖
∅)〉) |
108 | 86, 5, 107 | sylancl 694 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉) → (𝐴𝑀∅) = 〈𝐴, (1𝑜 ∖
∅)〉) |
109 | | df-ov 6653 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴𝑀∅) = (𝑀‘〈𝐴, ∅〉) |
110 | | dif0 3950 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(1𝑜 ∖ ∅) =
1𝑜 |
111 | 110 | opeq2i 4406 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
〈𝐴,
(1𝑜 ∖ ∅)〉 = 〈𝐴,
1𝑜〉 |
112 | 108, 109,
111 | 3eqtr3g 2679 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉) → (𝑀‘〈𝐴, ∅〉) = 〈𝐴,
1𝑜〉) |
113 | 97, 106, 112 | 3eqtr2rd 2663 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉) → 〈𝐴, 1𝑜〉 =
〈𝐵,
∅〉) |
114 | | opthg 4946 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ∈ 𝐼 ∧ 1𝑜 ∈ On)
→ (〈𝐴,
1𝑜〉 = 〈𝐵, ∅〉 ↔ (𝐴 = 𝐵 ∧ 1𝑜 =
∅))) |
115 | 114 | simplbda 654 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ 𝐼 ∧ 1𝑜 ∈ On)
∧ 〈𝐴,
1𝑜〉 = 〈𝐵, ∅〉) →
1𝑜 = ∅) |
116 | 86, 88, 113, 115 | syl21anc 1325 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉) → 1𝑜
= ∅) |
117 | 116 | ex 450 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
(〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = 〈“𝑏(𝑀‘𝑏)”〉 → 1𝑜 =
∅)) |
118 | 85, 117 | sylbid 230 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) →
(〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 = (∅ splice 〈𝑎, 𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉) →
1𝑜 = ∅)) |
119 | 118 | rexlimdvva 3038 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (∃𝑎 ∈
(0...(#‘∅))∃𝑏 ∈ (𝐼 ×
2𝑜)〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 = (∅
splice 〈𝑎, 𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉) →
1𝑜 = ∅)) |
120 | 53, 119 | syl5bi 232 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ∅ ∈ 𝑊) →
(〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 ∈ ran (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦
(∅ splice 〈𝑎,
𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉)) →
1𝑜 = ∅)) |
121 | 50, 120 | sylbid 230 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ∅ ∈ 𝑊) →
(〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 ∈ ran (𝑇‘∅) → 1𝑜
= ∅)) |
122 | 121 | expimpd 629 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((∅ ∈ 𝑊 ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ ran (𝑇‘∅)) →
1𝑜 = ∅)) |
123 | | vex 3203 |
. . . . . . . . . . . . . . . 16
⊢ 𝑥 ∈ V |
124 | | hasheq0 13154 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ V → ((#‘𝑥) = 0 ↔ 𝑥 = ∅)) |
125 | 123, 124 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
((#‘𝑥) = 0
↔ 𝑥 =
∅) |
126 | | eleq1 2689 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑊 ↔ ∅ ∈ 𝑊)) |
127 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = ∅ → (𝑇‘𝑥) = (𝑇‘∅)) |
128 | 127 | rneqd 5353 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = ∅ → ran (𝑇‘𝑥) = ran (𝑇‘∅)) |
129 | 128 | eleq2d 2687 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = ∅ →
(〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 ∈ ran (𝑇‘𝑥) ↔ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑇‘∅))) |
130 | 126, 129 | anbi12d 747 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑊 ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑇‘∅)))) |
131 | 125, 130 | sylbi 207 |
. . . . . . . . . . . . . 14
⊢
((#‘𝑥) = 0
→ ((𝑥 ∈ 𝑊 ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ ran (𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ ran (𝑇‘∅)))) |
132 | 131 | eqcoms 2630 |
. . . . . . . . . . . . 13
⊢ (0 =
(#‘𝑥) → ((𝑥 ∈ 𝑊 ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑇‘∅)))) |
133 | 132 | imbi1d 331 |
. . . . . . . . . . . 12
⊢ (0 =
(#‘𝑥) → (((𝑥 ∈ 𝑊 ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑇‘𝑥)) → 1𝑜 = ∅)
↔ ((∅ ∈ 𝑊
∧ 〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 ∈ ran (𝑇‘∅)) →
1𝑜 = ∅))) |
134 | 122, 133 | syl5ibrcom 237 |
. . . . . . . . . . 11
⊢ (𝜑 → (0 = (#‘𝑥) → ((𝑥 ∈ 𝑊 ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑇‘𝑥)) → 1𝑜 =
∅))) |
135 | 134 | com23 86 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑥 ∈ 𝑊 ∧ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑇‘𝑥)) → (0 = (#‘𝑥) → 1𝑜 =
∅))) |
136 | 135 | expdimp 453 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ ran (𝑇‘𝑥) → (0 = (#‘𝑥) → 1𝑜 =
∅))) |
137 | 45, 136 | mpdd 43 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ ran (𝑇‘𝑥) → 1𝑜 =
∅)) |
138 | 137 | necon3ad 2807 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (1𝑜 ≠ ∅
→ ¬ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑇‘𝑥))) |
139 | 22, 138 | mpi 20 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ¬ 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ ran (𝑇‘𝑥)) |
140 | 139 | nrexdv 3001 |
. . . . 5
⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝑊 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑇‘𝑥)) |
141 | | eliun 4524 |
. . . . 5
⊢
(〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ↔ ∃𝑥 ∈ 𝑊 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ ran
(𝑇‘𝑥)) |
142 | 140, 141 | sylnibr 319 |
. . . 4
⊢ (𝜑 → ¬
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉 ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) |
143 | 21, 142 | eldifd 3585 |
. . 3
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))) |
144 | | frgpnabl.d |
. . 3
⊢ 𝐷 = (𝑊 ∖ ∪
𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) |
145 | 143, 144 | syl6eleqr 2712 |
. 2
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ 𝐷) |
146 | | df-s2 13593 |
. . . . 5
⊢
〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 =
(〈“〈𝐴,
∅〉”〉 ++ 〈“〈𝐵,
∅〉”〉) |
147 | 12, 27 | efger 18131 |
. . . . . . 7
⊢ ∼ Er
𝑊 |
148 | 147 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ∼ Er 𝑊) |
149 | 148, 21 | erref 7762 |
. . . . 5
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∼
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉) |
150 | 146, 149 | syl5eqbrr 4689 |
. . . 4
⊢ (𝜑 → (〈“〈𝐴, ∅〉”〉 ++
〈“〈𝐵,
∅〉”〉) ∼
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉) |
151 | | ovex 6678 |
. . . . . 6
⊢
(〈“〈𝐴, ∅〉”〉 ++
〈“〈𝐵,
∅〉”〉) ∈ V |
152 | 146, 151 | eqeltri 2697 |
. . . . 5
⊢
〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈
V |
153 | 152, 151 | elec 7786 |
. . . 4
⊢
(〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈
[(〈“〈𝐴,
∅〉”〉 ++ 〈“〈𝐵, ∅〉”〉)] ∼ ↔
(〈“〈𝐴,
∅〉”〉 ++ 〈“〈𝐵, ∅〉”〉) ∼
〈“〈𝐴,
∅〉〈𝐵,
∅〉”〉) |
154 | 150, 153 | sylibr 224 |
. . 3
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ [(〈“〈𝐴, ∅〉”〉 ++
〈“〈𝐵,
∅〉”〉)] ∼ ) |
155 | | frgpnabl.u |
. . . . . . 7
⊢ 𝑈 =
(varFGrp‘𝐼) |
156 | 27, 155 | vrgpval 18180 |
. . . . . 6
⊢ ((𝐼 ∈ V ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼
) |
157 | 13, 1, 156 | syl2anc 693 |
. . . . 5
⊢ (𝜑 → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼
) |
158 | 27, 155 | vrgpval 18180 |
. . . . . 6
⊢ ((𝐼 ∈ V ∧ 𝐵 ∈ 𝐼) → (𝑈‘𝐵) = [〈“〈𝐵, ∅〉”〉] ∼
) |
159 | 13, 8, 158 | syl2anc 693 |
. . . . 5
⊢ (𝜑 → (𝑈‘𝐵) = [〈“〈𝐵, ∅〉”〉] ∼
) |
160 | 157, 159 | oveq12d 6668 |
. . . 4
⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = ([〈“〈𝐴, ∅〉”〉] ∼ +
[〈“〈𝐵,
∅〉”〉] ∼ )) |
161 | 7 | s1cld 13383 |
. . . . . 6
⊢ (𝜑 → 〈“〈𝐴, ∅〉”〉
∈ Word (𝐼 ×
2𝑜)) |
162 | 161, 20 | eleqtrrd 2704 |
. . . . 5
⊢ (𝜑 → 〈“〈𝐴, ∅〉”〉
∈ 𝑊) |
163 | 10 | s1cld 13383 |
. . . . . 6
⊢ (𝜑 → 〈“〈𝐵, ∅〉”〉
∈ Word (𝐼 ×
2𝑜)) |
164 | 163, 20 | eleqtrrd 2704 |
. . . . 5
⊢ (𝜑 → 〈“〈𝐵, ∅〉”〉
∈ 𝑊) |
165 | | frgpnabl.g |
. . . . . 6
⊢ 𝐺 = (freeGrp‘𝐼) |
166 | | frgpnabl.p |
. . . . . 6
⊢ + =
(+g‘𝐺) |
167 | 12, 165, 27, 166 | frgpadd 18176 |
. . . . 5
⊢
((〈“〈𝐴, ∅〉”〉 ∈ 𝑊 ∧ 〈“〈𝐵, ∅〉”〉
∈ 𝑊) →
([〈“〈𝐴,
∅〉”〉] ∼ + [〈“〈𝐵, ∅〉”〉]
∼
) = [(〈“〈𝐴,
∅〉”〉 ++ 〈“〈𝐵, ∅〉”〉)] ∼
) |
168 | 162, 164,
167 | syl2anc 693 |
. . . 4
⊢ (𝜑 → ([〈“〈𝐴, ∅〉”〉]
∼
+
[〈“〈𝐵,
∅〉”〉] ∼ ) =
[(〈“〈𝐴,
∅〉”〉 ++ 〈“〈𝐵, ∅〉”〉)] ∼
) |
169 | 160, 168 | eqtrd 2656 |
. . 3
⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = [(〈“〈𝐴, ∅〉”〉 ++
〈“〈𝐵,
∅〉”〉)] ∼ ) |
170 | 154, 169 | eleqtrrd 2704 |
. 2
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ ((𝑈‘𝐴) + (𝑈‘𝐵))) |
171 | 145, 170 | elind 3798 |
1
⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉
∈ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵)))) |