MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgpnabllem1 Structured version   Visualization version   GIF version

Theorem frgpnabllem1 18989
Description: Lemma for frgpnabl 18991. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 25-Apr-2024.)
Hypotheses
Ref Expression
frgpnabl.g 𝐺 = (freeGrp‘𝐼)
frgpnabl.w 𝑊 = ( I ‘Word (𝐼 × 2o))
frgpnabl.r = ( ~FG𝐼)
frgpnabl.p + = (+g𝐺)
frgpnabl.m 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
frgpnabl.t 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
frgpnabl.d 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
frgpnabl.u 𝑈 = (varFGrp𝐼)
frgpnabl.i (𝜑𝐼𝑉)
frgpnabl.a (𝜑𝐴𝐼)
frgpnabl.b (𝜑𝐵𝐼)
Assertion
Ref Expression
frgpnabllem1 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈𝐴) + (𝑈𝐵))))
Distinct variable groups:   𝑥,𝐴   𝑣,𝑛,𝑤,𝑥,𝑦,𝑧,𝐼   𝜑,𝑥   𝑥, ,𝑦,𝑧   𝑥,𝐵   𝑛,𝑊,𝑣,𝑤,𝑥,𝑦,𝑧   𝑥,𝐺   𝑛,𝑀,𝑣,𝑤,𝑥   𝑥,𝑇
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑤,𝑣,𝑛)   𝐴(𝑦,𝑧,𝑤,𝑣,𝑛)   𝐵(𝑦,𝑧,𝑤,𝑣,𝑛)   𝐷(𝑥,𝑦,𝑧,𝑤,𝑣,𝑛)   + (𝑥,𝑦,𝑧,𝑤,𝑣,𝑛)   (𝑤,𝑣,𝑛)   𝑇(𝑦,𝑧,𝑤,𝑣,𝑛)   𝑈(𝑥,𝑦,𝑧,𝑤,𝑣,𝑛)   𝐺(𝑦,𝑧,𝑤,𝑣,𝑛)   𝑀(𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧,𝑤,𝑣,𝑛)

Proof of Theorem frgpnabllem1
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgpnabl.a . . . . . . 7 (𝜑𝐴𝐼)
2 0ex 5178 . . . . . . . . 9 ∅ ∈ V
32prid1 4661 . . . . . . . 8 ∅ ∈ {∅, 1o}
4 df2o3 8104 . . . . . . . 8 2o = {∅, 1o}
53, 4eleqtrri 2892 . . . . . . 7 ∅ ∈ 2o
6 opelxpi 5560 . . . . . . 7 ((𝐴𝐼 ∧ ∅ ∈ 2o) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2o))
71, 5, 6sylancl 589 . . . . . 6 (𝜑 → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2o))
8 frgpnabl.b . . . . . . 7 (𝜑𝐵𝐼)
9 opelxpi 5560 . . . . . . 7 ((𝐵𝐼 ∧ ∅ ∈ 2o) → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2o))
108, 5, 9sylancl 589 . . . . . 6 (𝜑 → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2o))
117, 10s2cld 14228 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
12 frgpnabl.w . . . . . 6 𝑊 = ( I ‘Word (𝐼 × 2o))
13 frgpnabl.i . . . . . . . 8 (𝜑𝐼𝑉)
14 2on 8098 . . . . . . . 8 2o ∈ On
15 xpexg 7457 . . . . . . . 8 ((𝐼𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
1613, 14, 15sylancl 589 . . . . . . 7 (𝜑 → (𝐼 × 2o) ∈ V)
17 wrdexg 13871 . . . . . . 7 ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
18 fvi 6719 . . . . . . 7 (Word (𝐼 × 2o) ∈ V → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o))
1916, 17, 183syl 18 . . . . . 6 (𝜑 → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o))
2012, 19syl5eq 2848 . . . . 5 (𝜑𝑊 = Word (𝐼 × 2o))
2111, 20eleqtrrd 2896 . . . 4 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
22 1n0 8106 . . . . . . 7 1o ≠ ∅
23 2cn 11704 . . . . . . . . . . . . . 14 2 ∈ ℂ
2423addid2i 10821 . . . . . . . . . . . . 13 (0 + 2) = 2
25 s2len 14246 . . . . . . . . . . . . 13 (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = 2
2624, 25eqtr4i 2827 . . . . . . . . . . . 12 (0 + 2) = (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
27 frgpnabl.r . . . . . . . . . . . . . 14 = ( ~FG𝐼)
28 frgpnabl.m . . . . . . . . . . . . . 14 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
29 frgpnabl.t . . . . . . . . . . . . . 14 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
3012, 27, 28, 29efgtlen 18847 . . . . . . . . . . . . 13 ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((♯‘𝑥) + 2))
3130adantll 713 . . . . . . . . . . . 12 (((𝜑𝑥𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((♯‘𝑥) + 2))
3226, 31syl5eq 2848 . . . . . . . . . . 11 (((𝜑𝑥𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (0 + 2) = ((♯‘𝑥) + 2))
3332ex 416 . . . . . . . . . 10 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → (0 + 2) = ((♯‘𝑥) + 2)))
34 0cnd 10627 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → 0 ∈ ℂ)
35 simpr 488 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑊) → 𝑥𝑊)
3612efgrcl 18836 . . . . . . . . . . . . . . . 16 (𝑥𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
3736simprd 499 . . . . . . . . . . . . . . 15 (𝑥𝑊𝑊 = Word (𝐼 × 2o))
3837adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑊) → 𝑊 = Word (𝐼 × 2o))
3935, 38eleqtrd 2895 . . . . . . . . . . . . 13 ((𝜑𝑥𝑊) → 𝑥 ∈ Word (𝐼 × 2o))
40 lencl 13880 . . . . . . . . . . . . 13 (𝑥 ∈ Word (𝐼 × 2o) → (♯‘𝑥) ∈ ℕ0)
4139, 40syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝑊) → (♯‘𝑥) ∈ ℕ0)
4241nn0cnd 11949 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → (♯‘𝑥) ∈ ℂ)
43 2cnd 11707 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → 2 ∈ ℂ)
4434, 42, 43addcan2d 10837 . . . . . . . . . 10 ((𝜑𝑥𝑊) → ((0 + 2) = ((♯‘𝑥) + 2) ↔ 0 = (♯‘𝑥)))
4533, 44sylibd 242 . . . . . . . . 9 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → 0 = (♯‘𝑥)))
4612, 27, 28, 29efgtf 18843 . . . . . . . . . . . . . . . . . 18 (∅ ∈ 𝑊 → ((𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(♯‘∅)) × (𝐼 × 2o))⟶𝑊))
4746adantl 485 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ∅ ∈ 𝑊) → ((𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(♯‘∅)) × (𝐼 × 2o))⟶𝑊))
4847simpld 498 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∅ ∈ 𝑊) → (𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
4948rneqd 5776 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ∅ ∈ 𝑊) → ran (𝑇‘∅) = ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
5049eleq2d 2878 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))))
51 eqid 2801 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
52 ovex 7172 . . . . . . . . . . . . . . . 16 (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ V
5351, 52elrnmpo 7270 . . . . . . . . . . . . . . 15 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ↔ ∃𝑎 ∈ (0...(♯‘∅))∃𝑏 ∈ (𝐼 × 2o)⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
54 wrd0 13886 . . . . . . . . . . . . . . . . . . . . 21 ∅ ∈ Word (𝐼 × 2o)
5554a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∅ ∈ Word (𝐼 × 2o))
56 simprr 772 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑏 ∈ (𝐼 × 2o))
5728efgmf 18834 . . . . . . . . . . . . . . . . . . . . . . 23 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
5857ffvelrni 6831 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 ∈ (𝐼 × 2o) → (𝑀𝑏) ∈ (𝐼 × 2o))
5956, 58syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀𝑏) ∈ (𝐼 × 2o))
6056, 59s2cld 14228 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o))
61 ccatidid 13939 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ ++ ∅) = ∅
6261oveq1i 7149 . . . . . . . . . . . . . . . . . . . . . 22 ((∅ ++ ∅) ++ ∅) = (∅ ++ ∅)
6362, 61eqtr2i 2825 . . . . . . . . . . . . . . . . . . . . 21 ∅ = ((∅ ++ ∅) ++ ∅)
6463a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∅ = ((∅ ++ ∅) ++ ∅))
65 simprl 770 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...(♯‘∅)))
66 hash0 13728 . . . . . . . . . . . . . . . . . . . . . . . 24 (♯‘∅) = 0
6766oveq2i 7150 . . . . . . . . . . . . . . . . . . . . . . 23 (0...(♯‘∅)) = (0...0)
6865, 67eleqtrdi 2903 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...0))
69 elfz1eq 12917 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ (0...0) → 𝑎 = 0)
7068, 69syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = 0)
7170, 66eqtr4di 2854 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (♯‘∅))
7266oveq2i 7150 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 + (♯‘∅)) = (𝑎 + 0)
73 0cn 10626 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℂ
7470, 73eqeltrdi 2901 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ ℂ)
7574addid1d 10833 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 + 0) = 𝑎)
7672, 75syl5req 2849 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (𝑎 + (♯‘∅)))
7755, 55, 55, 60, 64, 71, 76splval2 14114 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = ((∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ ∅))
78 ccatlid 13935 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o) → (∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) = ⟨“𝑏(𝑀𝑏)”⟩)
7978oveq1d 7154 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o) → ((∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ ∅) = (⟨“𝑏(𝑀𝑏)”⟩ ++ ∅))
80 ccatrid 13936 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o) → (⟨“𝑏(𝑀𝑏)”⟩ ++ ∅) = ⟨“𝑏(𝑀𝑏)”⟩)
8179, 80eqtrd 2836 . . . . . . . . . . . . . . . . . . . 20 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o) → ((∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ ∅) = ⟨“𝑏(𝑀𝑏)”⟩)
8260, 81syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ ∅) = ⟨“𝑏(𝑀𝑏)”⟩)
8377, 82eqtrd 2836 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = ⟨“𝑏(𝑀𝑏)”⟩)
8483eqeq2d 2812 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩))
851ad3antrrr 729 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → 𝐴𝐼)
86 1on 8096 . . . . . . . . . . . . . . . . . . . 20 1o ∈ On
8786a1i 11 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → 1o ∈ On)
88 simpr 488 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩)
8988fveq1d 6651 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = (⟨“𝑏(𝑀𝑏)”⟩‘1))
90 opex 5324 . . . . . . . . . . . . . . . . . . . . . 22 𝐵, ∅⟩ ∈ V
91 s2fv1 14245 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝐵, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩)
9290, 91ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩
93 fvex 6662 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀𝑏) ∈ V
94 s2fv1 14245 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀𝑏) ∈ V → (⟨“𝑏(𝑀𝑏)”⟩‘1) = (𝑀𝑏))
9593, 94ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩‘1) = (𝑀𝑏)
9689, 92, 953eqtr3g 2859 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐵, ∅⟩ = (𝑀𝑏))
9788fveq1d 6651 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = (⟨“𝑏(𝑀𝑏)”⟩‘0))
98 opex 5324 . . . . . . . . . . . . . . . . . . . . . . 23 𝐴, ∅⟩ ∈ V
99 s2fv0 14244 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝐴, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩)
10098, 99ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩
101 s2fv0 14244 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 ∈ V → (⟨“𝑏(𝑀𝑏)”⟩‘0) = 𝑏)
102101elv 3449 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“𝑏(𝑀𝑏)”⟩‘0) = 𝑏
10397, 100, 1023eqtr3g 2859 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐴, ∅⟩ = 𝑏)
104103fveq2d 6653 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = (𝑀𝑏))
10528efgmval 18833 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝐼 ∧ ∅ ∈ 2o) → (𝐴𝑀∅) = ⟨𝐴, (1o ∖ ∅)⟩)
10685, 5, 105sylancl 589 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (𝐴𝑀∅) = ⟨𝐴, (1o ∖ ∅)⟩)
107 df-ov 7142 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝑀∅) = (𝑀‘⟨𝐴, ∅⟩)
108 dif0 4289 . . . . . . . . . . . . . . . . . . . . . 22 (1o ∖ ∅) = 1o
109108opeq2i 4772 . . . . . . . . . . . . . . . . . . . . 21 𝐴, (1o ∖ ∅)⟩ = ⟨𝐴, 1o
110106, 107, 1093eqtr3g 2859 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = ⟨𝐴, 1o⟩)
11196, 104, 1103eqtr2rd 2843 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩)
112 opthg 5337 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝐼 ∧ 1o ∈ On) → (⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩ ↔ (𝐴 = 𝐵 ∧ 1o = ∅)))
113112simplbda 503 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝐼 ∧ 1o ∈ On) ∧ ⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩) → 1o = ∅)
11485, 87, 111, 113syl21anc 836 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → 1o = ∅)
115114ex 416 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩ → 1o = ∅))
11684, 115sylbid 243 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → 1o = ∅))
117116rexlimdvva 3256 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ∅ ∈ 𝑊) → (∃𝑎 ∈ (0...(♯‘∅))∃𝑏 ∈ (𝐼 × 2o)⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → 1o = ∅))
11853, 117syl5bi 245 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) → 1o = ∅))
11950, 118sylbid 243 . . . . . . . . . . . . 13 ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) → 1o = ∅))
120119expimpd 457 . . . . . . . . . . . 12 (𝜑 → ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1o = ∅))
121 hasheq0 13724 . . . . . . . . . . . . . . . 16 (𝑥 ∈ V → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
122121elv 3449 . . . . . . . . . . . . . . 15 ((♯‘𝑥) = 0 ↔ 𝑥 = ∅)
123 eleq1 2880 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (𝑥𝑊 ↔ ∅ ∈ 𝑊))
124 fveq2 6649 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∅ → (𝑇𝑥) = (𝑇‘∅))
125124rneqd 5776 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → ran (𝑇𝑥) = ran (𝑇‘∅))
126125eleq2d 2878 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)))
127123, 126anbi12d 633 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
128122, 127sylbi 220 . . . . . . . . . . . . . 14 ((♯‘𝑥) = 0 → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
129128eqcoms 2809 . . . . . . . . . . . . 13 (0 = (♯‘𝑥) → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
130129imbi1d 345 . . . . . . . . . . . 12 (0 = (♯‘𝑥) → (((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → 1o = ∅) ↔ ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1o = ∅)))
131120, 130syl5ibrcom 250 . . . . . . . . . . 11 (𝜑 → (0 = (♯‘𝑥) → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → 1o = ∅)))
132131com23 86 . . . . . . . . . 10 (𝜑 → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (0 = (♯‘𝑥) → 1o = ∅)))
133132expdimp 456 . . . . . . . . 9 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → (0 = (♯‘𝑥) → 1o = ∅)))
13445, 133mpdd 43 . . . . . . . 8 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → 1o = ∅))
135134necon3ad 3003 . . . . . . 7 ((𝜑𝑥𝑊) → (1o ≠ ∅ → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)))
13622, 135mpi 20 . . . . . 6 ((𝜑𝑥𝑊) → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
137136nrexdv 3232 . . . . 5 (𝜑 → ¬ ∃𝑥𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
138 eliun 4888 . . . . 5 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑥𝑊 ran (𝑇𝑥) ↔ ∃𝑥𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
139137, 138sylnibr 332 . . . 4 (𝜑 → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑥𝑊 ran (𝑇𝑥))
14021, 139eldifd 3895 . . 3 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝑊 𝑥𝑊 ran (𝑇𝑥)))
141 frgpnabl.d . . 3 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
142140, 141eleqtrrdi 2904 . 2 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝐷)
143 df-s2 14205 . . . . 5 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)
14412, 27efger 18839 . . . . . . 7 Er 𝑊
145144a1i 11 . . . . . 6 (𝜑 Er 𝑊)
146145, 21erref 8296 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
147143, 146eqbrtrrid 5069 . . . 4 (𝜑 → (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
148143ovexi 7173 . . . . 5 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ V
149 ovex 7172 . . . . 5 (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∈ V
150148, 149elec 8320 . . . 4 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ↔ (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
151147, 150sylibr 237 . . 3 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
152 frgpnabl.u . . . . . . 7 𝑈 = (varFGrp𝐼)
15327, 152vrgpval 18888 . . . . . 6 ((𝐼𝑉𝐴𝐼) → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
15413, 1, 153syl2anc 587 . . . . 5 (𝜑 → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
15527, 152vrgpval 18888 . . . . . 6 ((𝐼𝑉𝐵𝐼) → (𝑈𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] )
15613, 8, 155syl2anc 587 . . . . 5 (𝜑 → (𝑈𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] )
157154, 156oveq12d 7157 . . . 4 (𝜑 → ((𝑈𝐴) + (𝑈𝐵)) = ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ))
1587s1cld 13952 . . . . . 6 (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
159158, 20eleqtrrd 2896 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊)
16010s1cld 13952 . . . . . 6 (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
161160, 20eleqtrrd 2896 . . . . 5 (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
162 frgpnabl.g . . . . . 6 𝐺 = (freeGrp‘𝐼)
163 frgpnabl.p . . . . . 6 + = (+g𝐺)
16412, 162, 27, 163frgpadd 18884 . . . . 5 ((⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊 ∧ ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊) → ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
165159, 161, 164syl2anc 587 . . . 4 (𝜑 → ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
166157, 165eqtrd 2836 . . 3 (𝜑 → ((𝑈𝐴) + (𝑈𝐵)) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
167151, 166eleqtrrd 2896 . 2 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ((𝑈𝐴) + (𝑈𝐵)))
168142, 167elind 4124 1 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈𝐴) + (𝑈𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wne 2990  wrex 3110  Vcvv 3444  cdif 3881  cin 3883  c0 4246  {cpr 4530  cop 4534  cotp 4536   ciun 4884   class class class wbr 5033  cmpt 5113   I cid 5427   × cxp 5521  ran crn 5524  Oncon0 6163  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141  1oc1o 8082  2oc2o 8083   Er wer 8273  [cec 8274  cc 10528  0cc0 10530  1c1 10531   + caddc 10533  2c2 11684  0cn0 11889  ...cfz 12889  chash 13690  Word cword 13861   ++ cconcat 13917  ⟨“cs1 13944   splice csplice 14106  ⟨“cs2 14198  +gcplusg 16560   ~FG cefg 18827  freeGrpcfrgp 18828  varFGrpcvrgp 18829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-ot 4537  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-ec 8278  df-qs 8282  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-fzo 13033  df-hash 13691  df-word 13862  df-concat 13918  df-s1 13945  df-substr 13998  df-pfx 14028  df-splice 14107  df-s2 14205  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-ds 16582  df-imas 16776  df-qus 16777  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-frmd 18009  df-efg 18830  df-frgp 18831  df-vrgp 18832
This theorem is referenced by:  frgpnabllem2  18990
  Copyright terms: Public domain W3C validator