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Theorem frgpnabllem1 19546
Description: Lemma for frgpnabl 19548. (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 5245 . . . . . . . . 9 ∅ ∈ V
32prid1 4707 . . . . . . . 8 ∅ ∈ {∅, 1o}
4 df2o3 8353 . . . . . . . 8 2o = {∅, 1o}
53, 4eleqtrri 2836 . . . . . . 7 ∅ ∈ 2o
6 opelxpi 5644 . . . . . . 7 ((𝐴𝐼 ∧ ∅ ∈ 2o) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2o))
71, 5, 6sylancl 586 . . . . . 6 (𝜑 → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2o))
8 frgpnabl.b . . . . . . 7 (𝜑𝐵𝐼)
9 opelxpi 5644 . . . . . . 7 ((𝐵𝐼 ∧ ∅ ∈ 2o) → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2o))
108, 5, 9sylancl 586 . . . . . 6 (𝜑 → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2o))
117, 10s2cld 14660 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
12 frgpnabl.w . . . . . 6 𝑊 = ( I ‘Word (𝐼 × 2o))
13 frgpnabl.i . . . . . . . 8 (𝜑𝐼𝑉)
14 2on 8359 . . . . . . . 8 2o ∈ On
15 xpexg 7641 . . . . . . . 8 ((𝐼𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
1613, 14, 15sylancl 586 . . . . . . 7 (𝜑 → (𝐼 × 2o) ∈ V)
17 wrdexg 14305 . . . . . . 7 ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
18 fvi 6883 . . . . . . 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, 19eqtrid 2788 . . . . 5 (𝜑𝑊 = Word (𝐼 × 2o))
2111, 20eleqtrrd 2840 . . . 4 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
22 1n0 8367 . . . . . . 7 1o ≠ ∅
23 2cn 12127 . . . . . . . . . . . . . 14 2 ∈ ℂ
2423addid2i 11242 . . . . . . . . . . . . 13 (0 + 2) = 2
25 s2len 14678 . . . . . . . . . . . . 13 (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = 2
2624, 25eqtr4i 2767 . . . . . . . . . . . 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 19404 . . . . . . . . . . . . 13 ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((♯‘𝑥) + 2))
3130adantll 711 . . . . . . . . . . . 12 (((𝜑𝑥𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((♯‘𝑥) + 2))
3226, 31eqtrid 2788 . . . . . . . . . . 11 (((𝜑𝑥𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (0 + 2) = ((♯‘𝑥) + 2))
3332ex 413 . . . . . . . . . 10 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → (0 + 2) = ((♯‘𝑥) + 2)))
34 0cnd 11047 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → 0 ∈ ℂ)
35 simpr 485 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑊) → 𝑥𝑊)
3612efgrcl 19393 . . . . . . . . . . . . . . . 16 (𝑥𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
3736simprd 496 . . . . . . . . . . . . . . 15 (𝑥𝑊𝑊 = Word (𝐼 × 2o))
3837adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑊) → 𝑊 = Word (𝐼 × 2o))
3935, 38eleqtrd 2839 . . . . . . . . . . . . 13 ((𝜑𝑥𝑊) → 𝑥 ∈ Word (𝐼 × 2o))
40 lencl 14314 . . . . . . . . . . . . 13 (𝑥 ∈ Word (𝐼 × 2o) → (♯‘𝑥) ∈ ℕ0)
4139, 40syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝑊) → (♯‘𝑥) ∈ ℕ0)
4241nn0cnd 12374 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → (♯‘𝑥) ∈ ℂ)
43 2cnd 12130 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → 2 ∈ ℂ)
4434, 42, 43addcan2d 11258 . . . . . . . . . 10 ((𝜑𝑥𝑊) → ((0 + 2) = ((♯‘𝑥) + 2) ↔ 0 = (♯‘𝑥)))
4533, 44sylibd 238 . . . . . . . . 9 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → 0 = (♯‘𝑥)))
4612, 27, 28, 29efgtf 19400 . . . . . . . . . . . . . . . . . 18 (∅ ∈ 𝑊 → ((𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(♯‘∅)) × (𝐼 × 2o))⟶𝑊))
4746adantl 482 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ∅ ∈ 𝑊) → ((𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(♯‘∅)) × (𝐼 × 2o))⟶𝑊))
4847simpld 495 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∅ ∈ 𝑊) → (𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
4948rneqd 5866 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ∅ ∈ 𝑊) → ran (𝑇‘∅) = ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
5049eleq2d 2822 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))))
51 eqid 2736 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
52 ovex 7349 . . . . . . . . . . . . . . . 16 (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ V
5351, 52elrnmpo 7451 . . . . . . . . . . . . . . 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 14320 . . . . . . . . . . . . . . . . . . . . 21 ∅ ∈ Word (𝐼 × 2o)
5554a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∅ ∈ Word (𝐼 × 2o))
56 simprr 770 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑏 ∈ (𝐼 × 2o))
5728efgmf 19391 . . . . . . . . . . . . . . . . . . . . . . 23 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
5857ffvelcdmi 6999 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 ∈ (𝐼 × 2o) → (𝑀𝑏) ∈ (𝐼 × 2o))
5956, 58syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀𝑏) ∈ (𝐼 × 2o))
6056, 59s2cld 14660 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o))
61 ccatidid 14372 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ ++ ∅) = ∅
6261oveq1i 7326 . . . . . . . . . . . . . . . . . . . . . 22 ((∅ ++ ∅) ++ ∅) = (∅ ++ ∅)
6362, 61eqtr2i 2765 . . . . . . . . . . . . . . . . . . . . 21 ∅ = ((∅ ++ ∅) ++ ∅)
6463a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∅ = ((∅ ++ ∅) ++ ∅))
65 simprl 768 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...(♯‘∅)))
66 hash0 14160 . . . . . . . . . . . . . . . . . . . . . . . 24 (♯‘∅) = 0
6766oveq2i 7327 . . . . . . . . . . . . . . . . . . . . . . 23 (0...(♯‘∅)) = (0...0)
6865, 67eleqtrdi 2847 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...0))
69 elfz1eq 13346 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ (0...0) → 𝑎 = 0)
7068, 69syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = 0)
7170, 66eqtr4di 2794 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (♯‘∅))
7266oveq2i 7327 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 + (♯‘∅)) = (𝑎 + 0)
73 0cn 11046 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℂ
7470, 73eqeltrdi 2845 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ ℂ)
7574addid1d 11254 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 + 0) = 𝑎)
7672, 75eqtr2id 2789 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (𝑎 + (♯‘∅)))
7755, 55, 55, 60, 64, 71, 76splval2 14546 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = ((∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ ∅))
78 ccatlid 14368 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o) → (∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) = ⟨“𝑏(𝑀𝑏)”⟩)
7978oveq1d 7331 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o) → ((∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ ∅) = (⟨“𝑏(𝑀𝑏)”⟩ ++ ∅))
80 ccatrid 14369 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o) → (⟨“𝑏(𝑀𝑏)”⟩ ++ ∅) = ⟨“𝑏(𝑀𝑏)”⟩)
8179, 80eqtrd 2776 . . . . . . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = ⟨“𝑏(𝑀𝑏)”⟩)
8483eqeq2d 2747 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩))
851ad3antrrr 727 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → 𝐴𝐼)
86 1on 8357 . . . . . . . . . . . . . . . . . . . 20 1o ∈ On
8786a1i 11 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → 1o ∈ On)
88 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩)
8988fveq1d 6813 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = (⟨“𝑏(𝑀𝑏)”⟩‘1))
90 opex 5397 . . . . . . . . . . . . . . . . . . . . . 22 𝐵, ∅⟩ ∈ V
91 s2fv1 14677 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝐵, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩)
9290, 91ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩
93 fvex 6824 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀𝑏) ∈ V
94 s2fv1 14677 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀𝑏) ∈ V → (⟨“𝑏(𝑀𝑏)”⟩‘1) = (𝑀𝑏))
9593, 94ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩‘1) = (𝑀𝑏)
9689, 92, 953eqtr3g 2799 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐵, ∅⟩ = (𝑀𝑏))
9788fveq1d 6813 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = (⟨“𝑏(𝑀𝑏)”⟩‘0))
98 opex 5397 . . . . . . . . . . . . . . . . . . . . . . 23 𝐴, ∅⟩ ∈ V
99 s2fv0 14676 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝐴, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩)
10098, 99ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩
101 s2fv0 14676 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 ∈ V → (⟨“𝑏(𝑀𝑏)”⟩‘0) = 𝑏)
102101elv 3446 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“𝑏(𝑀𝑏)”⟩‘0) = 𝑏
10397, 100, 1023eqtr3g 2799 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐴, ∅⟩ = 𝑏)
104103fveq2d 6815 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = (𝑀𝑏))
10528efgmval 19390 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝐼 ∧ ∅ ∈ 2o) → (𝐴𝑀∅) = ⟨𝐴, (1o ∖ ∅)⟩)
10685, 5, 105sylancl 586 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (𝐴𝑀∅) = ⟨𝐴, (1o ∖ ∅)⟩)
107 df-ov 7319 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝑀∅) = (𝑀‘⟨𝐴, ∅⟩)
108 dif0 4316 . . . . . . . . . . . . . . . . . . . . . 22 (1o ∖ ∅) = 1o
109108opeq2i 4818 . . . . . . . . . . . . . . . . . . . . 21 𝐴, (1o ∖ ∅)⟩ = ⟨𝐴, 1o
110106, 107, 1093eqtr3g 2799 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = ⟨𝐴, 1o⟩)
11196, 104, 1103eqtr2rd 2783 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩)
112 opthg 5410 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝐼 ∧ 1o ∈ On) → (⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩ ↔ (𝐴 = 𝐵 ∧ 1o = ∅)))
113112simplbda 500 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝐼 ∧ 1o ∈ On) ∧ ⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩) → 1o = ∅)
11485, 87, 111, 113syl21anc 835 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → 1o = ∅)
115114ex 413 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩ → 1o = ∅))
11684, 115sylbid 239 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → 1o = ∅))
117116rexlimdvva 3201 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ∅ ∈ 𝑊) → (∃𝑎 ∈ (0...(♯‘∅))∃𝑏 ∈ (𝐼 × 2o)⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → 1o = ∅))
11853, 117biimtrid 241 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) → 1o = ∅))
11950, 118sylbid 239 . . . . . . . . . . . . 13 ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) → 1o = ∅))
120119expimpd 454 . . . . . . . . . . . 12 (𝜑 → ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1o = ∅))
121 hasheq0 14156 . . . . . . . . . . . . . . . 16 (𝑥 ∈ V → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
122121elv 3446 . . . . . . . . . . . . . . 15 ((♯‘𝑥) = 0 ↔ 𝑥 = ∅)
123 eleq1 2824 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (𝑥𝑊 ↔ ∅ ∈ 𝑊))
124 fveq2 6811 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∅ → (𝑇𝑥) = (𝑇‘∅))
125124rneqd 5866 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → ran (𝑇𝑥) = ran (𝑇‘∅))
126125eleq2d 2822 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)))
127123, 126anbi12d 631 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
128122, 127sylbi 216 . . . . . . . . . . . . . 14 ((♯‘𝑥) = 0 → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
129128eqcoms 2744 . . . . . . . . . . . . 13 (0 = (♯‘𝑥) → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
130129imbi1d 341 . . . . . . . . . . . 12 (0 = (♯‘𝑥) → (((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → 1o = ∅) ↔ ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1o = ∅)))
131120, 130syl5ibrcom 246 . . . . . . . . . . 11 (𝜑 → (0 = (♯‘𝑥) → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → 1o = ∅)))
132131com23 86 . . . . . . . . . 10 (𝜑 → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (0 = (♯‘𝑥) → 1o = ∅)))
133132expdimp 453 . . . . . . . . 9 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → (0 = (♯‘𝑥) → 1o = ∅)))
13445, 133mpdd 43 . . . . . . . 8 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → 1o = ∅))
135134necon3ad 2953 . . . . . . 7 ((𝜑𝑥𝑊) → (1o ≠ ∅ → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)))
13622, 135mpi 20 . . . . . 6 ((𝜑𝑥𝑊) → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
137136nrexdv 3142 . . . . 5 (𝜑 → ¬ ∃𝑥𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
138 eliun 4940 . . . . 5 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑥𝑊 ran (𝑇𝑥) ↔ ∃𝑥𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
139137, 138sylnibr 328 . . . 4 (𝜑 → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑥𝑊 ran (𝑇𝑥))
14021, 139eldifd 3907 . . 3 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝑊 𝑥𝑊 ran (𝑇𝑥)))
141 frgpnabl.d . . 3 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
142140, 141eleqtrrdi 2848 . 2 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝐷)
143 df-s2 14637 . . . . 5 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)
14412, 27efger 19396 . . . . . . 7 Er 𝑊
145144a1i 11 . . . . . 6 (𝜑 Er 𝑊)
146145, 21erref 8567 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
147143, 146eqbrtrrid 5122 . . . 4 (𝜑 → (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
148143ovexi 7350 . . . . 5 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ V
149 ovex 7349 . . . . 5 (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∈ V
150148, 149elec 8591 . . . 4 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ↔ (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
151147, 150sylibr 233 . . 3 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
152 frgpnabl.u . . . . . . 7 𝑈 = (varFGrp𝐼)
15327, 152vrgpval 19445 . . . . . 6 ((𝐼𝑉𝐴𝐼) → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
15413, 1, 153syl2anc 584 . . . . 5 (𝜑 → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
15527, 152vrgpval 19445 . . . . . 6 ((𝐼𝑉𝐵𝐼) → (𝑈𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] )
15613, 8, 155syl2anc 584 . . . . 5 (𝜑 → (𝑈𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] )
157154, 156oveq12d 7334 . . . 4 (𝜑 → ((𝑈𝐴) + (𝑈𝐵)) = ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ))
1587s1cld 14385 . . . . . 6 (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
159158, 20eleqtrrd 2840 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊)
16010s1cld 14385 . . . . . 6 (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
161160, 20eleqtrrd 2840 . . . . 5 (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
162 frgpnabl.g . . . . . 6 𝐺 = (freeGrp‘𝐼)
163 frgpnabl.p . . . . . 6 + = (+g𝐺)
16412, 162, 27, 163frgpadd 19441 . . . . 5 ((⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊 ∧ ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊) → ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
165159, 161, 164syl2anc 584 . . . 4 (𝜑 → ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
166157, 165eqtrd 2776 . . 3 (𝜑 → ((𝑈𝐴) + (𝑈𝐵)) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
167151, 166eleqtrrd 2840 . 2 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ((𝑈𝐴) + (𝑈𝐵)))
168142, 167elind 4138 1 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈𝐴) + (𝑈𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wne 2940  wrex 3070  Vcvv 3440  cdif 3893  cin 3895  c0 4266  {cpr 4572  cop 4576  cotp 4578   ciun 4936   class class class wbr 5086  cmpt 5169   I cid 5505   × cxp 5605  ran crn 5608  Oncon0 6288  wf 6461  cfv 6465  (class class class)co 7316  cmpo 7318  1oc1o 8338  2oc2o 8339   Er wer 8544  [cec 8545  cc 10948  0cc0 10950  1c1 10951   + caddc 10953  2c2 12107  0cn0 12312  ...cfz 13318  chash 14123  Word cword 14295   ++ cconcat 14351  ⟨“cs1 14377   splice csplice 14538  ⟨“cs2 14630  +gcplusg 17036   ~FG cefg 19384  freeGrpcfrgp 19385  varFGrpcvrgp 19386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629  ax-cnex 11006  ax-resscn 11007  ax-1cn 11008  ax-icn 11009  ax-addcl 11010  ax-addrcl 11011  ax-mulcl 11012  ax-mulrcl 11013  ax-mulcom 11014  ax-addass 11015  ax-mulass 11016  ax-distr 11017  ax-i2m1 11018  ax-1ne0 11019  ax-1rid 11020  ax-rnegex 11021  ax-rrecex 11022  ax-cnre 11023  ax-pre-lttri 11024  ax-pre-lttrn 11025  ax-pre-ltadd 11026  ax-pre-mulgt0 11027
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-ot 4579  df-uni 4850  df-int 4892  df-iun 4938  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5170  df-tr 5204  df-id 5506  df-eprel 5512  df-po 5520  df-so 5521  df-fr 5562  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-riota 7273  df-ov 7319  df-oprab 7320  df-mpo 7321  df-om 7759  df-1st 7877  df-2nd 7878  df-frecs 8145  df-wrecs 8176  df-recs 8250  df-rdg 8289  df-1o 8345  df-2o 8346  df-er 8547  df-ec 8549  df-qs 8553  df-map 8666  df-en 8783  df-dom 8784  df-sdom 8785  df-fin 8786  df-sup 9277  df-inf 9278  df-card 9774  df-pnf 11090  df-mnf 11091  df-xr 11092  df-ltxr 11093  df-le 11094  df-sub 11286  df-neg 11287  df-nn 12053  df-2 12115  df-3 12116  df-4 12117  df-5 12118  df-6 12119  df-7 12120  df-8 12121  df-9 12122  df-n0 12313  df-z 12399  df-dec 12517  df-uz 12662  df-fz 13319  df-fzo 13462  df-hash 14124  df-word 14296  df-concat 14352  df-s1 14378  df-substr 14430  df-pfx 14460  df-splice 14539  df-s2 14637  df-struct 16922  df-slot 16957  df-ndx 16969  df-base 16987  df-plusg 17049  df-mulr 17050  df-sca 17052  df-vsca 17053  df-ip 17054  df-tset 17055  df-ple 17056  df-ds 17058  df-imas 17293  df-qus 17294  df-mgm 18400  df-sgrp 18449  df-mnd 18460  df-frmd 18561  df-efg 19387  df-frgp 19388  df-vrgp 19389
This theorem is referenced by:  frgpnabllem2  19547
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