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Theorem frgpnabllem1 19867
Description: Lemma for frgpnabl 19869. (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 5304 . . . . . . . . 9 ∅ ∈ V
32prid1 4761 . . . . . . . 8 ∅ ∈ {∅, 1o}
4 df2o3 8496 . . . . . . . 8 2o = {∅, 1o}
53, 4eleqtrri 2825 . . . . . . 7 ∅ ∈ 2o
6 opelxpi 5711 . . . . . . 7 ((𝐴𝐼 ∧ ∅ ∈ 2o) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2o))
71, 5, 6sylancl 584 . . . . . 6 (𝜑 → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2o))
8 frgpnabl.b . . . . . . 7 (𝜑𝐵𝐼)
9 opelxpi 5711 . . . . . . 7 ((𝐵𝐼 ∧ ∅ ∈ 2o) → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2o))
108, 5, 9sylancl 584 . . . . . 6 (𝜑 → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2o))
117, 10s2cld 14875 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
12 frgpnabl.w . . . . . 6 𝑊 = ( I ‘Word (𝐼 × 2o))
13 frgpnabl.i . . . . . . . 8 (𝜑𝐼𝑉)
14 2on 8502 . . . . . . . 8 2o ∈ On
15 xpexg 7750 . . . . . . . 8 ((𝐼𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
1613, 14, 15sylancl 584 . . . . . . 7 (𝜑 → (𝐼 × 2o) ∈ V)
17 wrdexg 14527 . . . . . . 7 ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
18 fvi 6970 . . . . . . 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 2778 . . . . 5 (𝜑𝑊 = Word (𝐼 × 2o))
2111, 20eleqtrrd 2829 . . . 4 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
22 1n0 8510 . . . . . . 7 1o ≠ ∅
23 2cn 12333 . . . . . . . . . . . . . 14 2 ∈ ℂ
2423addlidi 11443 . . . . . . . . . . . . 13 (0 + 2) = 2
25 s2len 14893 . . . . . . . . . . . . 13 (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = 2
2624, 25eqtr4i 2757 . . . . . . . . . . . 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 19720 . . . . . . . . . . . . 13 ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((♯‘𝑥) + 2))
3130adantll 712 . . . . . . . . . . . 12 (((𝜑𝑥𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((♯‘𝑥) + 2))
3226, 31eqtrid 2778 . . . . . . . . . . 11 (((𝜑𝑥𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (0 + 2) = ((♯‘𝑥) + 2))
3332ex 411 . . . . . . . . . 10 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → (0 + 2) = ((♯‘𝑥) + 2)))
34 0cnd 11248 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → 0 ∈ ℂ)
35 simpr 483 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑊) → 𝑥𝑊)
3612efgrcl 19709 . . . . . . . . . . . . . . . 16 (𝑥𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
3736simprd 494 . . . . . . . . . . . . . . 15 (𝑥𝑊𝑊 = Word (𝐼 × 2o))
3837adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑊) → 𝑊 = Word (𝐼 × 2o))
3935, 38eleqtrd 2828 . . . . . . . . . . . . 13 ((𝜑𝑥𝑊) → 𝑥 ∈ Word (𝐼 × 2o))
40 lencl 14536 . . . . . . . . . . . . 13 (𝑥 ∈ Word (𝐼 × 2o) → (♯‘𝑥) ∈ ℕ0)
4139, 40syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝑊) → (♯‘𝑥) ∈ ℕ0)
4241nn0cnd 12580 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → (♯‘𝑥) ∈ ℂ)
43 2cnd 12336 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → 2 ∈ ℂ)
4434, 42, 43addcan2d 11459 . . . . . . . . . 10 ((𝜑𝑥𝑊) → ((0 + 2) = ((♯‘𝑥) + 2) ↔ 0 = (♯‘𝑥)))
4533, 44sylibd 238 . . . . . . . . 9 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → 0 = (♯‘𝑥)))
4612, 27, 28, 29efgtf 19716 . . . . . . . . . . . . . . . . . 18 (∅ ∈ 𝑊 → ((𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(♯‘∅)) × (𝐼 × 2o))⟶𝑊))
4746adantl 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ∅ ∈ 𝑊) → ((𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(♯‘∅)) × (𝐼 × 2o))⟶𝑊))
4847simpld 493 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∅ ∈ 𝑊) → (𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
4948rneqd 5936 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ∅ ∈ 𝑊) → ran (𝑇‘∅) = ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
5049eleq2d 2812 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))))
51 eqid 2726 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
52 ovex 7449 . . . . . . . . . . . . . . . 16 (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ V
5351, 52elrnmpo 7554 . . . . . . . . . . . . . . 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 14542 . . . . . . . . . . . . . . . . . . . . 21 ∅ ∈ Word (𝐼 × 2o)
5554a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∅ ∈ Word (𝐼 × 2o))
56 simprr 771 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑏 ∈ (𝐼 × 2o))
5728efgmf 19707 . . . . . . . . . . . . . . . . . . . . . . 23 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
5857ffvelcdmi 7089 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 ∈ (𝐼 × 2o) → (𝑀𝑏) ∈ (𝐼 × 2o))
5956, 58syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀𝑏) ∈ (𝐼 × 2o))
6056, 59s2cld 14875 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o))
61 ccatidid 14593 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ ++ ∅) = ∅
6261oveq1i 7426 . . . . . . . . . . . . . . . . . . . . . 22 ((∅ ++ ∅) ++ ∅) = (∅ ++ ∅)
6362, 61eqtr2i 2755 . . . . . . . . . . . . . . . . . . . . 21 ∅ = ((∅ ++ ∅) ++ ∅)
6463a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∅ = ((∅ ++ ∅) ++ ∅))
65 simprl 769 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...(♯‘∅)))
66 hash0 14379 . . . . . . . . . . . . . . . . . . . . . . . 24 (♯‘∅) = 0
6766oveq2i 7427 . . . . . . . . . . . . . . . . . . . . . . 23 (0...(♯‘∅)) = (0...0)
6865, 67eleqtrdi 2836 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...0))
69 elfz1eq 13560 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ (0...0) → 𝑎 = 0)
7068, 69syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = 0)
7170, 66eqtr4di 2784 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (♯‘∅))
7266oveq2i 7427 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 + (♯‘∅)) = (𝑎 + 0)
73 0cn 11247 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℂ
7470, 73eqeltrdi 2834 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ ℂ)
7574addridd 11455 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 + 0) = 𝑎)
7672, 75eqtr2id 2779 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (𝑎 + (♯‘∅)))
7755, 55, 55, 60, 64, 71, 76splval2 14760 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = ((∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ ∅))
78 ccatlid 14589 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o) → (∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) = ⟨“𝑏(𝑀𝑏)”⟩)
7978oveq1d 7431 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o) → ((∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ ∅) = (⟨“𝑏(𝑀𝑏)”⟩ ++ ∅))
80 ccatrid 14590 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o) → (⟨“𝑏(𝑀𝑏)”⟩ ++ ∅) = ⟨“𝑏(𝑀𝑏)”⟩)
8179, 80eqtrd 2766 . . . . . . . . . . . . . . . . . . . 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 2766 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = ⟨“𝑏(𝑀𝑏)”⟩)
8483eqeq2d 2737 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩))
851ad3antrrr 728 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → 𝐴𝐼)
86 1on 8500 . . . . . . . . . . . . . . . . . . . 20 1o ∈ On
8786a1i 11 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → 1o ∈ On)
88 simpr 483 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩)
8988fveq1d 6895 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = (⟨“𝑏(𝑀𝑏)”⟩‘1))
90 opex 5462 . . . . . . . . . . . . . . . . . . . . . 22 𝐵, ∅⟩ ∈ V
91 s2fv1 14892 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝐵, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩)
9290, 91ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩
93 fvex 6906 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀𝑏) ∈ V
94 s2fv1 14892 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀𝑏) ∈ V → (⟨“𝑏(𝑀𝑏)”⟩‘1) = (𝑀𝑏))
9593, 94ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩‘1) = (𝑀𝑏)
9689, 92, 953eqtr3g 2789 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐵, ∅⟩ = (𝑀𝑏))
9788fveq1d 6895 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = (⟨“𝑏(𝑀𝑏)”⟩‘0))
98 opex 5462 . . . . . . . . . . . . . . . . . . . . . . 23 𝐴, ∅⟩ ∈ V
99 s2fv0 14891 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝐴, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩)
10098, 99ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩
101 s2fv0 14891 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 ∈ V → (⟨“𝑏(𝑀𝑏)”⟩‘0) = 𝑏)
102101elv 3468 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“𝑏(𝑀𝑏)”⟩‘0) = 𝑏
10397, 100, 1023eqtr3g 2789 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐴, ∅⟩ = 𝑏)
104103fveq2d 6897 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = (𝑀𝑏))
10528efgmval 19706 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝐼 ∧ ∅ ∈ 2o) → (𝐴𝑀∅) = ⟨𝐴, (1o ∖ ∅)⟩)
10685, 5, 105sylancl 584 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (𝐴𝑀∅) = ⟨𝐴, (1o ∖ ∅)⟩)
107 df-ov 7419 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝑀∅) = (𝑀‘⟨𝐴, ∅⟩)
108 dif0 4370 . . . . . . . . . . . . . . . . . . . . . 22 (1o ∖ ∅) = 1o
109108opeq2i 4875 . . . . . . . . . . . . . . . . . . . . 21 𝐴, (1o ∖ ∅)⟩ = ⟨𝐴, 1o
110106, 107, 1093eqtr3g 2789 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = ⟨𝐴, 1o⟩)
11196, 104, 1103eqtr2rd 2773 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩)
112 opthg 5475 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝐼 ∧ 1o ∈ On) → (⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩ ↔ (𝐴 = 𝐵 ∧ 1o = ∅)))
113112simplbda 498 . . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . . 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 3202 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . 12 (𝜑 → ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1o = ∅))
121 hasheq0 14375 . . . . . . . . . . . . . . . 16 (𝑥 ∈ V → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
122121elv 3468 . . . . . . . . . . . . . . 15 ((♯‘𝑥) = 0 ↔ 𝑥 = ∅)
123 eleq1 2814 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (𝑥𝑊 ↔ ∅ ∈ 𝑊))
124 fveq2 6893 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∅ → (𝑇𝑥) = (𝑇‘∅))
125124rneqd 5936 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → ran (𝑇𝑥) = ran (𝑇‘∅))
126125eleq2d 2812 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)))
127123, 126anbi12d 630 . . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . 13 (0 = (♯‘𝑥) → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
130129imbi1d 340 . . . . . . . . . . . 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 451 . . . . . . . . 9 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → (0 = (♯‘𝑥) → 1o = ∅)))
13445, 133mpdd 43 . . . . . . . 8 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → 1o = ∅))
135134necon3ad 2943 . . . . . . 7 ((𝜑𝑥𝑊) → (1o ≠ ∅ → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)))
13622, 135mpi 20 . . . . . 6 ((𝜑𝑥𝑊) → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
137136nrexdv 3139 . . . . 5 (𝜑 → ¬ ∃𝑥𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
138 eliun 4997 . . . . 5 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑥𝑊 ran (𝑇𝑥) ↔ ∃𝑥𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
139137, 138sylnibr 328 . . . 4 (𝜑 → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑥𝑊 ran (𝑇𝑥))
14021, 139eldifd 3957 . . 3 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝑊 𝑥𝑊 ran (𝑇𝑥)))
141 frgpnabl.d . . 3 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
142140, 141eleqtrrdi 2837 . 2 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝐷)
143 df-s2 14852 . . . . 5 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)
14412, 27efger 19712 . . . . . . 7 Er 𝑊
145144a1i 11 . . . . . 6 (𝜑 Er 𝑊)
146145, 21erref 8746 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
147143, 146eqbrtrrid 5181 . . . 4 (𝜑 → (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
148143ovexi 7450 . . . . 5 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ V
149 ovex 7449 . . . . 5 (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∈ V
150148, 149elec 8772 . . . 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 19761 . . . . . 6 ((𝐼𝑉𝐴𝐼) → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
15413, 1, 153syl2anc 582 . . . . 5 (𝜑 → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
15527, 152vrgpval 19761 . . . . . 6 ((𝐼𝑉𝐵𝐼) → (𝑈𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] )
15613, 8, 155syl2anc 582 . . . . 5 (𝜑 → (𝑈𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] )
157154, 156oveq12d 7434 . . . 4 (𝜑 → ((𝑈𝐴) + (𝑈𝐵)) = ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ))
1587s1cld 14606 . . . . . 6 (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
159158, 20eleqtrrd 2829 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊)
16010s1cld 14606 . . . . . 6 (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
161160, 20eleqtrrd 2829 . . . . 5 (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
162 frgpnabl.g . . . . . 6 𝐺 = (freeGrp‘𝐼)
163 frgpnabl.p . . . . . 6 + = (+g𝐺)
16412, 162, 27, 163frgpadd 19757 . . . . 5 ((⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊 ∧ ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊) → ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
165159, 161, 164syl2anc 582 . . . 4 (𝜑 → ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
166157, 165eqtrd 2766 . . 3 (𝜑 → ((𝑈𝐴) + (𝑈𝐵)) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
167151, 166eleqtrrd 2829 . 2 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ((𝑈𝐴) + (𝑈𝐵)))
168142, 167elind 4192 1 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈𝐴) + (𝑈𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wne 2930  wrex 3060  Vcvv 3462  cdif 3943  cin 3945  c0 4322  {cpr 4625  cop 4629  cotp 4631   ciun 4993   class class class wbr 5145  cmpt 5228   I cid 5571   × cxp 5672  ran crn 5675  Oncon0 6368  wf 6542  cfv 6546  (class class class)co 7416  cmpo 7418  1oc1o 8481  2oc2o 8482   Er wer 8723  [cec 8724  cc 11147  0cc0 11149  1c1 11150   + caddc 11152  2c2 12313  0cn0 12518  ...cfz 13532  chash 14342  Word cword 14517   ++ cconcat 14573  ⟨“cs1 14598   splice csplice 14752  ⟨“cs2 14845  +gcplusg 17261   ~FG cefg 19700  freeGrpcfrgp 19701  varFGrpcvrgp 19702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4906  df-int 4947  df-iun 4995  df-iin 4996  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-er 8726  df-ec 8728  df-qs 8732  df-map 8849  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9478  df-inf 9479  df-card 9975  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-nn 12259  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12519  df-z 12605  df-dec 12724  df-uz 12869  df-fz 13533  df-fzo 13676  df-hash 14343  df-word 14518  df-concat 14574  df-s1 14599  df-substr 14644  df-pfx 14674  df-splice 14753  df-s2 14852  df-struct 17144  df-slot 17179  df-ndx 17191  df-base 17209  df-plusg 17274  df-mulr 17275  df-sca 17277  df-vsca 17278  df-ip 17279  df-tset 17280  df-ple 17281  df-ds 17283  df-imas 17518  df-qus 17519  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-frmd 18834  df-efg 19703  df-frgp 19704  df-vrgp 19705
This theorem is referenced by:  frgpnabllem2  19868
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