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Theorem frgpnabllem1 18204
Description: Lemma for frgpnabl 18206. (Contributed by Mario Carneiro, 21-Apr-2016.)
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 (𝜑𝐼 ∈ V)
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)

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 4755 . . . . . . . . 9 ∅ ∈ V
32prid1 4272 . . . . . . . 8 ∅ ∈ {∅, 1𝑜}
4 df2o3 7525 . . . . . . . 8 2𝑜 = {∅, 1𝑜}
53, 4eleqtrri 2697 . . . . . . 7 ∅ ∈ 2𝑜
6 opelxpi 5113 . . . . . . 7 ((𝐴𝐼 ∧ ∅ ∈ 2𝑜) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2𝑜))
71, 5, 6sylancl 693 . . . . . 6 (𝜑 → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2𝑜))
8 frgpnabl.b . . . . . . 7 (𝜑𝐵𝐼)
9 opelxpi 5113 . . . . . . 7 ((𝐵𝐼 ∧ ∅ ∈ 2𝑜) → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2𝑜))
108, 5, 9sylancl 693 . . . . . 6 (𝜑 → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2𝑜))
117, 10s2cld 13559 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2𝑜))
12 frgpnabl.w . . . . . 6 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
13 frgpnabl.i . . . . . . . 8 (𝜑𝐼 ∈ V)
14 2on 7520 . . . . . . . 8 2𝑜 ∈ On
15 xpexg 6920 . . . . . . . 8 ((𝐼 ∈ V ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
1613, 14, 15sylancl 693 . . . . . . 7 (𝜑 → (𝐼 × 2𝑜) ∈ V)
17 wrdexg 13261 . . . . . . 7 ((𝐼 × 2𝑜) ∈ V → Word (𝐼 × 2𝑜) ∈ V)
18 fvi 6217 . . . . . . 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 2667 . . . . 5 (𝜑𝑊 = Word (𝐼 × 2𝑜))
2111, 20eleqtrrd 2701 . . . 4 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
22 1n0 7527 . . . . . . 7 1𝑜 ≠ ∅
23 2cn 11042 . . . . . . . . . . . . . 14 2 ∈ ℂ
2423addid2i 10175 . . . . . . . . . . . . 13 (0 + 2) = 2
25 s2len 13577 . . . . . . . . . . . . 13 (#‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = 2
2624, 25eqtr4i 2646 . . . . . . . . . . . 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 18067 . . . . . . . . . . . . 13 ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (#‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((#‘𝑥) + 2))
3130adantll 749 . . . . . . . . . . . 12 (((𝜑𝑥𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (#‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((#‘𝑥) + 2))
3226, 31syl5eq 2667 . . . . . . . . . . 11 (((𝜑𝑥𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (0 + 2) = ((#‘𝑥) + 2))
3332ex 450 . . . . . . . . . 10 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → (0 + 2) = ((#‘𝑥) + 2)))
34 0cnd 9984 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → 0 ∈ ℂ)
35 simpr 477 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑊) → 𝑥𝑊)
3612efgrcl 18056 . . . . . . . . . . . . . . . 16 (𝑥𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
3736simprd 479 . . . . . . . . . . . . . . 15 (𝑥𝑊𝑊 = Word (𝐼 × 2𝑜))
3837adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑊) → 𝑊 = Word (𝐼 × 2𝑜))
3935, 38eleqtrd 2700 . . . . . . . . . . . . 13 ((𝜑𝑥𝑊) → 𝑥 ∈ Word (𝐼 × 2𝑜))
40 lencl 13270 . . . . . . . . . . . . 13 (𝑥 ∈ Word (𝐼 × 2𝑜) → (#‘𝑥) ∈ ℕ0)
4139, 40syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝑊) → (#‘𝑥) ∈ ℕ0)
4241nn0cnd 11304 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → (#‘𝑥) ∈ ℂ)
43 2cnd 11044 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → 2 ∈ ℂ)
4434, 42, 43addcan2d 10191 . . . . . . . . . 10 ((𝜑𝑥𝑊) → ((0 + 2) = ((#‘𝑥) + 2) ↔ 0 = (#‘𝑥)))
4533, 44sylibd 229 . . . . . . . . 9 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → 0 = (#‘𝑥)))
4612, 27, 28, 29efgtf 18063 . . . . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∅ ∈ 𝑊) → (𝑇‘∅) = (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
4948rneqd 5318 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ∅ ∈ 𝑊) → ran (𝑇‘∅) = ran (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
5049eleq2d 2684 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))))
51 eqid 2621 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
52 ovex 6638 . . . . . . . . . . . . . . . 16 (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ V
5351, 52elrnmpt2 6733 . . . . . . . . . . . . . . 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 13276 . . . . . . . . . . . . . . . . . . . . 21 ∅ ∈ Word (𝐼 × 2𝑜)
5554a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ∅ ∈ Word (𝐼 × 2𝑜))
56 simprr 795 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑏 ∈ (𝐼 × 2𝑜))
5728efgmf 18054 . . . . . . . . . . . . . . . . . . . . . . 23 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
5857ffvelrni 6319 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 ∈ (𝐼 × 2𝑜) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
5956, 58syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
6056, 59s2cld 13559 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
61 ccatlid 13315 . . . . . . . . . . . . . . . . . . . . . . . 24 (∅ ∈ Word (𝐼 × 2𝑜) → (∅ ++ ∅) = ∅)
6254, 61ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ ++ ∅) = ∅
6362oveq1i 6620 . . . . . . . . . . . . . . . . . . . . . 22 ((∅ ++ ∅) ++ ∅) = (∅ ++ ∅)
6463, 62eqtr2i 2644 . . . . . . . . . . . . . . . . . . . . 21 ∅ = ((∅ ++ ∅) ++ ∅)
6564a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ∅ = ((∅ ++ ∅) ++ ∅))
66 simprl 793 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ (0...(#‘∅)))
67 hash0 13105 . . . . . . . . . . . . . . . . . . . . . . . 24 (#‘∅) = 0
6867oveq2i 6621 . . . . . . . . . . . . . . . . . . . . . . 23 (0...(#‘∅)) = (0...0)
6966, 68syl6eleq 2708 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ (0...0))
70 elfz1eq 12301 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ (0...0) → 𝑎 = 0)
7169, 70syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 = 0)
7271, 67syl6eqr 2673 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 = (#‘∅))
7367oveq2i 6621 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 + (#‘∅)) = (𝑎 + 0)
74 0cn 9983 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℂ
7571, 74syl6eqel 2706 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ ℂ)
7675addid1d 10187 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 + 0) = 𝑎)
7773, 76syl5req 2668 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 = (𝑎 + (#‘∅)))
7855, 55, 55, 60, 65, 72, 77splval2 13452 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = ((∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ ∅))
79 ccatlid 13315 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜) → (∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) = ⟨“𝑏(𝑀𝑏)”⟩)
8079oveq1d 6625 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜) → ((∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ ∅) = (⟨“𝑏(𝑀𝑏)”⟩ ++ ∅))
81 ccatrid 13316 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜) → (⟨“𝑏(𝑀𝑏)”⟩ ++ ∅) = ⟨“𝑏(𝑀𝑏)”⟩)
8280, 81eqtrd 2655 . . . . . . . . . . . . . . . . . . . 20 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜) → ((∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ ∅) = ⟨“𝑏(𝑀𝑏)”⟩)
8360, 82syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ ∅) = ⟨“𝑏(𝑀𝑏)”⟩)
8478, 83eqtrd 2655 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = ⟨“𝑏(𝑀𝑏)”⟩)
8584eqeq2d 2631 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩))
861ad3antrrr 765 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → 𝐴𝐼)
87 1on 7519 . . . . . . . . . . . . . . . . . . . 20 1𝑜 ∈ On
8887a1i 11 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → 1𝑜 ∈ On)
89 simpr 477 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩)
9089fveq1d 6155 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = (⟨“𝑏(𝑀𝑏)”⟩‘1))
91 opex 4898 . . . . . . . . . . . . . . . . . . . . . 22 𝐵, ∅⟩ ∈ V
92 s2fv1 13576 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝐵, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩)
9391, 92ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩
94 fvex 6163 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀𝑏) ∈ V
95 s2fv1 13576 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀𝑏) ∈ V → (⟨“𝑏(𝑀𝑏)”⟩‘1) = (𝑀𝑏))
9694, 95ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩‘1) = (𝑀𝑏)
9790, 93, 963eqtr3g 2678 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐵, ∅⟩ = (𝑀𝑏))
9889fveq1d 6155 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = (⟨“𝑏(𝑀𝑏)”⟩‘0))
99 opex 4898 . . . . . . . . . . . . . . . . . . . . . . 23 𝐴, ∅⟩ ∈ V
100 s2fv0 13575 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝐴, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩)
10199, 100ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩
102 vex 3192 . . . . . . . . . . . . . . . . . . . . . . 23 𝑏 ∈ V
103 s2fv0 13575 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 ∈ V → (⟨“𝑏(𝑀𝑏)”⟩‘0) = 𝑏)
104102, 103ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“𝑏(𝑀𝑏)”⟩‘0) = 𝑏
10598, 101, 1043eqtr3g 2678 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐴, ∅⟩ = 𝑏)
106105fveq2d 6157 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = (𝑀𝑏))
10728efgmval 18053 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝐼 ∧ ∅ ∈ 2𝑜) → (𝐴𝑀∅) = ⟨𝐴, (1𝑜 ∖ ∅)⟩)
10886, 5, 107sylancl 693 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (𝐴𝑀∅) = ⟨𝐴, (1𝑜 ∖ ∅)⟩)
109 df-ov 6613 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝑀∅) = (𝑀‘⟨𝐴, ∅⟩)
110 dif0 3929 . . . . . . . . . . . . . . . . . . . . . 22 (1𝑜 ∖ ∅) = 1𝑜
111110opeq2i 4379 . . . . . . . . . . . . . . . . . . . . 21 𝐴, (1𝑜 ∖ ∅)⟩ = ⟨𝐴, 1𝑜
112108, 109, 1113eqtr3g 2678 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = ⟨𝐴, 1𝑜⟩)
11397, 106, 1123eqtr2rd 2662 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐴, 1𝑜⟩ = ⟨𝐵, ∅⟩)
114 opthg 4911 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝐼 ∧ 1𝑜 ∈ On) → (⟨𝐴, 1𝑜⟩ = ⟨𝐵, ∅⟩ ↔ (𝐴 = 𝐵 ∧ 1𝑜 = ∅)))
115114simplbda 653 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝐼 ∧ 1𝑜 ∈ On) ∧ ⟨𝐴, 1𝑜⟩ = ⟨𝐵, ∅⟩) → 1𝑜 = ∅)
11686, 88, 113, 115syl21anc 1322 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → 1𝑜 = ∅)
117116ex 450 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩ → 1𝑜 = ∅))
11885, 117sylbid 230 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → 1𝑜 = ∅))
119118rexlimdvva 3032 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ∅ ∈ 𝑊) → (∃𝑎 ∈ (0...(#‘∅))∃𝑏 ∈ (𝐼 × 2𝑜)⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → 1𝑜 = ∅))
12053, 119syl5bi 232 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) → 1𝑜 = ∅))
12150, 120sylbid 230 . . . . . . . . . . . . 13 ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) → 1𝑜 = ∅))
122121expimpd 628 . . . . . . . . . . . 12 (𝜑 → ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1𝑜 = ∅))
123 vex 3192 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
124 hasheq0 13101 . . . . . . . . . . . . . . . 16 (𝑥 ∈ V → ((#‘𝑥) = 0 ↔ 𝑥 = ∅))
125123, 124ax-mp 5 . . . . . . . . . . . . . . 15 ((#‘𝑥) = 0 ↔ 𝑥 = ∅)
126 eleq1 2686 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (𝑥𝑊 ↔ ∅ ∈ 𝑊))
127 fveq2 6153 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∅ → (𝑇𝑥) = (𝑇‘∅))
128127rneqd 5318 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → ran (𝑇𝑥) = ran (𝑇‘∅))
129128eleq2d 2684 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)))
130126, 129anbi12d 746 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
131125, 130sylbi 207 . . . . . . . . . . . . . 14 ((#‘𝑥) = 0 → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
132131eqcoms 2629 . . . . . . . . . . . . 13 (0 = (#‘𝑥) → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
133132imbi1d 331 . . . . . . . . . . . 12 (0 = (#‘𝑥) → (((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → 1𝑜 = ∅) ↔ ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1𝑜 = ∅)))
134122, 133syl5ibrcom 237 . . . . . . . . . . 11 (𝜑 → (0 = (#‘𝑥) → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → 1𝑜 = ∅)))
135134com23 86 . . . . . . . . . 10 (𝜑 → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (0 = (#‘𝑥) → 1𝑜 = ∅)))
136135expdimp 453 . . . . . . . . 9 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → (0 = (#‘𝑥) → 1𝑜 = ∅)))
13745, 136mpdd 43 . . . . . . . 8 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → 1𝑜 = ∅))
138137necon3ad 2803 . . . . . . 7 ((𝜑𝑥𝑊) → (1𝑜 ≠ ∅ → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)))
13922, 138mpi 20 . . . . . 6 ((𝜑𝑥𝑊) → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
140139nrexdv 2996 . . . . 5 (𝜑 → ¬ ∃𝑥𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
141 eliun 4495 . . . . 5 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑥𝑊 ran (𝑇𝑥) ↔ ∃𝑥𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
142140, 141sylnibr 319 . . . 4 (𝜑 → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑥𝑊 ran (𝑇𝑥))
14321, 142eldifd 3570 . . 3 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝑊 𝑥𝑊 ran (𝑇𝑥)))
144 frgpnabl.d . . 3 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
145143, 144syl6eleqr 2709 . 2 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝐷)
146 df-s2 13537 . . . . 5 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)
14712, 27efger 18059 . . . . . . 7 Er 𝑊
148147a1i 11 . . . . . 6 (𝜑 Er 𝑊)
149148, 21erref 7714 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
150146, 149syl5eqbrr 4654 . . . 4 (𝜑 → (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
151 ovex 6638 . . . . . 6 (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∈ V
152146, 151eqeltri 2694 . . . . 5 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ V
153152, 151elec 7738 . . . 4 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ↔ (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
154150, 153sylibr 224 . . 3 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
155 frgpnabl.u . . . . . . 7 𝑈 = (varFGrp𝐼)
15627, 155vrgpval 18108 . . . . . 6 ((𝐼 ∈ V ∧ 𝐴𝐼) → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
15713, 1, 156syl2anc 692 . . . . 5 (𝜑 → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
15827, 155vrgpval 18108 . . . . . 6 ((𝐼 ∈ V ∧ 𝐵𝐼) → (𝑈𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] )
15913, 8, 158syl2anc 692 . . . . 5 (𝜑 → (𝑈𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] )
160157, 159oveq12d 6628 . . . 4 (𝜑 → ((𝑈𝐴) + (𝑈𝐵)) = ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ))
1617s1cld 13329 . . . . . 6 (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ Word (𝐼 × 2𝑜))
162161, 20eleqtrrd 2701 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊)
16310s1cld 13329 . . . . . 6 (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2𝑜))
164163, 20eleqtrrd 2701 . . . . 5 (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
165 frgpnabl.g . . . . . 6 𝐺 = (freeGrp‘𝐼)
166 frgpnabl.p . . . . . 6 + = (+g𝐺)
16712, 165, 27, 166frgpadd 18104 . . . . 5 ((⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊 ∧ ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊) → ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
168162, 164, 167syl2anc 692 . . . 4 (𝜑 → ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
169160, 168eqtrd 2655 . . 3 (𝜑 → ((𝑈𝐴) + (𝑈𝐵)) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
170154, 169eleqtrrd 2701 . 2 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ((𝑈𝐴) + (𝑈𝐵)))
171145, 170elind 3781 1 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈𝐴) + (𝑈𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wrex 2908  Vcvv 3189  cdif 3556  cin 3558  c0 3896  {cpr 4155  cop 4159  cotp 4161   ciun 4490   class class class wbr 4618  cmpt 4678   I cid 4989   × cxp 5077  ran crn 5080  Oncon0 5687  wf 5848  cfv 5852  (class class class)co 6610  cmpt2 6612  1𝑜c1o 7505  2𝑜c2o 7506   Er wer 7691  [cec 7692  cc 9885  0cc0 9887  1c1 9888   + caddc 9890  2c2 11021  0cn0 11243  ...cfz 12275  #chash 13064  Word cword 13237   ++ cconcat 13239  ⟨“cs1 13240   splice csplice 13242  ⟨“cs2 13530  +gcplusg 15869   ~FG cefg 18047  freeGrpcfrgp 18048  varFGrpcvrgp 18049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-ot 4162  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-ec 7696  df-qs 7700  df-map 7811  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-sup 8299  df-inf 8300  df-card 8716  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-3 11031  df-4 11032  df-5 11033  df-6 11034  df-7 11035  df-8 11036  df-9 11037  df-n0 11244  df-z 11329  df-dec 11445  df-uz 11639  df-fz 12276  df-fzo 12414  df-hash 13065  df-word 13245  df-concat 13247  df-s1 13248  df-substr 13249  df-splice 13250  df-s2 13537  df-struct 15790  df-ndx 15791  df-slot 15792  df-base 15793  df-plusg 15882  df-mulr 15883  df-sca 15885  df-vsca 15886  df-ip 15887  df-tset 15888  df-ple 15889  df-ds 15892  df-imas 16096  df-qus 16097  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-frmd 17314  df-efg 18050  df-frgp 18051  df-vrgp 18052
This theorem is referenced by:  frgpnabllem2  18205
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