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Theorem frgpnabllem1 18099
 Description: Lemma for frgpnabl 18101. (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 4718 . . . . . . . . 9 ∅ ∈ V
32prid1 4241 . . . . . . . 8 ∅ ∈ {∅, 1𝑜}
4 df2o3 7460 . . . . . . . 8 2𝑜 = {∅, 1𝑜}
53, 4eleqtrri 2687 . . . . . . 7 ∅ ∈ 2𝑜
6 opelxpi 5072 . . . . . . 7 ((𝐴𝐼 ∧ ∅ ∈ 2𝑜) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2𝑜))
71, 5, 6sylancl 693 . . . . . 6 (𝜑 → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2𝑜))
8 frgpnabl.b . . . . . . 7 (𝜑𝐵𝐼)
9 opelxpi 5072 . . . . . . 7 ((𝐵𝐼 ∧ ∅ ∈ 2𝑜) → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2𝑜))
108, 5, 9sylancl 693 . . . . . 6 (𝜑 → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2𝑜))
117, 10s2cld 13466 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2𝑜))
12 frgpnabl.w . . . . . 6 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
13 frgpnabl.i . . . . . . . 8 (𝜑𝐼 ∈ V)
14 2on 7455 . . . . . . . 8 2𝑜 ∈ On
15 xpexg 6858 . . . . . . . 8 ((𝐼 ∈ V ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
1613, 14, 15sylancl 693 . . . . . . 7 (𝜑 → (𝐼 × 2𝑜) ∈ V)
17 wrdexg 13170 . . . . . . 7 ((𝐼 × 2𝑜) ∈ V → Word (𝐼 × 2𝑜) ∈ V)
18 fvi 6165 . . . . . . 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 2656 . . . . 5 (𝜑𝑊 = Word (𝐼 × 2𝑜))
2111, 20eleqtrrd 2691 . . . 4 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
22 1n0 7462 . . . . . . 7 1𝑜 ≠ ∅
23 2cn 10968 . . . . . . . . . . . . . 14 2 ∈ ℂ
2423addid2i 10103 . . . . . . . . . . . . 13 (0 + 2) = 2
25 s2len 13484 . . . . . . . . . . . . 13 (#‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = 2
2624, 25eqtr4i 2635 . . . . . . . . . . . 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 17962 . . . . . . . . . . . . 13 ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (#‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((#‘𝑥) + 2))
3130adantll 746 . . . . . . . . . . . 12 (((𝜑𝑥𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (#‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((#‘𝑥) + 2))
3226, 31syl5eq 2656 . . . . . . . . . . 11 (((𝜑𝑥𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (0 + 2) = ((#‘𝑥) + 2))
3332ex 449 . . . . . . . . . 10 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → (0 + 2) = ((#‘𝑥) + 2)))
34 0cnd 9912 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → 0 ∈ ℂ)
35 simpr 476 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑊) → 𝑥𝑊)
3612efgrcl 17951 . . . . . . . . . . . . . . . 16 (𝑥𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
3736simprd 478 . . . . . . . . . . . . . . 15 (𝑥𝑊𝑊 = Word (𝐼 × 2𝑜))
3837adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑊) → 𝑊 = Word (𝐼 × 2𝑜))
3935, 38eleqtrd 2690 . . . . . . . . . . . . 13 ((𝜑𝑥𝑊) → 𝑥 ∈ Word (𝐼 × 2𝑜))
40 lencl 13179 . . . . . . . . . . . . 13 (𝑥 ∈ Word (𝐼 × 2𝑜) → (#‘𝑥) ∈ ℕ0)
4139, 40syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝑊) → (#‘𝑥) ∈ ℕ0)
4241nn0cnd 11230 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → (#‘𝑥) ∈ ℂ)
43 2cnd 10970 . . . . . . . . . . 11 ((𝜑𝑥𝑊) → 2 ∈ ℂ)
4434, 42, 43addcan2d 10119 . . . . . . . . . 10 ((𝜑𝑥𝑊) → ((0 + 2) = ((#‘𝑥) + 2) ↔ 0 = (#‘𝑥)))
4533, 44sylibd 228 . . . . . . . . 9 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → 0 = (#‘𝑥)))
4612, 27, 28, 29efgtf 17958 . . . . . . . . . . . . . . . . . 18 (∅ ∈ 𝑊 → ((𝑇‘∅) = (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(#‘∅)) × (𝐼 × 2𝑜))⟶𝑊))
4746adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ∅ ∈ 𝑊) → ((𝑇‘∅) = (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(#‘∅)) × (𝐼 × 2𝑜))⟶𝑊))
4847simpld 474 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∅ ∈ 𝑊) → (𝑇‘∅) = (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
4948rneqd 5274 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ∅ ∈ 𝑊) → ran (𝑇‘∅) = ran (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
5049eleq2d 2673 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))))
51 eqid 2610 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
52 ovex 6577 . . . . . . . . . . . . . . . 16 (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ V
5351, 52elrnmpt2 6671 . . . . . . . . . . . . . . 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 13185 . . . . . . . . . . . . . . . . . . . . 21 ∅ ∈ Word (𝐼 × 2𝑜)
5554a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ∅ ∈ Word (𝐼 × 2𝑜))
56 simprr 792 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑏 ∈ (𝐼 × 2𝑜))
5728efgmf 17949 . . . . . . . . . . . . . . . . . . . . . . 23 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
5857ffvelrni 6266 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 ∈ (𝐼 × 2𝑜) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
5956, 58syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
6056, 59s2cld 13466 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
61 ccatlid 13222 . . . . . . . . . . . . . . . . . . . . . . . 24 (∅ ∈ Word (𝐼 × 2𝑜) → (∅ ++ ∅) = ∅)
6254, 61ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ ++ ∅) = ∅
6362oveq1i 6559 . . . . . . . . . . . . . . . . . . . . . 22 ((∅ ++ ∅) ++ ∅) = (∅ ++ ∅)
6463, 62eqtr2i 2633 . . . . . . . . . . . . . . . . . . . . 21 ∅ = ((∅ ++ ∅) ++ ∅)
6564a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ∅ = ((∅ ++ ∅) ++ ∅))
66 simprl 790 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ (0...(#‘∅)))
67 hash0 13019 . . . . . . . . . . . . . . . . . . . . . . . 24 (#‘∅) = 0
6867oveq2i 6560 . . . . . . . . . . . . . . . . . . . . . . 23 (0...(#‘∅)) = (0...0)
6966, 68syl6eleq 2698 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ (0...0))
70 elfz1eq 12223 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ (0...0) → 𝑎 = 0)
7169, 70syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 = 0)
7271, 67syl6eqr 2662 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 = (#‘∅))
7367oveq2i 6560 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 + (#‘∅)) = (𝑎 + 0)
74 0cn 9911 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℂ
7571, 74syl6eqel 2696 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ ℂ)
7675addid1d 10115 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 + 0) = 𝑎)
7773, 76syl5req 2657 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 = (𝑎 + (#‘∅)))
7855, 55, 55, 60, 65, 72, 77splval2 13359 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = ((∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ ∅))
79 ccatlid 13222 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜) → (∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) = ⟨“𝑏(𝑀𝑏)”⟩)
8079oveq1d 6564 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜) → ((∅ ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ ∅) = (⟨“𝑏(𝑀𝑏)”⟩ ++ ∅))
81 ccatrid 13223 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜) → (⟨“𝑏(𝑀𝑏)”⟩ ++ ∅) = ⟨“𝑏(𝑀𝑏)”⟩)
8280, 81eqtrd 2644 . . . . . . . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = ⟨“𝑏(𝑀𝑏)”⟩)
8584eqeq2d 2620 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩))
861ad3antrrr 762 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → 𝐴𝐼)
87 1on 7454 . . . . . . . . . . . . . . . . . . . 20 1𝑜 ∈ On
8887a1i 11 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → 1𝑜 ∈ On)
89 simpr 476 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩)
9089fveq1d 6105 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = (⟨“𝑏(𝑀𝑏)”⟩‘1))
91 opex 4859 . . . . . . . . . . . . . . . . . . . . . 22 𝐵, ∅⟩ ∈ V
92 s2fv1 13483 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝐵, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩)
9391, 92ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩
94 fvex 6113 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀𝑏) ∈ V
95 s2fv1 13483 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀𝑏) ∈ V → (⟨“𝑏(𝑀𝑏)”⟩‘1) = (𝑀𝑏))
9694, 95ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (⟨“𝑏(𝑀𝑏)”⟩‘1) = (𝑀𝑏)
9790, 93, 963eqtr3g 2667 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐵, ∅⟩ = (𝑀𝑏))
9889fveq1d 6105 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = (⟨“𝑏(𝑀𝑏)”⟩‘0))
99 opex 4859 . . . . . . . . . . . . . . . . . . . . . . 23 𝐴, ∅⟩ ∈ V
100 s2fv0 13482 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝐴, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩)
10199, 100ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩
102 vex 3176 . . . . . . . . . . . . . . . . . . . . . . 23 𝑏 ∈ V
103 s2fv0 13482 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 ∈ V → (⟨“𝑏(𝑀𝑏)”⟩‘0) = 𝑏)
104102, 103ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (⟨“𝑏(𝑀𝑏)”⟩‘0) = 𝑏
10598, 101, 1043eqtr3g 2667 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐴, ∅⟩ = 𝑏)
106105fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = (𝑀𝑏))
10728efgmval 17948 . . . . . . . . . . . . . . . . . . . . . 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 6552 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝑀∅) = (𝑀‘⟨𝐴, ∅⟩)
110 dif0 3904 . . . . . . . . . . . . . . . . . . . . . 22 (1𝑜 ∖ ∅) = 1𝑜
111110opeq2i 4344 . . . . . . . . . . . . . . . . . . . . 21 𝐴, (1𝑜 ∖ ∅)⟩ = ⟨𝐴, 1𝑜
112108, 109, 1113eqtr3g 2667 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = ⟨𝐴, 1𝑜⟩)
11397, 106, 1123eqtr2rd 2651 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → ⟨𝐴, 1𝑜⟩ = ⟨𝐵, ∅⟩)
114 opthg 4872 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝐼 ∧ 1𝑜 ∈ On) → (⟨𝐴, 1𝑜⟩ = ⟨𝐵, ∅⟩ ↔ (𝐴 = 𝐵 ∧ 1𝑜 = ∅)))
115114simplbda 652 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝐼 ∧ 1𝑜 ∈ On) ∧ ⟨𝐴, 1𝑜⟩ = ⟨𝐵, ∅⟩) → 1𝑜 = ∅)
11686, 88, 113, 115syl21anc 1317 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩) → 1𝑜 = ∅)
117116ex 449 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀𝑏)”⟩ → 1𝑜 = ∅))
11885, 117sylbid 229 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → 1𝑜 = ∅))
119118rexlimdvva 3020 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ∅ ∈ 𝑊) → (∃𝑎 ∈ (0...(#‘∅))∃𝑏 ∈ (𝐼 × 2𝑜)⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → 1𝑜 = ∅))
12053, 119syl5bi 231 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) → 1𝑜 = ∅))
12150, 120sylbid 229 . . . . . . . . . . . . 13 ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) → 1𝑜 = ∅))
122121expimpd 627 . . . . . . . . . . . 12 (𝜑 → ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1𝑜 = ∅))
123 vex 3176 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
124 hasheq0 13015 . . . . . . . . . . . . . . . 16 (𝑥 ∈ V → ((#‘𝑥) = 0 ↔ 𝑥 = ∅))
125123, 124ax-mp 5 . . . . . . . . . . . . . . 15 ((#‘𝑥) = 0 ↔ 𝑥 = ∅)
126 eleq1 2676 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (𝑥𝑊 ↔ ∅ ∈ 𝑊))
127 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∅ → (𝑇𝑥) = (𝑇‘∅))
128127rneqd 5274 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → ran (𝑇𝑥) = ran (𝑇‘∅))
129128eleq2d 2673 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)))
130126, 129anbi12d 743 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
131125, 130sylbi 206 . . . . . . . . . . . . . 14 ((#‘𝑥) = 0 → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
132131eqcoms 2618 . . . . . . . . . . . . 13 (0 = (#‘𝑥) → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
133132imbi1d 330 . . . . . . . . . . . 12 (0 = (#‘𝑥) → (((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → 1𝑜 = ∅) ↔ ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1𝑜 = ∅)))
134122, 133syl5ibrcom 236 . . . . . . . . . . 11 (𝜑 → (0 = (#‘𝑥) → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → 1𝑜 = ∅)))
135134com23 84 . . . . . . . . . 10 (𝜑 → ((𝑥𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)) → (0 = (#‘𝑥) → 1𝑜 = ∅)))
136135expdimp 452 . . . . . . . . 9 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → (0 = (#‘𝑥) → 1𝑜 = ∅)))
13745, 136mpdd 42 . . . . . . . 8 ((𝜑𝑥𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥) → 1𝑜 = ∅))
138137necon3ad 2795 . . . . . . 7 ((𝜑𝑥𝑊) → (1𝑜 ≠ ∅ → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥)))
13922, 138mpi 20 . . . . . 6 ((𝜑𝑥𝑊) → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
140139nrexdv 2984 . . . . 5 (𝜑 → ¬ ∃𝑥𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
141 eliun 4460 . . . . 5 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑥𝑊 ran (𝑇𝑥) ↔ ∃𝑥𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇𝑥))
142140, 141sylnibr 318 . . . 4 (𝜑 → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑥𝑊 ran (𝑇𝑥))
14321, 142eldifd 3551 . . 3 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝑊 𝑥𝑊 ran (𝑇𝑥)))
144 frgpnabl.d . . 3 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
145143, 144syl6eleqr 2699 . 2 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝐷)
146 df-s2 13444 . . . . 5 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)
14712, 27efger 17954 . . . . . . 7 Er 𝑊
148147a1i 11 . . . . . 6 (𝜑 Er 𝑊)
149148, 21erref 7649 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
150146, 149syl5eqbrr 4619 . . . 4 (𝜑 → (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
151 ovex 6577 . . . . . 6 (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∈ V
152146, 151eqeltri 2684 . . . . 5 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ V
153152, 151elec 7673 . . . 4 (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ↔ (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
154150, 153sylibr 223 . . 3 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
155 frgpnabl.u . . . . . . 7 𝑈 = (varFGrp𝐼)
15627, 155vrgpval 18003 . . . . . 6 ((𝐼 ∈ V ∧ 𝐴𝐼) → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
15713, 1, 156syl2anc 691 . . . . 5 (𝜑 → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )
15827, 155vrgpval 18003 . . . . . 6 ((𝐼 ∈ V ∧ 𝐵𝐼) → (𝑈𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] )
15913, 8, 158syl2anc 691 . . . . 5 (𝜑 → (𝑈𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] )
160157, 159oveq12d 6567 . . . 4 (𝜑 → ((𝑈𝐴) + (𝑈𝐵)) = ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ))
1617s1cld 13236 . . . . . 6 (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ Word (𝐼 × 2𝑜))
162161, 20eleqtrrd 2691 . . . . 5 (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊)
16310s1cld 13236 . . . . . 6 (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2𝑜))
164163, 20eleqtrrd 2691 . . . . 5 (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
165 frgpnabl.g . . . . . 6 𝐺 = (freeGrp‘𝐼)
166 frgpnabl.p . . . . . 6 + = (+g𝐺)
16712, 165, 27, 166frgpadd 17999 . . . . 5 ((⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊 ∧ ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊) → ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
168162, 164, 167syl2anc 691 . . . 4 (𝜑 → ([⟨“⟨𝐴, ∅⟩”⟩] + [⟨“⟨𝐵, ∅⟩”⟩] ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
169160, 168eqtrd 2644 . . 3 (𝜑 → ((𝑈𝐴) + (𝑈𝐵)) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] )
170154, 169eleqtrrd 2691 . 2 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ((𝑈𝐴) + (𝑈𝐵)))
171145, 170elind 3760 1 (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈𝐴) + (𝑈𝐵))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539  ∅c0 3874  {cpr 4127  ⟨cop 4131  ⟨cotp 4133  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   × cxp 5036  ran crn 5039  Oncon0 5640  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1𝑜c1o 7440  2𝑜c2o 7441   Er wer 7626  [cec 7627  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  ℕ0cn0 11169  ...cfz 12197  #chash 12979  Word cword 13146   ++ cconcat 13148  ⟨“cs1 13149   splice csplice 13151  ⟨“cs2 13437  +gcplusg 15768   ~FG cefg 17942  freeGrpcfrgp 17943  varFGrpcvrgp 17944 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-s2 13444  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-imas 15991  df-qus 15992  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-frmd 17209  df-efg 17945  df-frgp 17946  df-vrgp 17947 This theorem is referenced by:  frgpnabllem2  18100
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