Theorem frgpnabllem1 19810

Description: Lemma for frgpnabl 19812. (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	⊢ ∼ = ( ~FG ‘𝐼)
frgpnabl.p	⊢ + = (+g‘𝐺)
frgpnabl.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
frgpnabl.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
frgpnabl.d	⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
frgpnabl.u	⊢ 𝑈 = (varFGrp‘𝐼)
frgpnabl.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
frgpnabl.a	⊢ (𝜑 → 𝐴 ∈ 𝐼)
frgpnabl.b	⊢ (𝜑 → 𝐵 ∈ 𝐼)

Assertion

Ref	Expression
frgpnabllem1	⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵))))

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.

Step	Hyp	Ref	Expression
1		frgpnabl.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐼)
2		0ex 5265	. . . . . . . . 9 ⊢ ∅ ∈ V
3	2	prid1 4729	. . . . . . . 8 ⊢ ∅ ∈ {∅, 1o}
4		df2o3 8445	. . . . . . . 8 ⊢ 2o = {∅, 1o}
5	3, 4	eleqtrri 2828	. . . . . . 7 ⊢ ∅ ∈ 2o
6		opelxpi 5678	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2o))
7	1, 5, 6	sylancl 586	. . . . . 6 ⊢ (𝜑 → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2o))
8		frgpnabl.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝐼)
9		opelxpi 5678	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐼 ∧ ∅ ∈ 2o) → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2o))
10	8, 5, 9	sylancl 586	. . . . . 6 ⊢ (𝜑 → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2o))
11	7, 10	s2cld 14844	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
12		frgpnabl.w	. . . . . 6 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
13		frgpnabl.i	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
14		2on 8450	. . . . . . . 8 ⊢ 2o ∈ On
15		xpexg 7729	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
16	13, 14, 15	sylancl 586	. . . . . . 7 ⊢ (𝜑 → (𝐼 × 2o) ∈ V)
17		wrdexg 14496	. . . . . . 7 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
18		fvi 6940	. . . . . . 7 ⊢ (Word (𝐼 × 2o) ∈ V → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o))
19	16, 17, 18	3syl 18	. . . . . 6 ⊢ (𝜑 → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o))
20	12, 19	eqtrid 2777	. . . . 5 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o))
21	11, 20	eleqtrrd 2832	. . . 4 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
22		1n0 8455	. . . . . . 7 ⊢ 1o ≠ ∅
23		2cn 12268	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
24	23	addlidi 11369	. . . . . . . . . . . . 13 ⊢ (0 + 2) = 2
25		s2len 14862	. . . . . . . . . . . . 13 ⊢ (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = 2
26	24, 25	eqtr4i 2756	. . . . . . . . . . . 12 ⊢ (0 + 2) = (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
27		frgpnabl.r	. . . . . . . . . . . . . 14 ⊢ ∼ = ( ~FG ‘𝐼)
28		frgpnabl.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
29		frgpnabl.t	. . . . . . . . . . . . . 14 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
30	12, 27, 28, 29	efgtlen 19663	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((♯‘𝑥) + 2))
31	30	adantll 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((♯‘𝑥) + 2))
32	26, 31	eqtrid 2777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (0 + 2) = ((♯‘𝑥) + 2))
33	32	ex 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) → (0 + 2) = ((♯‘𝑥) + 2)))
34		0cnd 11174	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 0 ∈ ℂ)
35		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑥 ∈ 𝑊)
36	12	efgrcl 19652	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
37	36	simprd 495	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o))
38	37	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑊 = Word (𝐼 × 2o))
39	35, 38	eleqtrd 2831	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑥 ∈ Word (𝐼 × 2o))
40		lencl 14505	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (♯‘𝑥) ∈ ℕ0)
41	39, 40	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (♯‘𝑥) ∈ ℕ0)
42	41	nn0cnd 12512	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (♯‘𝑥) ∈ ℂ)
43		2cnd 12271	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 2 ∈ ℂ)
44	34, 42, 43	addcan2d 11385	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ((0 + 2) = ((♯‘𝑥) + 2) ↔ 0 = (♯‘𝑥)))
45	33, 44	sylibd 239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) → 0 = (♯‘𝑥)))
46	12, 27, 28, 29	efgtf 19659	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∈ 𝑊 → ((𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(♯‘∅)) × (𝐼 × 2o))⟶𝑊))
47	46	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → ((𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(♯‘∅)) × (𝐼 × 2o))⟶𝑊))
48	47	simpld 494	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
49	48	rneqd 5905	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → ran (𝑇‘∅) = ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
50	49	eleq2d 2815	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))))
51		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
52		ovex 7423	. . . . . . . . . . . . . . . 16 ⊢ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ V
53	51, 52	elrnmpo 7528	. . . . . . . . . . . . . . 15 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ↔ ∃𝑎 ∈ (0...(♯‘∅))∃𝑏 ∈ (𝐼 × 2o)⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
54		wrd0 14511	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ ∈ Word (𝐼 × 2o)
55	54	a1i 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))
57	28	efgmf 19650	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
58	57	ffvelcdmi 7058	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 ∈ (𝐼 × 2o) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
59	56, 58	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
60	56, 59	s2cld 14844	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
61		ccatidid 14562	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ ++ ∅) = ∅
62	61	oveq1i 7400	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∅ ++ ∅) ++ ∅) = (∅ ++ ∅)
63	62, 61	eqtr2i 2754	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ = ((∅ ++ ∅) ++ ∅)
64	63	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∅ = ((∅ ++ ∅) ++ ∅))
65		simprl 770	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...(♯‘∅)))
66		hash0 14339	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (♯‘∅) = 0
67	66	oveq2i 7401	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0...(♯‘∅)) = (0...0)
68	65, 67	eleqtrdi 2839	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...0))
69		elfz1eq 13503	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 ∈ (0...0) → 𝑎 = 0)
70	68, 69	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = 0)
71	70, 66	eqtr4di 2783	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (♯‘∅))
72	66	oveq2i 7401	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 + (♯‘∅)) = (𝑎 + 0)
73		0cn 11173	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℂ
74	70, 73	eqeltrdi 2837	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ ℂ)
75	74	addridd 11381	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 + 0) = 𝑎)
76	72, 75	eqtr2id 2778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (𝑎 + (♯‘∅)))
77	55, 55, 55, 60, 64, 71, 76	splval2 14729	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = ((∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ ∅))
78		ccatlid 14558	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o) → (∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) = ⟨“𝑏(𝑀‘𝑏)”⟩)
79	78	oveq1d 7405	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o) → ((∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ ∅) = (⟨“𝑏(𝑀‘𝑏)”⟩ ++ ∅))
80		ccatrid 14559	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o) → (⟨“𝑏(𝑀‘𝑏)”⟩ ++ ∅) = ⟨“𝑏(𝑀‘𝑏)”⟩)
81	79, 80	eqtrd 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o) → ((∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ ∅) = ⟨“𝑏(𝑀‘𝑏)”⟩)
82	60, 81	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ ∅) = ⟨“𝑏(𝑀‘𝑏)”⟩)
83	77, 82	eqtrd 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = ⟨“𝑏(𝑀‘𝑏)”⟩)
84	83	eqeq2d 2741	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩))
85	1	ad3antrrr 730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → 𝐴 ∈ 𝐼)
86		1on 8449	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1o ∈ On
87	86	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → 1o ∈ On)
88		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩)
89	88	fveq1d 6863	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = (⟨“𝑏(𝑀‘𝑏)”⟩‘1))
90		opex 5427	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨𝐵, ∅⟩ ∈ V
91		s2fv1 14861	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝐵, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩)
92	90, 91	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩
93		fvex 6874	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀‘𝑏) ∈ V
94		s2fv1 14861	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑀‘𝑏) ∈ V → (⟨“𝑏(𝑀‘𝑏)”⟩‘1) = (𝑀‘𝑏))
95	93, 94	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩‘1) = (𝑀‘𝑏)
96	89, 92, 95	3eqtr3g 2788	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐵, ∅⟩ = (𝑀‘𝑏))
97	88	fveq1d 6863	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = (⟨“𝑏(𝑀‘𝑏)”⟩‘0))
98		opex 5427	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨𝐴, ∅⟩ ∈ V
99		s2fv0 14860	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝐴, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩)
100	98, 99	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩
101		s2fv0 14860	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 ∈ V → (⟨“𝑏(𝑀‘𝑏)”⟩‘0) = 𝑏)
102	101	elv 3455	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩‘0) = 𝑏
103	97, 100, 102	3eqtr3g 2788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐴, ∅⟩ = 𝑏)
104	103	fveq2d 6865	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = (𝑀‘𝑏))
105	28	efgmval 19649	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → (𝐴𝑀∅) = ⟨𝐴, (1o ∖ ∅)⟩)
106	85, 5, 105	sylancl 586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (𝐴𝑀∅) = ⟨𝐴, (1o ∖ ∅)⟩)
107		df-ov 7393	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴𝑀∅) = (𝑀‘⟨𝐴, ∅⟩)
108		dif0 4344	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1o ∖ ∅) = 1o
109	108	opeq2i 4844	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨𝐴, (1o ∖ ∅)⟩ = ⟨𝐴, 1o⟩
110	106, 107, 109	3eqtr3g 2788	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = ⟨𝐴, 1o⟩)
111	96, 104, 110	3eqtr2rd 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩)
112		opthg 5440	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝐼 ∧ 1o ∈ On) → (⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩ ↔ (𝐴 = 𝐵 ∧ 1o = ∅)))
113	112	simplbda 499	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ 𝐼 ∧ 1o ∈ On) ∧ ⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩) → 1o = ∅)
114	85, 87, 111, 113	syl21anc 837	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → 1o = ∅)
115	114	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩ → 1o = ∅))
116	84, 115	sylbid 240	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) → 1o = ∅))
117	116	rexlimdvva 3195	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (∃𝑎 ∈ (0...(♯‘∅))∃𝑏 ∈ (𝐼 × 2o)⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) → 1o = ∅))
118	53, 117	biimtrid 242	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) → 1o = ∅))
119	50, 118	sylbid 240	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) → 1o = ∅))
120	119	expimpd 453	. . . . . . . . . . . 12 ⊢ (𝜑 → ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1o = ∅))
121		hasheq0 14335	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
122	121	elv 3455	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) = 0 ↔ 𝑥 = ∅)
123		eleq1 2817	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑊 ↔ ∅ ∈ 𝑊))
124		fveq2 6861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → (𝑇‘𝑥) = (𝑇‘∅))
125	124	rneqd 5905	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → ran (𝑇‘𝑥) = ran (𝑇‘∅))
126	125	eleq2d 2815	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)))
127	123, 126	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
128	122, 127	sylbi 217	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) = 0 → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
129	128	eqcoms 2738	. . . . . . . . . . . . 13 ⊢ (0 = (♯‘𝑥) → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
130	129	imbi1d 341	. . . . . . . . . . . 12 ⊢ (0 = (♯‘𝑥) → (((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → 1o = ∅) ↔ ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1o = ∅)))
131	120, 130	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ (𝜑 → (0 = (♯‘𝑥) → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → 1o = ∅)))
132	131	com23 86	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (0 = (♯‘𝑥) → 1o = ∅)))
133	132	expdimp 452	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) → (0 = (♯‘𝑥) → 1o = ∅)))
134	45, 133	mpdd 43	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) → 1o = ∅))
135	134	necon3ad 2939	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (1o ≠ ∅ → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)))
136	22, 135	mpi 20	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥))
137	136	nrexdv 3129	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥))
138		eliun 4962	. . . . 5 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ↔ ∃𝑥 ∈ 𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥))
139	137, 138	sylnibr 329	. . . 4 ⊢ (𝜑 → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
140	21, 139	eldifd 3928	. . 3 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)))
141		frgpnabl.d	. . 3 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
142	140, 141	eleqtrrdi 2840	. 2 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝐷)
143		df-s2 14821	. . . . 5 ⊢ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)
144	12, 27	efger 19655	. . . . . . 7 ⊢ ∼ Er 𝑊
145	144	a1i 11	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝑊)
146	145, 21	erref 8694	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∼ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
147	143, 146	eqbrtrrid 5146	. . . 4 ⊢ (𝜑 → (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∼ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
148	143	ovexi 7424	. . . . 5 ⊢ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ V
149		ovex 7423	. . . . 5 ⊢ (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∈ V
150	148, 149	elec 8720	. . . 4 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ ↔ (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∼ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
151	147, 150	sylibr 234	. . 3 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
152		frgpnabl.u	. . . . . . 7 ⊢ 𝑈 = (varFGrp‘𝐼)
153	27, 152	vrgpval 19704	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
154	13, 1, 153	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
155	27, 152	vrgpval 19704	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐵 ∈ 𝐼) → (𝑈‘𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] ∼ )
156	13, 8, 155	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑈‘𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] ∼ )
157	154, 156	oveq12d 7408	. . . 4 ⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = ([⟨“⟨𝐴, ∅⟩”⟩] ∼ + [⟨“⟨𝐵, ∅⟩”⟩] ∼ ))
158	7	s1cld 14575	. . . . . 6 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
159	158, 20	eleqtrrd 2832	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊)
160	10	s1cld 14575	. . . . . 6 ⊢ (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
161	160, 20	eleqtrrd 2832	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
162		frgpnabl.g	. . . . . 6 ⊢ 𝐺 = (freeGrp‘𝐼)
163		frgpnabl.p	. . . . . 6 ⊢ + = (+g‘𝐺)
164	12, 162, 27, 163	frgpadd 19700	. . . . 5 ⊢ ((⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊 ∧ ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊) → ([⟨“⟨𝐴, ∅⟩”⟩] ∼ + [⟨“⟨𝐵, ∅⟩”⟩] ∼ ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
165	159, 161, 164	syl2anc 584	. . . 4 ⊢ (𝜑 → ([⟨“⟨𝐴, ∅⟩”⟩] ∼ + [⟨“⟨𝐵, ∅⟩”⟩] ∼ ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
166	157, 165	eqtrd 2765	. . 3 ⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
167	151, 166	eleqtrrd 2832	. 2 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ((𝑈‘𝐴) + (𝑈‘𝐵)))
168	142, 167	elind 4166	1 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054  Vcvv 3450   ∖ cdif 3914   ∩ cin 3916  ∅c0 4299  {cpr 4594  ⟨cop 4598  ⟨cotp 4600  ∪ ciun 4958   class class class wbr 5110   ↦ cmpt 5191   I cid 5535   × cxp 5639  ran crn 5642  Oncon0 6335  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  1_oc1o 8430  2_oc2o 8431   Er wer 8671  [cec 8672  ℂcc 11073  0cc0 11075  1c1 11076   + caddc 11078  2c2 12248  ℕ₀cn0 12449  ...cfz 13475  ♯chash 14302  Word cword 14485   ++ cconcat 14542  ⟨“cs1 14567   splice csplice 14721  ⟨“cs2 14814  +_gcplusg 17227   ~_FG cefg 19643  freeGrpcfrgp 19644  var_FGrpcvrgp 19645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-ec 8676  df-qs 8680  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-fzo 13623  df-hash 14303  df-word 14486  df-concat 14543  df-s1 14568  df-substr 14613  df-pfx 14643  df-splice 14722  df-s2 14821  df-struct 17124  df-slot 17159  df-ndx 17171  df-base 17187  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-imas 17478  df-qus 17479  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-frmd 18783  df-efg 19646  df-frgp 19647  df-vrgp 19648

This theorem is referenced by:  frgpnabllem2  19811