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	⊢ ∼ = ( ~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 5245	. . . . . . . . 9 ⊢ ∅ ∈ V
3	2	prid1 4707	. . . . . . . 8 ⊢ ∅ ∈ {∅, 1o}
4		df2o3 8353	. . . . . . . 8 ⊢ 2o = {∅, 1o}
5	3, 4	eleqtrri 2836	. . . . . . 7 ⊢ ∅ ∈ 2o
6		opelxpi 5644	. . . . . . 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 5644	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐼 ∧ ∅ ∈ 2o) → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2o))
10	8, 5, 9	sylancl 586	. . . . . 6 ⊢ (𝜑 → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2o))
11	7, 10	s2cld 14660	. . . . 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 8359	. . . . . . . 8 ⊢ 2o ∈ On
15		xpexg 7641	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
16	13, 14, 15	sylancl 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))
19	16, 17, 18	3syl 18	. . . . . 6 ⊢ (𝜑 → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o))
20	12, 19	eqtrid 2788	. . . . 5 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o))
21	11, 20	eleqtrrd 2840	. . . 4 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
22		1n0 8367	. . . . . . 7 ⊢ 1o ≠ ∅
23		2cn 12127	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
24	23	addid2i 11242	. . . . . . . . . . . . 13 ⊢ (0 + 2) = 2
25		s2len 14678	. . . . . . . . . . . . 13 ⊢ (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = 2
26	24, 25	eqtr4i 2767	. . . . . . . . . . . 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 19404	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((♯‘𝑥) + 2))
31	30	adantll 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((♯‘𝑥) + 2))
32	26, 31	eqtrid 2788	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (0 + 2) = ((♯‘𝑥) + 2))
33	32	ex 413	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) → (0 + 2) = ((♯‘𝑥) + 2)))
34		0cnd 11047	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 0 ∈ ℂ)
35		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑥 ∈ 𝑊)
36	12	efgrcl 19393	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
37	36	simprd 496	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o))
38	37	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑊 = Word (𝐼 × 2o))
39	35, 38	eleqtrd 2839	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑥 ∈ Word (𝐼 × 2o))
40		lencl 14314	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (♯‘𝑥) ∈ ℕ0)
41	39, 40	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (♯‘𝑥) ∈ ℕ0)
42	41	nn0cnd 12374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (♯‘𝑥) ∈ ℂ)
43		2cnd 12130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 2 ∈ ℂ)
44	34, 42, 43	addcan2d 11258	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ((0 + 2) = ((♯‘𝑥) + 2) ↔ 0 = (♯‘𝑥)))
45	33, 44	sylibd 238	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) → 0 = (♯‘𝑥)))
46	12, 27, 28, 29	efgtf 19400	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∈ 𝑊 → ((𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(♯‘∅)) × (𝐼 × 2o))⟶𝑊))
47	46	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → ((𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(♯‘∅)) × (𝐼 × 2o))⟶𝑊))
48	47	simpld 495	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
49	48	rneqd 5866	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → ran (𝑇‘∅) = ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
50	49	eleq2d 2822	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))))
51		eqid 2736	. . . . . . . . . . . . . . . 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 7349	. . . . . . . . . . . . . . . 16 ⊢ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ V
53	51, 52	elrnmpo 7451	. . . . . . . . . . . . . . 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 14320	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ ∈ Word (𝐼 × 2o)
55	54	a1i 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))
57	28	efgmf 19391	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
58	57	ffvelcdmi 6999	. . . . . . . . . . . . . . . . . . . . . 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 14660	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
61		ccatidid 14372	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ ++ ∅) = ∅
62	61	oveq1i 7326	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∅ ++ ∅) ++ ∅) = (∅ ++ ∅)
63	62, 61	eqtr2i 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ = ((∅ ++ ∅) ++ ∅)
64	63	a1i 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
67	66	oveq2i 7327	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0...(♯‘∅)) = (0...0)
68	65, 67	eleqtrdi 2847	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...0))
69		elfz1eq 13346	. . . . . . . . . . . . . . . . . . . . . 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 2794	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (♯‘∅))
72	66	oveq2i 7327	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 + (♯‘∅)) = (𝑎 + 0)
73		0cn 11046	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℂ
74	70, 73	eqeltrdi 2845	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ ℂ)
75	74	addid1d 11254	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 + 0) = 𝑎)
76	72, 75	eqtr2id 2789	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (𝑎 + (♯‘∅)))
77	55, 55, 55, 60, 64, 71, 76	splval2 14546	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = ((∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ ∅))
78		ccatlid 14368	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o) → (∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) = ⟨“𝑏(𝑀‘𝑏)”⟩)
79	78	oveq1d 7331	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o) → ((∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ ∅) = (⟨“𝑏(𝑀‘𝑏)”⟩ ++ ∅))
80		ccatrid 14369	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o) → (⟨“𝑏(𝑀‘𝑏)”⟩ ++ ∅) = ⟨“𝑏(𝑀‘𝑏)”⟩)
81	79, 80	eqtrd 2776	. . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = ⟨“𝑏(𝑀‘𝑏)”⟩)
84	83	eqeq2d 2747	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩))
85	1	ad3antrrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → 𝐴 ∈ 𝐼)
86		1on 8357	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1o ∈ On
87	86	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → 1o ∈ On)
88		simpr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩)
89	88	fveq1d 6813	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = (⟨“𝑏(𝑀‘𝑏)”⟩‘1))
90		opex 5397	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨𝐵, ∅⟩ ∈ V
91		s2fv1 14677	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝐵, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩)
92	90, 91	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩
93		fvex 6824	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀‘𝑏) ∈ V
94		s2fv1 14677	. . . . . . . . . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐵, ∅⟩ = (𝑀‘𝑏))
97	88	fveq1d 6813	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = (⟨“𝑏(𝑀‘𝑏)”⟩‘0))
98		opex 5397	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨𝐴, ∅⟩ ∈ V
99		s2fv0 14676	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝐴, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩)
100	98, 99	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩
101		s2fv0 14676	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 ∈ V → (⟨“𝑏(𝑀‘𝑏)”⟩‘0) = 𝑏)
102	101	elv 3446	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩‘0) = 𝑏
103	97, 100, 102	3eqtr3g 2799	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐴, ∅⟩ = 𝑏)
104	103	fveq2d 6815	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = (𝑀‘𝑏))
105	28	efgmval 19390	. . . . . . . . . . . . . . . . . . . . . 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 7319	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴𝑀∅) = (𝑀‘⟨𝐴, ∅⟩)
108		dif0 4316	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1o ∖ ∅) = 1o
109	108	opeq2i 4818	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨𝐴, (1o ∖ ∅)⟩ = ⟨𝐴, 1o⟩
110	106, 107, 109	3eqtr3g 2799	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = ⟨𝐴, 1o⟩)
111	96, 104, 110	3eqtr2rd 2783	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩)
112		opthg 5410	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝐼 ∧ 1o ∈ On) → (⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩ ↔ (𝐴 = 𝐵 ∧ 1o = ∅)))
113	112	simplbda 500	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ 𝐼 ∧ 1o ∈ On) ∧ ⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩) → 1o = ∅)
114	85, 87, 111, 113	syl21anc 835	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → 1o = ∅)
115	114	ex 413	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩ → 1o = ∅))
116	84, 115	sylbid 239	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) → 1o = ∅))
117	116	rexlimdvva 3201	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (∃𝑎 ∈ (0...(♯‘∅))∃𝑏 ∈ (𝐼 × 2o)⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) → 1o = ∅))
118	53, 117	biimtrid 241	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) → 1o = ∅))
119	50, 118	sylbid 239	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) → 1o = ∅))
120	119	expimpd 454	. . . . . . . . . . . 12 ⊢ (𝜑 → ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1o = ∅))
121		hasheq0 14156	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
122	121	elv 3446	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) = 0 ↔ 𝑥 = ∅)
123		eleq1 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑊 ↔ ∅ ∈ 𝑊))
124		fveq2 6811	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → (𝑇‘𝑥) = (𝑇‘∅))
125	124	rneqd 5866	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → ran (𝑇‘𝑥) = ran (𝑇‘∅))
126	125	eleq2d 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)))
127	123, 126	anbi12d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
128	122, 127	sylbi 216	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) = 0 → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
129	128	eqcoms 2744	. . . . . . . . . . . . 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 246	. . . . . . . . . . 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 453	. . . . . . . . 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 2953	. . . . . . 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 3142	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥))
138		eliun 4940	. . . . 5 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ↔ ∃𝑥 ∈ 𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥))
139	137, 138	sylnibr 328	. . . 4 ⊢ (𝜑 → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
140	21, 139	eldifd 3907	. . 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 2848	. 2 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝐷)
143		df-s2 14637	. . . . 5 ⊢ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)
144	12, 27	efger 19396	. . . . . . 7 ⊢ ∼ Er 𝑊
145	144	a1i 11	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝑊)
146	145, 21	erref 8567	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∼ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
147	143, 146	eqbrtrrid 5122	. . . 4 ⊢ (𝜑 → (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∼ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
148	143	ovexi 7350	. . . . 5 ⊢ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ V
149		ovex 7349	. . . . 5 ⊢ (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∈ V
150	148, 149	elec 8591	. . . 4 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ ↔ (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∼ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
151	147, 150	sylibr 233	. . 3 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
152		frgpnabl.u	. . . . . . 7 ⊢ 𝑈 = (varFGrp‘𝐼)
153	27, 152	vrgpval 19445	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
154	13, 1, 153	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
155	27, 152	vrgpval 19445	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐵 ∈ 𝐼) → (𝑈‘𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] ∼ )
156	13, 8, 155	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑈‘𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] ∼ )
157	154, 156	oveq12d 7334	. . . 4 ⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = ([⟨“⟨𝐴, ∅⟩”⟩] ∼ + [⟨“⟨𝐵, ∅⟩”⟩] ∼ ))
158	7	s1cld 14385	. . . . . 6 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
159	158, 20	eleqtrrd 2840	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊)
160	10	s1cld 14385	. . . . . 6 ⊢ (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
161	160, 20	eleqtrrd 2840	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
162		frgpnabl.g	. . . . . 6 ⊢ 𝐺 = (freeGrp‘𝐼)
163		frgpnabl.p	. . . . . 6 ⊢ + = (+g‘𝐺)
164	12, 162, 27, 163	frgpadd 19441	. . . . 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 2776	. . 3 ⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
167	151, 166	eleqtrrd 2840	. 2 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ((𝑈‘𝐴) + (𝑈‘𝐵)))
168	142, 167	elind 4138	1 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵))))

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  1_oc1o 8338  2_oc2o 8339   Er wer 8544  [cec 8545  ℂcc 10948  0cc0 10950  1c1 10951   + caddc 10953  2c2 12107  ℕ₀cn0 12312  ...cfz 13318  ♯chash 14123  Word cword 14295   ++ cconcat 14351  ⟨“cs1 14377   splice csplice 14538  ⟨“cs2 14630  +_gcplusg 17036   ~_FG cefg 19384  freeGrpcfrgp 19385  var_FGrpcvrgp 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