Theorem frgpnabllem1 19230

Description: Lemma for frgpnabl 19232. (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 5189	. . . . . . . . 9 ⊢ ∅ ∈ V
3	2	prid1 4668	. . . . . . . 8 ⊢ ∅ ∈ {∅, 1o}
4		df2o3 8206	. . . . . . . 8 ⊢ 2o = {∅, 1o}
5	3, 4	eleqtrri 2833	. . . . . . 7 ⊢ ∅ ∈ 2o
6		opelxpi 5577	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2o))
7	1, 5, 6	sylancl 589	. . . . . 6 ⊢ (𝜑 → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2o))
8		frgpnabl.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝐼)
9		opelxpi 5577	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐼 ∧ ∅ ∈ 2o) → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2o))
10	8, 5, 9	sylancl 589	. . . . . 6 ⊢ (𝜑 → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2o))
11	7, 10	s2cld 14419	. . . . 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 8199	. . . . . . . 8 ⊢ 2o ∈ On
15		xpexg 7524	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
16	13, 14, 15	sylancl 589	. . . . . . 7 ⊢ (𝜑 → (𝐼 × 2o) ∈ V)
17		wrdexg 14062	. . . . . . 7 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
18		fvi 6776	. . . . . . 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	syl5eq 2786	. . . . 5 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o))
21	11, 20	eleqtrrd 2837	. . . 4 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
22		1n0 8210	. . . . . . 7 ⊢ 1o ≠ ∅
23		2cn 11888	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
24	23	addid2i 11003	. . . . . . . . . . . . 13 ⊢ (0 + 2) = 2
25		s2len 14437	. . . . . . . . . . . . 13 ⊢ (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = 2
26	24, 25	eqtr4i 2765	. . . . . . . . . . . 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 19088	. . . . . . . . . . . . 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	syl5eq 2786	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (0 + 2) = ((♯‘𝑥) + 2))
33	32	ex 416	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) → (0 + 2) = ((♯‘𝑥) + 2)))
34		0cnd 10809	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 0 ∈ ℂ)
35		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑥 ∈ 𝑊)
36	12	efgrcl 19077	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
37	36	simprd 499	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o))
38	37	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑊 = Word (𝐼 × 2o))
39	35, 38	eleqtrd 2836	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑥 ∈ Word (𝐼 × 2o))
40		lencl 14071	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (♯‘𝑥) ∈ ℕ0)
41	39, 40	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (♯‘𝑥) ∈ ℕ0)
42	41	nn0cnd 12135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (♯‘𝑥) ∈ ℂ)
43		2cnd 11891	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 2 ∈ ℂ)
44	34, 42, 43	addcan2d 11019	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ((0 + 2) = ((♯‘𝑥) + 2) ↔ 0 = (♯‘𝑥)))
45	33, 44	sylibd 242	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) → 0 = (♯‘𝑥)))
46	12, 27, 28, 29	efgtf 19084	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∈ 𝑊 → ((𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(♯‘∅)) × (𝐼 × 2o))⟶𝑊))
47	46	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → ((𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(♯‘∅)) × (𝐼 × 2o))⟶𝑊))
48	47	simpld 498	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
49	48	rneqd 5796	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → ran (𝑇‘∅) = ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
50	49	eleq2d 2819	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))))
51		eqid 2734	. . . . . . . . . . . . . . . 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 7235	. . . . . . . . . . . . . . . 16 ⊢ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ V
53	51, 52	elrnmpo 7335	. . . . . . . . . . . . . . 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 14077	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ ∈ Word (𝐼 × 2o)
55	54	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∅ ∈ Word (𝐼 × 2o))
56		simprr 773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑏 ∈ (𝐼 × 2o))
57	28	efgmf 19075	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
58	57	ffvelrni 6892	. . . . . . . . . . . . . . . . . . . . . 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 14419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
61		ccatidid 14130	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ ++ ∅) = ∅
62	61	oveq1i 7212	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∅ ++ ∅) ++ ∅) = (∅ ++ ∅)
63	62, 61	eqtr2i 2763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ = ((∅ ++ ∅) ++ ∅)
64	63	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∅ = ((∅ ++ ∅) ++ ∅))
65		simprl 771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...(♯‘∅)))
66		hash0 13917	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (♯‘∅) = 0
67	66	oveq2i 7213	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0...(♯‘∅)) = (0...0)
68	65, 67	eleqtrdi 2844	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...0))
69		elfz1eq 13106	. . . . . . . . . . . . . . . . . . . . . 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 2792	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (♯‘∅))
72	66	oveq2i 7213	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 + (♯‘∅)) = (𝑎 + 0)
73		0cn 10808	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℂ
74	70, 73	eqeltrdi 2842	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ ℂ)
75	74	addid1d 11015	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 + 0) = 𝑎)
76	72, 75	eqtr2id 2787	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (𝑎 + (♯‘∅)))
77	55, 55, 55, 60, 64, 71, 76	splval2 14305	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = ((∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ ∅))
78		ccatlid 14126	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o) → (∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) = ⟨“𝑏(𝑀‘𝑏)”⟩)
79	78	oveq1d 7217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o) → ((∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ ∅) = (⟨“𝑏(𝑀‘𝑏)”⟩ ++ ∅))
80		ccatrid 14127	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o) → (⟨“𝑏(𝑀‘𝑏)”⟩ ++ ∅) = ⟨“𝑏(𝑀‘𝑏)”⟩)
81	79, 80	eqtrd 2774	. . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = ⟨“𝑏(𝑀‘𝑏)”⟩)
84	83	eqeq2d 2745	. . . . . . . . . . . . . . . . 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 8198	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1o ∈ On
87	86	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → 1o ∈ On)
88		simpr 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩)
89	88	fveq1d 6708	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = (⟨“𝑏(𝑀‘𝑏)”⟩‘1))
90		opex 5337	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨𝐵, ∅⟩ ∈ V
91		s2fv1 14436	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝐵, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩)
92	90, 91	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩
93		fvex 6719	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀‘𝑏) ∈ V
94		s2fv1 14436	. . . . . . . . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐵, ∅⟩ = (𝑀‘𝑏))
97	88	fveq1d 6708	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = (⟨“𝑏(𝑀‘𝑏)”⟩‘0))
98		opex 5337	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨𝐴, ∅⟩ ∈ V
99		s2fv0 14435	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝐴, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩)
100	98, 99	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩
101		s2fv0 14435	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 ∈ V → (⟨“𝑏(𝑀‘𝑏)”⟩‘0) = 𝑏)
102	101	elv 3407	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩‘0) = 𝑏
103	97, 100, 102	3eqtr3g 2797	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐴, ∅⟩ = 𝑏)
104	103	fveq2d 6710	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = (𝑀‘𝑏))
105	28	efgmval 19074	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → (𝐴𝑀∅) = ⟨𝐴, (1o ∖ ∅)⟩)
106	85, 5, 105	sylancl 589	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (𝐴𝑀∅) = ⟨𝐴, (1o ∖ ∅)⟩)
107		df-ov 7205	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴𝑀∅) = (𝑀‘⟨𝐴, ∅⟩)
108		dif0 4277	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1o ∖ ∅) = 1o
109	108	opeq2i 4778	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨𝐴, (1o ∖ ∅)⟩ = ⟨𝐴, 1o⟩
110	106, 107, 109	3eqtr3g 2797	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = ⟨𝐴, 1o⟩)
111	96, 104, 110	3eqtr2rd 2781	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩)
112		opthg 5350	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝐼 ∧ 1o ∈ On) → (⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩ ↔ (𝐴 = 𝐵 ∧ 1o = ∅)))
113	112	simplbda 503	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ 𝐼 ∧ 1o ∈ On) ∧ ⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩) → 1o = ∅)
114	85, 87, 111, 113	syl21anc 838	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → 1o = ∅)
115	114	ex 416	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩ → 1o = ∅))
116	84, 115	sylbid 243	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) → 1o = ∅))
117	116	rexlimdvva 3206	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (∃𝑎 ∈ (0...(♯‘∅))∃𝑏 ∈ (𝐼 × 2o)⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) → 1o = ∅))
118	53, 117	syl5bi 245	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) → 1o = ∅))
119	50, 118	sylbid 243	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) → 1o = ∅))
120	119	expimpd 457	. . . . . . . . . . . 12 ⊢ (𝜑 → ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1o = ∅))
121		hasheq0 13913	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
122	121	elv 3407	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) = 0 ↔ 𝑥 = ∅)
123		eleq1 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑊 ↔ ∅ ∈ 𝑊))
124		fveq2 6706	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → (𝑇‘𝑥) = (𝑇‘∅))
125	124	rneqd 5796	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → ran (𝑇‘𝑥) = ran (𝑇‘∅))
126	125	eleq2d 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)))
127	123, 126	anbi12d 634	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
128	122, 127	sylbi 220	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) = 0 → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
129	128	eqcoms 2742	. . . . . . . . . . . . 13 ⊢ (0 = (♯‘𝑥) → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
130	129	imbi1d 345	. . . . . . . . . . . 12 ⊢ (0 = (♯‘𝑥) → (((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → 1o = ∅) ↔ ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1o = ∅)))
131	120, 130	syl5ibrcom 250	. . . . . . . . . . 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 456	. . . . . . . . 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 2948	. . . . . . 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 3182	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥))
138		eliun 4898	. . . . 5 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ↔ ∃𝑥 ∈ 𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥))
139	137, 138	sylnibr 332	. . . 4 ⊢ (𝜑 → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
140	21, 139	eldifd 3868	. . 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 2845	. 2 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝐷)
143		df-s2 14396	. . . . 5 ⊢ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)
144	12, 27	efger 19080	. . . . . . 7 ⊢ ∼ Er 𝑊
145	144	a1i 11	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝑊)
146	145, 21	erref 8400	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∼ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
147	143, 146	eqbrtrrid 5079	. . . 4 ⊢ (𝜑 → (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∼ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
148	143	ovexi 7236	. . . . 5 ⊢ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ V
149		ovex 7235	. . . . 5 ⊢ (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∈ V
150	148, 149	elec 8424	. . . 4 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ ↔ (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∼ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
151	147, 150	sylibr 237	. . 3 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
152		frgpnabl.u	. . . . . . 7 ⊢ 𝑈 = (varFGrp‘𝐼)
153	27, 152	vrgpval 19129	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
154	13, 1, 153	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
155	27, 152	vrgpval 19129	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐵 ∈ 𝐼) → (𝑈‘𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] ∼ )
156	13, 8, 155	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝑈‘𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] ∼ )
157	154, 156	oveq12d 7220	. . . 4 ⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = ([⟨“⟨𝐴, ∅⟩”⟩] ∼ + [⟨“⟨𝐵, ∅⟩”⟩] ∼ ))
158	7	s1cld 14143	. . . . . 6 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
159	158, 20	eleqtrrd 2837	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊)
160	10	s1cld 14143	. . . . . 6 ⊢ (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
161	160, 20	eleqtrrd 2837	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
162		frgpnabl.g	. . . . . 6 ⊢ 𝐺 = (freeGrp‘𝐼)
163		frgpnabl.p	. . . . . 6 ⊢ + = (+g‘𝐺)
164	12, 162, 27, 163	frgpadd 19125	. . . . 5 ⊢ ((⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊 ∧ ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊) → ([⟨“⟨𝐴, ∅⟩”⟩] ∼ + [⟨“⟨𝐵, ∅⟩”⟩] ∼ ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
165	159, 161, 164	syl2anc 587	. . . 4 ⊢ (𝜑 → ([⟨“⟨𝐴, ∅⟩”⟩] ∼ + [⟨“⟨𝐵, ∅⟩”⟩] ∼ ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
166	157, 165	eqtrd 2774	. . 3 ⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
167	151, 166	eleqtrrd 2837	. 2 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ((𝑈‘𝐴) + (𝑈‘𝐵)))
168	142, 167	elind 4098	1 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ≠ wne 2935  ∃wrex 3055  Vcvv 3401   ∖ cdif 3854   ∩ cin 3856  ∅c0 4227  {cpr 4533  ⟨cop 4537  ⟨cotp 4539  ∪ ciun 4894   class class class wbr 5043   ↦ cmpt 5124   I cid 5443   × cxp 5538  ran crn 5541  Oncon0 6202  ⟶wf 6365  ‘cfv 6369  (class class class)co 7202   ∈ cmpo 7204  1_oc1o 8184  2_oc2o 8185   Er wer 8377  [cec 8378  ℂcc 10710  0cc0 10712  1c1 10713   + caddc 10715  2c2 11868  ℕ₀cn0 12073  ...cfz 13078  ♯chash 13879  Word cword 14052   ++ cconcat 14108  ⟨“cs1 14135   splice csplice 14297  ⟨“cs2 14389  +_gcplusg 16767   ~_FG cefg 19068  freeGrpcfrgp 19069  var_FGrpcvrgp 19070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-ot 4540  df-uni 4810  df-int 4850  df-iun 4896  df-iin 4897  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-2o 8192  df-er 8380  df-ec 8382  df-qs 8386  df-map 8499  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-sup 9047  df-inf 9048  df-card 9538  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-nn 11814  df-2 11876  df-3 11877  df-4 11878  df-5 11879  df-6 11880  df-7 11881  df-8 11882  df-9 11883  df-n0 12074  df-z 12160  df-dec 12277  df-uz 12422  df-fz 13079  df-fzo 13222  df-hash 13880  df-word 14053  df-concat 14109  df-s1 14136  df-substr 14189  df-pfx 14219  df-splice 14298  df-s2 14396  df-struct 16686  df-ndx 16687  df-slot 16688  df-base 16690  df-plusg 16780  df-mulr 16781  df-sca 16783  df-vsca 16784  df-ip 16785  df-tset 16786  df-ple 16787  df-ds 16789  df-imas 16985  df-qus 16986  df-mgm 18086  df-sgrp 18135  df-mnd 18146  df-frmd 18248  df-efg 19071  df-frgp 19072  df-vrgp 19073

This theorem is referenced by:  frgpnabllem2  19231