Theorem frgpnabllem1 19867

Description: Lemma for frgpnabl 19869. (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 5304	. . . . . . . . 9 ⊢ ∅ ∈ V
3	2	prid1 4761	. . . . . . . 8 ⊢ ∅ ∈ {∅, 1o}
4		df2o3 8496	. . . . . . . 8 ⊢ 2o = {∅, 1o}
5	3, 4	eleqtrri 2825	. . . . . . 7 ⊢ ∅ ∈ 2o
6		opelxpi 5711	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2o))
7	1, 5, 6	sylancl 584	. . . . . 6 ⊢ (𝜑 → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2o))
8		frgpnabl.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝐼)
9		opelxpi 5711	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐼 ∧ ∅ ∈ 2o) → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2o))
10	8, 5, 9	sylancl 584	. . . . . 6 ⊢ (𝜑 → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2o))
11	7, 10	s2cld 14875	. . . . 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 8502	. . . . . . . 8 ⊢ 2o ∈ On
15		xpexg 7750	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
16	13, 14, 15	sylancl 584	. . . . . . 7 ⊢ (𝜑 → (𝐼 × 2o) ∈ V)
17		wrdexg 14527	. . . . . . 7 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
18		fvi 6970	. . . . . . 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 2778	. . . . 5 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o))
21	11, 20	eleqtrrd 2829	. . . 4 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
22		1n0 8510	. . . . . . 7 ⊢ 1o ≠ ∅
23		2cn 12333	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
24	23	addlidi 11443	. . . . . . . . . . . . 13 ⊢ (0 + 2) = 2
25		s2len 14893	. . . . . . . . . . . . 13 ⊢ (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = 2
26	24, 25	eqtr4i 2757	. . . . . . . . . . . 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 19720	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((♯‘𝑥) + 2))
31	30	adantll 712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (♯‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((♯‘𝑥) + 2))
32	26, 31	eqtrid 2778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (0 + 2) = ((♯‘𝑥) + 2))
33	32	ex 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) → (0 + 2) = ((♯‘𝑥) + 2)))
34		0cnd 11248	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 0 ∈ ℂ)
35		simpr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑥 ∈ 𝑊)
36	12	efgrcl 19709	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
37	36	simprd 494	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o))
38	37	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑊 = Word (𝐼 × 2o))
39	35, 38	eleqtrd 2828	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑥 ∈ Word (𝐼 × 2o))
40		lencl 14536	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (♯‘𝑥) ∈ ℕ0)
41	39, 40	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (♯‘𝑥) ∈ ℕ0)
42	41	nn0cnd 12580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (♯‘𝑥) ∈ ℂ)
43		2cnd 12336	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 2 ∈ ℂ)
44	34, 42, 43	addcan2d 11459	. . . . . . . . . 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 19716	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∈ 𝑊 → ((𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(♯‘∅)) × (𝐼 × 2o))⟶𝑊))
47	46	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → ((𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(♯‘∅)) × (𝐼 × 2o))⟶𝑊))
48	47	simpld 493	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (𝑇‘∅) = (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
49	48	rneqd 5936	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → ran (𝑇‘∅) = ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
50	49	eleq2d 2812	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(♯‘∅)), 𝑏 ∈ (𝐼 × 2o) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))))
51		eqid 2726	. . . . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . . . . 16 ⊢ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ V
53	51, 52	elrnmpo 7554	. . . . . . . . . . . . . . 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 14542	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ ∈ Word (𝐼 × 2o)
55	54	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∅ ∈ Word (𝐼 × 2o))
56		simprr 771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑏 ∈ (𝐼 × 2o))
57	28	efgmf 19707	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
58	57	ffvelcdmi 7089	. . . . . . . . . . . . . . . . . . . . . 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 14875	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
61		ccatidid 14593	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ ++ ∅) = ∅
62	61	oveq1i 7426	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∅ ++ ∅) ++ ∅) = (∅ ++ ∅)
63	62, 61	eqtr2i 2755	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ = ((∅ ++ ∅) ++ ∅)
64	63	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∅ = ((∅ ++ ∅) ++ ∅))
65		simprl 769	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...(♯‘∅)))
66		hash0 14379	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (♯‘∅) = 0
67	66	oveq2i 7427	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0...(♯‘∅)) = (0...0)
68	65, 67	eleqtrdi 2836	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...0))
69		elfz1eq 13560	. . . . . . . . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (♯‘∅))
72	66	oveq2i 7427	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 + (♯‘∅)) = (𝑎 + 0)
73		0cn 11247	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℂ
74	70, 73	eqeltrdi 2834	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ ℂ)
75	74	addridd 11455	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 + 0) = 𝑎)
76	72, 75	eqtr2id 2779	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (𝑎 + (♯‘∅)))
77	55, 55, 55, 60, 64, 71, 76	splval2 14760	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = ((∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ ∅))
78		ccatlid 14589	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o) → (∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) = ⟨“𝑏(𝑀‘𝑏)”⟩)
79	78	oveq1d 7431	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o) → ((∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ ∅) = (⟨“𝑏(𝑀‘𝑏)”⟩ ++ ∅))
80		ccatrid 14590	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o) → (⟨“𝑏(𝑀‘𝑏)”⟩ ++ ∅) = ⟨“𝑏(𝑀‘𝑏)”⟩)
81	79, 80	eqtrd 2766	. . . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = ⟨“𝑏(𝑀‘𝑏)”⟩)
84	83	eqeq2d 2737	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩))
85	1	ad3antrrr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → 𝐴 ∈ 𝐼)
86		1on 8500	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1o ∈ On
87	86	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → 1o ∈ On)
88		simpr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩)
89	88	fveq1d 6895	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = (⟨“𝑏(𝑀‘𝑏)”⟩‘1))
90		opex 5462	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨𝐵, ∅⟩ ∈ V
91		s2fv1 14892	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝐵, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩)
92	90, 91	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩
93		fvex 6906	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀‘𝑏) ∈ V
94		s2fv1 14892	. . . . . . . . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐵, ∅⟩ = (𝑀‘𝑏))
97	88	fveq1d 6895	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = (⟨“𝑏(𝑀‘𝑏)”⟩‘0))
98		opex 5462	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨𝐴, ∅⟩ ∈ V
99		s2fv0 14891	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝐴, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩)
100	98, 99	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩
101		s2fv0 14891	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 ∈ V → (⟨“𝑏(𝑀‘𝑏)”⟩‘0) = 𝑏)
102	101	elv 3468	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩‘0) = 𝑏
103	97, 100, 102	3eqtr3g 2789	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐴, ∅⟩ = 𝑏)
104	103	fveq2d 6897	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = (𝑀‘𝑏))
105	28	efgmval 19706	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → (𝐴𝑀∅) = ⟨𝐴, (1o ∖ ∅)⟩)
106	85, 5, 105	sylancl 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (𝐴𝑀∅) = ⟨𝐴, (1o ∖ ∅)⟩)
107		df-ov 7419	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴𝑀∅) = (𝑀‘⟨𝐴, ∅⟩)
108		dif0 4370	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1o ∖ ∅) = 1o
109	108	opeq2i 4875	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨𝐴, (1o ∖ ∅)⟩ = ⟨𝐴, 1o⟩
110	106, 107, 109	3eqtr3g 2789	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = ⟨𝐴, 1o⟩)
111	96, 104, 110	3eqtr2rd 2773	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩)
112		opthg 5475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝐼 ∧ 1o ∈ On) → (⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩ ↔ (𝐴 = 𝐵 ∧ 1o = ∅)))
113	112	simplbda 498	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ 𝐼 ∧ 1o ∈ On) ∧ ⟨𝐴, 1o⟩ = ⟨𝐵, ∅⟩) → 1o = ∅)
114	85, 87, 111, 113	syl21anc 836	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(♯‘∅)) ∧ 𝑏 ∈ (𝐼 × 2o))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → 1o = ∅)
115	114	ex 411	. . . . . . . . . . . . . . . . 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 3202	. . . . . . . . . . . . . . 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 452	. . . . . . . . . . . 12 ⊢ (𝜑 → ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1o = ∅))
121		hasheq0 14375	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
122	121	elv 3468	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) = 0 ↔ 𝑥 = ∅)
123		eleq1 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑊 ↔ ∅ ∈ 𝑊))
124		fveq2 6893	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → (𝑇‘𝑥) = (𝑇‘∅))
125	124	rneqd 5936	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → ran (𝑇‘𝑥) = ran (𝑇‘∅))
126	125	eleq2d 2812	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)))
127	123, 126	anbi12d 630	. . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . 13 ⊢ (0 = (♯‘𝑥) → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
130	129	imbi1d 340	. . . . . . . . . . . 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 451	. . . . . . . . 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 2943	. . . . . . 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 3139	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥))
138		eliun 4997	. . . . 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 3957	. . 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 2837	. 2 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝐷)
143		df-s2 14852	. . . . 5 ⊢ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)
144	12, 27	efger 19712	. . . . . . 7 ⊢ ∼ Er 𝑊
145	144	a1i 11	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝑊)
146	145, 21	erref 8746	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∼ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
147	143, 146	eqbrtrrid 5181	. . . 4 ⊢ (𝜑 → (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∼ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
148	143	ovexi 7450	. . . . 5 ⊢ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ V
149		ovex 7449	. . . . 5 ⊢ (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∈ V
150	148, 149	elec 8772	. . . 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 19761	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
154	13, 1, 153	syl2anc 582	. . . . 5 ⊢ (𝜑 → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
155	27, 152	vrgpval 19761	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐵 ∈ 𝐼) → (𝑈‘𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] ∼ )
156	13, 8, 155	syl2anc 582	. . . . 5 ⊢ (𝜑 → (𝑈‘𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] ∼ )
157	154, 156	oveq12d 7434	. . . 4 ⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = ([⟨“⟨𝐴, ∅⟩”⟩] ∼ + [⟨“⟨𝐵, ∅⟩”⟩] ∼ ))
158	7	s1cld 14606	. . . . . 6 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
159	158, 20	eleqtrrd 2829	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊)
160	10	s1cld 14606	. . . . . 6 ⊢ (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2o))
161	160, 20	eleqtrrd 2829	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
162		frgpnabl.g	. . . . . 6 ⊢ 𝐺 = (freeGrp‘𝐼)
163		frgpnabl.p	. . . . . 6 ⊢ + = (+g‘𝐺)
164	12, 162, 27, 163	frgpadd 19757	. . . . 5 ⊢ ((⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊 ∧ ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊) → ([⟨“⟨𝐴, ∅⟩”⟩] ∼ + [⟨“⟨𝐵, ∅⟩”⟩] ∼ ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
165	159, 161, 164	syl2anc 582	. . . 4 ⊢ (𝜑 → ([⟨“⟨𝐴, ∅⟩”⟩] ∼ + [⟨“⟨𝐵, ∅⟩”⟩] ∼ ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
166	157, 165	eqtrd 2766	. . 3 ⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
167	151, 166	eleqtrrd 2829	. 2 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ((𝑈‘𝐴) + (𝑈‘𝐵)))
168	142, 167	elind 4192	1 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∃wrex 3060  Vcvv 3462   ∖ cdif 3943   ∩ cin 3945  ∅c0 4322  {cpr 4625  ⟨cop 4629  ⟨cotp 4631  ∪ ciun 4993   class class class wbr 5145   ↦ cmpt 5228   I cid 5571   × cxp 5672  ran crn 5675  Oncon0 6368  ⟶wf 6542  ‘cfv 6546  (class class class)co 7416   ∈ cmpo 7418  1_oc1o 8481  2_oc2o 8482   Er wer 8723  [cec 8724  ℂcc 11147  0cc0 11149  1c1 11150   + caddc 11152  2c2 12313  ℕ₀cn0 12518  ...cfz 13532  ♯chash 14342  Word cword 14517   ++ cconcat 14573  ⟨“cs1 14598   splice csplice 14752  ⟨“cs2 14845  +_gcplusg 17261   ~_FG cefg 19700  freeGrpcfrgp 19701  var_FGrpcvrgp 19702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4906  df-int 4947  df-iun 4995  df-iin 4996  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-er 8726  df-ec 8728  df-qs 8732  df-map 8849  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9478  df-inf 9479  df-card 9975  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-nn 12259  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12519  df-z 12605  df-dec 12724  df-uz 12869  df-fz 13533  df-fzo 13676  df-hash 14343  df-word 14518  df-concat 14574  df-s1 14599  df-substr 14644  df-pfx 14674  df-splice 14753  df-s2 14852  df-struct 17144  df-slot 17179  df-ndx 17191  df-base 17209  df-plusg 17274  df-mulr 17275  df-sca 17277  df-vsca 17278  df-ip 17279  df-tset 17280  df-ple 17281  df-ds 17283  df-imas 17518  df-qus 17519  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-frmd 18834  df-efg 19703  df-frgp 19704  df-vrgp 19705

This theorem is referenced by:  frgpnabllem2  19868