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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 × 2_𝑜))
frgpnabl.r	⊢ ∼ = ( ~_FG ‘𝐼)
frgpnabl.p	⊢ + = (+_g‘𝐺)
frgpnabl.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
frgpnabl.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
frgpnabl.d	⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
frgpnabl.u	⊢ 𝑈 = (var_FGrp‘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)

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 4718	. . . . . . . . 9 ⊢ ∅ ∈ V
3	2	prid1 4241	. . . . . . . 8 ⊢ ∅ ∈ {∅, 1_𝑜}
4		df2o3 7460	. . . . . . . 8 ⊢ 2_𝑜 = {∅, 1_𝑜}
5	3, 4	eleqtrri 2687	. . . . . . 7 ⊢ ∅ ∈ 2_𝑜
6		opelxpi 5072	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2_𝑜) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2_𝑜))
7	1, 5, 6	sylancl 693	. . . . . 6 ⊢ (𝜑 → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2_𝑜))
8		frgpnabl.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝐼)
9		opelxpi 5072	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐼 ∧ ∅ ∈ 2_𝑜) → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2_𝑜))
10	8, 5, 9	sylancl 693	. . . . . 6 ⊢ (𝜑 → ⟨𝐵, ∅⟩ ∈ (𝐼 × 2_𝑜))
11	7, 10	s2cld 13466	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2_𝑜))
12		frgpnabl.w	. . . . . 6 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
13		frgpnabl.i	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ V)
14		2on 7455	. . . . . . . 8 ⊢ 2_𝑜 ∈ On
15		xpexg 6858	. . . . . . . 8 ⊢ ((𝐼 ∈ V ∧ 2_𝑜 ∈ On) → (𝐼 × 2_𝑜) ∈ V)
16	13, 14, 15	sylancl 693	. . . . . . 7 ⊢ (𝜑 → (𝐼 × 2_𝑜) ∈ V)
17		wrdexg 13170	. . . . . . 7 ⊢ ((𝐼 × 2_𝑜) ∈ V → Word (𝐼 × 2_𝑜) ∈ V)
18		fvi 6165	. . . . . . 7 ⊢ (Word (𝐼 × 2_𝑜) ∈ V → ( I ‘Word (𝐼 × 2_𝑜)) = Word (𝐼 × 2_𝑜))
19	16, 17, 18	3syl 18	. . . . . 6 ⊢ (𝜑 → ( I ‘Word (𝐼 × 2_𝑜)) = Word (𝐼 × 2_𝑜))
20	12, 19	syl5eq 2656	. . . . 5 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2_𝑜))
21	11, 20	eleqtrrd 2691	. . . 4 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
22		1n0 7462	. . . . . . 7 ⊢ 1_𝑜 ≠ ∅
23		2cn 10968	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
24	23	addid2i 10103	. . . . . . . . . . . . 13 ⊢ (0 + 2) = 2
25		s2len 13484	. . . . . . . . . . . . 13 ⊢ (#‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = 2
26	24, 25	eqtr4i 2635	. . . . . . . . . . . 12 ⊢ (0 + 2) = (#‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
27		frgpnabl.r	. . . . . . . . . . . . . 14 ⊢ ∼ = ( ~_FG ‘𝐼)
28		frgpnabl.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
29		frgpnabl.t	. . . . . . . . . . . . . 14 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
30	12, 27, 28, 29	efgtlen 17962	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (#‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((#‘𝑥) + 2))
31	30	adantll 746	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (#‘⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩) = ((#‘𝑥) + 2))
32	26, 31	syl5eq 2656	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (0 + 2) = ((#‘𝑥) + 2))
33	32	ex 449	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) → (0 + 2) = ((#‘𝑥) + 2)))
34		0cnd 9912	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 0 ∈ ℂ)
35		simpr 476	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑥 ∈ 𝑊)
36	12	efgrcl 17951	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2_𝑜)))
37	36	simprd 478	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2_𝑜))
38	37	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑊 = Word (𝐼 × 2_𝑜))
39	35, 38	eleqtrd 2690	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑥 ∈ Word (𝐼 × 2_𝑜))
40		lencl 13179	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word (𝐼 × 2_𝑜) → (#‘𝑥) ∈ ℕ₀)
41	39, 40	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (#‘𝑥) ∈ ℕ₀)
42	41	nn0cnd 11230	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (#‘𝑥) ∈ ℂ)
43		2cnd 10970	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 2 ∈ ℂ)
44	34, 42, 43	addcan2d 10119	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ((0 + 2) = ((#‘𝑥) + 2) ↔ 0 = (#‘𝑥)))
45	33, 44	sylibd 228	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) → 0 = (#‘𝑥)))
46	12, 27, 28, 29	efgtf 17958	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∈ 𝑊 → ((𝑇‘∅) = (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2_𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(#‘∅)) × (𝐼 × 2_𝑜))⟶𝑊))
47	46	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → ((𝑇‘∅) = (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2_𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘∅):((0...(#‘∅)) × (𝐼 × 2_𝑜))⟶𝑊))
48	47	simpld 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (𝑇‘∅) = (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2_𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
49	48	rneqd 5274	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → ran (𝑇‘∅) = ran (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2_𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
50	49	eleq2d 2673	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2_𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))))
51		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2_𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2_𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
52		ovex 6577	. . . . . . . . . . . . . . . 16 ⊢ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ V
53	51, 52	elrnmpt2 6671	. . . . . . . . . . . . . . 15 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2_𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ↔ ∃𝑎 ∈ (0...(#‘∅))∃𝑏 ∈ (𝐼 × 2_𝑜)⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
54		wrd0 13185	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ ∈ Word (𝐼 × 2_𝑜)
55	54	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ∅ ∈ Word (𝐼 × 2_𝑜))
56		simprr 792	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → 𝑏 ∈ (𝐼 × 2_𝑜))
57	28	efgmf 17949	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑀:(𝐼 × 2_𝑜)⟶(𝐼 × 2_𝑜)
58	57	ffvelrni 6266	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 ∈ (𝐼 × 2_𝑜) → (𝑀‘𝑏) ∈ (𝐼 × 2_𝑜))
59	56, 58	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (𝑀‘𝑏) ∈ (𝐼 × 2_𝑜))
60	56, 59	s2cld 13466	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2_𝑜))
61		ccatlid 13222	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∅ ∈ Word (𝐼 × 2_𝑜) → (∅ ++ ∅) = ∅)
62	54, 61	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ ++ ∅) = ∅
63	62	oveq1i 6559	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∅ ++ ∅) ++ ∅) = (∅ ++ ∅)
64	63, 62	eqtr2i 2633	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ = ((∅ ++ ∅) ++ ∅)
65	64	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ∅ = ((∅ ++ ∅) ++ ∅))
66		simprl 790	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → 𝑎 ∈ (0...(#‘∅)))
67		hash0 13019	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (#‘∅) = 0
68	67	oveq2i 6560	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0...(#‘∅)) = (0...0)
69	66, 68	syl6eleq 2698	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → 𝑎 ∈ (0...0))
70		elfz1eq 12223	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 ∈ (0...0) → 𝑎 = 0)
71	69, 70	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → 𝑎 = 0)
72	71, 67	syl6eqr 2662	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → 𝑎 = (#‘∅))
73	67	oveq2i 6560	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 + (#‘∅)) = (𝑎 + 0)
74		0cn 9911	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℂ
75	71, 74	syl6eqel 2696	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → 𝑎 ∈ ℂ)
76	75	addid1d 10115	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (𝑎 + 0) = 𝑎)
77	73, 76	syl5req 2657	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → 𝑎 = (𝑎 + (#‘∅)))
78	55, 55, 55, 60, 65, 72, 77	splval2 13359	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = ((∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ ∅))
79		ccatlid 13222	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2_𝑜) → (∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) = ⟨“𝑏(𝑀‘𝑏)”⟩)
80	79	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2_𝑜) → ((∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ ∅) = (⟨“𝑏(𝑀‘𝑏)”⟩ ++ ∅))
81		ccatrid 13223	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2_𝑜) → (⟨“𝑏(𝑀‘𝑏)”⟩ ++ ∅) = ⟨“𝑏(𝑀‘𝑏)”⟩)
82	80, 81	eqtrd 2644	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2_𝑜) → ((∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ ∅) = ⟨“𝑏(𝑀‘𝑏)”⟩)
83	60, 82	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ((∅ ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ ∅) = ⟨“𝑏(𝑀‘𝑏)”⟩)
84	78, 83	eqtrd 2644	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = ⟨“𝑏(𝑀‘𝑏)”⟩)
85	84	eqeq2d 2620	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩))
86	1	ad3antrrr 762	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → 𝐴 ∈ 𝐼)
87		1on 7454	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1_𝑜 ∈ On
88	87	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → 1_𝑜 ∈ On)
89		simpr 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩)
90	89	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = (⟨“𝑏(𝑀‘𝑏)”⟩‘1))
91		opex 4859	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨𝐵, ∅⟩ ∈ V
92		s2fv1 13483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝐵, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩)
93	91, 92	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘1) = ⟨𝐵, ∅⟩
94		fvex 6113	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀‘𝑏) ∈ V
95		s2fv1 13483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑀‘𝑏) ∈ V → (⟨“𝑏(𝑀‘𝑏)”⟩‘1) = (𝑀‘𝑏))
96	94, 95	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩‘1) = (𝑀‘𝑏)
97	90, 93, 96	3eqtr3g 2667	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐵, ∅⟩ = (𝑀‘𝑏))
98	89	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = (⟨“𝑏(𝑀‘𝑏)”⟩‘0))
99		opex 4859	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨𝐴, ∅⟩ ∈ V
100		s2fv0 13482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝐴, ∅⟩ ∈ V → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩)
101	99, 100	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩‘0) = ⟨𝐴, ∅⟩
102		vex 3176	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑏 ∈ V
103		s2fv0 13482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 ∈ V → (⟨“𝑏(𝑀‘𝑏)”⟩‘0) = 𝑏)
104	102, 103	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨“𝑏(𝑀‘𝑏)”⟩‘0) = 𝑏
105	98, 101, 104	3eqtr3g 2667	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐴, ∅⟩ = 𝑏)
106	105	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = (𝑀‘𝑏))
107	28	efgmval 17948	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2_𝑜) → (𝐴𝑀∅) = ⟨𝐴, (1_𝑜 ∖ ∅)⟩)
108	86, 5, 107	sylancl 693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (𝐴𝑀∅) = ⟨𝐴, (1_𝑜 ∖ ∅)⟩)
109		df-ov 6552	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴𝑀∅) = (𝑀‘⟨𝐴, ∅⟩)
110		dif0 3904	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1_𝑜 ∖ ∅) = 1_𝑜
111	110	opeq2i 4344	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨𝐴, (1_𝑜 ∖ ∅)⟩ = ⟨𝐴, 1_𝑜⟩
112	108, 109, 111	3eqtr3g 2667	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → (𝑀‘⟨𝐴, ∅⟩) = ⟨𝐴, 1_𝑜⟩)
113	97, 106, 112	3eqtr2rd 2651	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → ⟨𝐴, 1_𝑜⟩ = ⟨𝐵, ∅⟩)
114		opthg 4872	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝐼 ∧ 1_𝑜 ∈ On) → (⟨𝐴, 1_𝑜⟩ = ⟨𝐵, ∅⟩ ↔ (𝐴 = 𝐵 ∧ 1_𝑜 = ∅)))
115	114	simplbda 652	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ 𝐼 ∧ 1_𝑜 ∈ On) ∧ ⟨𝐴, 1_𝑜⟩ = ⟨𝐵, ∅⟩) → 1_𝑜 = ∅)
116	86, 88, 113, 115	syl21anc 1317	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩) → 1_𝑜 = ∅)
117	116	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩ → 1_𝑜 = ∅))
118	85, 117	sylbid 229	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ∅ ∈ 𝑊) ∧ (𝑎 ∈ (0...(#‘∅)) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) → 1_𝑜 = ∅))
119	118	rexlimdvva 3020	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (∃𝑎 ∈ (0...(#‘∅))∃𝑏 ∈ (𝐼 × 2_𝑜)⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) → 1_𝑜 = ∅))
120	53, 119	syl5bi 231	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑎 ∈ (0...(#‘∅)), 𝑏 ∈ (𝐼 × 2_𝑜) ↦ (∅ splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) → 1_𝑜 = ∅))
121	50, 120	sylbid 229	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ∅ ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅) → 1_𝑜 = ∅))
122	121	expimpd 627	. . . . . . . . . . . 12 ⊢ (𝜑 → ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1_𝑜 = ∅))
123		vex 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
124		hasheq0 13015	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → ((#‘𝑥) = 0 ↔ 𝑥 = ∅))
125	123, 124	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑥) = 0 ↔ 𝑥 = ∅)
126		eleq1 2676	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑊 ↔ ∅ ∈ 𝑊))
127		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → (𝑇‘𝑥) = (𝑇‘∅))
128	127	rneqd 5274	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → ran (𝑇‘𝑥) = ran (𝑇‘∅))
129	128	eleq2d 2673	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) ↔ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)))
130	126, 129	anbi12d 743	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
131	125, 130	sylbi 206	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑥) = 0 → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
132	131	eqcoms 2618	. . . . . . . . . . . . 13 ⊢ (0 = (#‘𝑥) → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) ↔ (∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅))))
133	132	imbi1d 330	. . . . . . . . . . . 12 ⊢ (0 = (#‘𝑥) → (((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → 1_𝑜 = ∅) ↔ ((∅ ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘∅)) → 1_𝑜 = ∅)))
134	122, 133	syl5ibrcom 236	. . . . . . . . . . 11 ⊢ (𝜑 → (0 = (#‘𝑥) → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → 1_𝑜 = ∅)))
135	134	com23 84	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝑊 ∧ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)) → (0 = (#‘𝑥) → 1_𝑜 = ∅)))
136	135	expdimp 452	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) → (0 = (#‘𝑥) → 1_𝑜 = ∅)))
137	45, 136	mpdd 42	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥) → 1_𝑜 = ∅))
138	137	necon3ad 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (1_𝑜 ≠ ∅ → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥)))
139	22, 138	mpi 20	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥))
140	139	nrexdv 2984	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥))
141		eliun 4460	. . . . 5 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ↔ ∃𝑥 ∈ 𝑊 ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ran (𝑇‘𝑥))
142	140, 141	sylnibr 318	. . . 4 ⊢ (𝜑 → ¬ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
143	21, 142	eldifd 3551	. . 3 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)))
144		frgpnabl.d	. . 3 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
145	143, 144	syl6eleqr 2699	. 2 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ 𝐷)
146		df-s2 13444	. . . . 5 ⊢ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ = (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)
147	12, 27	efger 17954	. . . . . . 7 ⊢ ∼ Er 𝑊
148	147	a1i 11	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝑊)
149	148, 21	erref 7649	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∼ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
150	146, 149	syl5eqbrr 4619	. . . 4 ⊢ (𝜑 → (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∼ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
151		ovex 6577	. . . . . 6 ⊢ (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∈ V
152	146, 151	eqeltri 2684	. . . . 5 ⊢ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ V
153	152, 151	elec 7673	. . . 4 ⊢ (⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ ↔ (⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩) ∼ ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩)
154	150, 153	sylibr 223	. . 3 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
155		frgpnabl.u	. . . . . . 7 ⊢ 𝑈 = (var_FGrp‘𝐼)
156	27, 155	vrgpval 18003	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
157	13, 1, 156	syl2anc 691	. . . . 5 ⊢ (𝜑 → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
158	27, 155	vrgpval 18003	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝐵 ∈ 𝐼) → (𝑈‘𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] ∼ )
159	13, 8, 158	syl2anc 691	. . . . 5 ⊢ (𝜑 → (𝑈‘𝐵) = [⟨“⟨𝐵, ∅⟩”⟩] ∼ )
160	157, 159	oveq12d 6567	. . . 4 ⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = ([⟨“⟨𝐴, ∅⟩”⟩] ∼ + [⟨“⟨𝐵, ∅⟩”⟩] ∼ ))
161	7	s1cld 13236	. . . . . 6 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ Word (𝐼 × 2_𝑜))
162	161, 20	eleqtrrd 2691	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊)
163	10	s1cld 13236	. . . . . 6 ⊢ (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ Word (𝐼 × 2_𝑜))
164	163, 20	eleqtrrd 2691	. . . . 5 ⊢ (𝜑 → ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊)
165		frgpnabl.g	. . . . . 6 ⊢ 𝐺 = (freeGrp‘𝐼)
166		frgpnabl.p	. . . . . 6 ⊢ + = (+_g‘𝐺)
167	12, 165, 27, 166	frgpadd 17999	. . . . 5 ⊢ ((⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊 ∧ ⟨“⟨𝐵, ∅⟩”⟩ ∈ 𝑊) → ([⟨“⟨𝐴, ∅⟩”⟩] ∼ + [⟨“⟨𝐵, ∅⟩”⟩] ∼ ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
168	162, 164, 167	syl2anc 691	. . . 4 ⊢ (𝜑 → ([⟨“⟨𝐴, ∅⟩”⟩] ∼ + [⟨“⟨𝐵, ∅⟩”⟩] ∼ ) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
169	160, 168	eqtrd 2644	. . 3 ⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = [(⟨“⟨𝐴, ∅⟩”⟩ ++ ⟨“⟨𝐵, ∅⟩”⟩)] ∼ )
170	154, 169	eleqtrrd 2691	. 2 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ ((𝑈‘𝐴) + (𝑈‘𝐵)))
171	145, 170	elind 3760	1 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵))))

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 1_𝑜c1o 7440 2_𝑜c2o 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 var_FGrpcvrgp 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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