Theorem frgp0 19674

Description: The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
frgp0.m	⊢ 𝐺 = (freeGrp‘𝐼)
frgp0.r	⊢ ∼ = ( ~FG ‘𝐼)

Assertion

Ref	Expression
frgp0	⊢ (𝐼 ∈ 𝑉 → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺)))

Proof of Theorem frgp0

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 𝑦 𝑧 𝑛 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgp0.m	. . 3 ⊢ 𝐺 = (freeGrp‘𝐼)
2		eqid 2729	. . 3 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o))
3		frgp0.r	. . 3 ⊢ ∼ = ( ~FG ‘𝐼)
4	1, 2, 3	frgpval 19672	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
5		2on 8424	. . . . 5 ⊢ 2o ∈ On
6		xpexg 7706	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
7	5, 6	mpan2 691	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × 2o) ∈ V)
8		eqid 2729	. . . . 5 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o)))
9	2, 8	frmdbas 18761	. . . 4 ⊢ ((𝐼 × 2o) ∈ V → (Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 × 2o))
10	7, 9	syl 17	. . 3 ⊢ (𝐼 ∈ 𝑉 → (Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 × 2o))
11	10	eqcomd 2735	. 2 ⊢ (𝐼 ∈ 𝑉 → Word (𝐼 × 2o) = (Base‘(freeMnd‘(𝐼 × 2o))))
12		eqidd 2730	. 2 ⊢ (𝐼 ∈ 𝑉 → (+g‘(freeMnd‘(𝐼 × 2o))) = (+g‘(freeMnd‘(𝐼 × 2o))))
13		eqid 2729	. . . 4 ⊢ ( I ‘Word (𝐼 × 2o)) = ( I ‘Word (𝐼 × 2o))
14	13, 3	efger 19632	. . 3 ⊢ ∼ Er ( I ‘Word (𝐼 × 2o))
15		wrdexg 14465	. . . . 5 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
16		fvi 6919	. . . . 5 ⊢ (Word (𝐼 × 2o) ∈ V → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o))
17	7, 15, 16	3syl 18	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o))
18		ereq2 8656	. . . 4 ⊢ (( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o) → ( ∼ Er ( I ‘Word (𝐼 × 2o)) ↔ ∼ Er Word (𝐼 × 2o)))
19	17, 18	syl 17	. . 3 ⊢ (𝐼 ∈ 𝑉 → ( ∼ Er ( I ‘Word (𝐼 × 2o)) ↔ ∼ Er Word (𝐼 × 2o)))
20	14, 19	mpbii 233	. 2 ⊢ (𝐼 ∈ 𝑉 → ∼ Er Word (𝐼 × 2o))
21		fvexd 6855	. 2 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2o)) ∈ V)
22		eqid 2729	. . . 4 ⊢ (+g‘(freeMnd‘(𝐼 × 2o))) = (+g‘(freeMnd‘(𝐼 × 2o)))
23	1, 2, 3, 22	frgpcpbl 19673	. . 3 ⊢ ((𝑎 ∼ 𝑏 ∧ 𝑐 ∼ 𝑑) → (𝑎(+g‘(freeMnd‘(𝐼 × 2o)))𝑐) ∼ (𝑏(+g‘(freeMnd‘(𝐼 × 2o)))𝑑))
24	23	a1i 11	. 2 ⊢ (𝐼 ∈ 𝑉 → ((𝑎 ∼ 𝑏 ∧ 𝑐 ∼ 𝑑) → (𝑎(+g‘(freeMnd‘(𝐼 × 2o)))𝑐) ∼ (𝑏(+g‘(freeMnd‘(𝐼 × 2o)))𝑑)))
25	2	frmdmnd 18768	. . . . . 6 ⊢ ((𝐼 × 2o) ∈ V → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
26	7, 25	syl 17	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
27	26	3ad2ant1 1133	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
28		simp2 1137	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
29	11	3ad2ant1 1133	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → Word (𝐼 × 2o) = (Base‘(freeMnd‘(𝐼 × 2o))))
30	28, 29	eleqtrd 2830	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
31		simp3 1138	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑦 ∈ Word (𝐼 × 2o))
32	31, 29	eleqtrd 2830	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
33	8, 22	mndcl 18651	. . . 4 ⊢ (((freeMnd‘(𝐼 × 2o)) ∈ Mnd ∧ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o))) ∧ 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2o)))) → (𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
34	27, 30, 32, 33	syl3anc 1373	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → (𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
35	34, 29	eleqtrrd 2831	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → (𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦) ∈ Word (𝐼 × 2o))
36	20	adantr 480	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → ∼ Er Word (𝐼 × 2o))
37	26	adantr 480	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
38	34	3adant3r3 1185	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → (𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
39		simpr3 1197	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → 𝑧 ∈ Word (𝐼 × 2o))
40	11	adantr 480	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → Word (𝐼 × 2o) = (Base‘(freeMnd‘(𝐼 × 2o))))
41	39, 40	eleqtrd 2830	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
42	8, 22	mndcl 18651	. . . . . 6 ⊢ (((freeMnd‘(𝐼 × 2o)) ∈ Mnd ∧ (𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2o))) ∧ 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2o)))) → ((𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦)(+g‘(freeMnd‘(𝐼 × 2o)))𝑧) ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
43	37, 38, 41, 42	syl3anc 1373	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → ((𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦)(+g‘(freeMnd‘(𝐼 × 2o)))𝑧) ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
44	43, 40	eleqtrrd 2831	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → ((𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦)(+g‘(freeMnd‘(𝐼 × 2o)))𝑧) ∈ Word (𝐼 × 2o))
45	36, 44	erref 8668	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → ((𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦)(+g‘(freeMnd‘(𝐼 × 2o)))𝑧) ∼ ((𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦)(+g‘(freeMnd‘(𝐼 × 2o)))𝑧))
46	30	3adant3r3 1185	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
47	32	3adant3r3 1185	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
48	8, 22	mndass 18652	. . . 4 ⊢ (((freeMnd‘(𝐼 × 2o)) ∈ Mnd ∧ (𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o))) ∧ 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2o))) ∧ 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))) → ((𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦)(+g‘(freeMnd‘(𝐼 × 2o)))𝑧) = (𝑥(+g‘(freeMnd‘(𝐼 × 2o)))(𝑦(+g‘(freeMnd‘(𝐼 × 2o)))𝑧)))
49	37, 46, 47, 41, 48	syl13anc 1374	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → ((𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦)(+g‘(freeMnd‘(𝐼 × 2o)))𝑧) = (𝑥(+g‘(freeMnd‘(𝐼 × 2o)))(𝑦(+g‘(freeMnd‘(𝐼 × 2o)))𝑧)))
50	45, 49	breqtrd 5128	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → ((𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦)(+g‘(freeMnd‘(𝐼 × 2o)))𝑧) ∼ (𝑥(+g‘(freeMnd‘(𝐼 × 2o)))(𝑦(+g‘(freeMnd‘(𝐼 × 2o)))𝑧)))
51		wrd0 14480	. . 3 ⊢ ∅ ∈ Word (𝐼 × 2o)
52	51	a1i 11	. 2 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ Word (𝐼 × 2o))
53	51, 11	eleqtrid 2834	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
54	53	adantr 480	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
55	11	eleq2d 2814	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ Word (𝐼 × 2o) ↔ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o)))))
56	55	biimpa 476	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
57	2, 8, 22	frmdadd 18764	. . . . 5 ⊢ ((∅ ∈ (Base‘(freeMnd‘(𝐼 × 2o))) ∧ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o)))) → (∅(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) = (∅ ++ 𝑥))
58	54, 56, 57	syl2anc 584	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) = (∅ ++ 𝑥))
59		ccatlid 14527	. . . . 5 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (∅ ++ 𝑥) = 𝑥)
60	59	adantl 481	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅ ++ 𝑥) = 𝑥)
61	58, 60	eqtrd 2764	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) = 𝑥)
62	20	adantr 480	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → ∼ Er Word (𝐼 × 2o))
63		simpr 484	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
64	62, 63	erref 8668	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → 𝑥 ∼ 𝑥)
65	61, 64	eqbrtrd 5124	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) ∼ 𝑥)
66		revcl 14702	. . . 4 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (reverse‘𝑥) ∈ Word (𝐼 × 2o))
67	66	adantl 481	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (reverse‘𝑥) ∈ Word (𝐼 × 2o))
68		eqid 2729	. . . . 5 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
69	68	efgmf 19627	. . . 4 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩):(𝐼 × 2o)⟶(𝐼 × 2o)
70	69	a1i 11	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩):(𝐼 × 2o)⟶(𝐼 × 2o))
71		wrdco 14773	. . 3 ⊢ (((reverse‘𝑥) ∈ Word (𝐼 × 2o) ∧ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩):(𝐼 × 2o)⟶(𝐼 × 2o)) → ((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ∈ Word (𝐼 × 2o))
72	67, 70, 71	syl2anc 584	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → ((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ∈ Word (𝐼 × 2o))
73	11	adantr 480	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → Word (𝐼 × 2o) = (Base‘(freeMnd‘(𝐼 × 2o))))
74	72, 73	eleqtrd 2830	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → ((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
75	2, 8, 22	frmdadd 18764	. . . 4 ⊢ ((((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ∈ (Base‘(freeMnd‘(𝐼 × 2o))) ∧ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o)))) → (((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) ∘ (reverse‘𝑥))(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) = (((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥))
76	74, 56, 75	syl2anc 584	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) ∘ (reverse‘𝑥))(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) = (((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥))
77	17	eleq2d 2814	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ ( I ‘Word (𝐼 × 2o)) ↔ 𝑥 ∈ Word (𝐼 × 2o)))
78	77	biimpar 477	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ ( I ‘Word (𝐼 × 2o)))
79		eqid 2729	. . . . 5 ⊢ (𝑣 ∈ ( I ‘Word (𝐼 × 2o)) ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘𝑤)”⟩⟩))) = (𝑣 ∈ ( I ‘Word (𝐼 × 2o)) ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘𝑤)”⟩⟩)))
80	13, 3, 68, 79	efginvrel1 19642	. . . 4 ⊢ (𝑥 ∈ ( I ‘Word (𝐼 × 2o)) → (((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥) ∼ ∅)
81	78, 80	syl 17	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥) ∼ ∅)
82	76, 81	eqbrtrd 5124	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) ∘ (reverse‘𝑥))(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) ∼ ∅)
83	4, 11, 12, 20, 21, 24, 35, 50, 52, 65, 72, 82	qusgrp2 18972	1 ⊢ (𝐼 ∈ 𝑉 → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ∖ cdif 3908  ∅c0 4292  ⟨cop 4591  ⟨cotp 4593   class class class wbr 5102   ↦ cmpt 5183   I cid 5525   × cxp 5629   ∘ ccom 5635  Oncon0 6320  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  1_oc1o 8404  2_oc2o 8405   Er wer 8645  [cec 8646  0cc0 11044  ...cfz 13444  ♯chash 14271  Word cword 14454   ++ cconcat 14511   splice csplice 14690  reversecreverse 14699  ⟨“cs2 14783  Basecbs 17155  +_gcplusg 17196  0_gc0g 17378  Mndcmnd 18643  freeMndcfrmd 18756  Grpcgrp 18847   ~_FG cefg 19620  freeGrpcfrgp 19621

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-ec 8650  df-qs 8654  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-inf 9370  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-xnn0 12492  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-fzo 13592  df-hash 14272  df-word 14455  df-lsw 14504  df-concat 14512  df-s1 14537  df-substr 14582  df-pfx 14612  df-splice 14691  df-reverse 14700  df-s2 14790  df-struct 17093  df-slot 17128  df-ndx 17140  df-base 17156  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-0g 17380  df-imas 17447  df-qus 17448  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-frmd 18758  df-grp 18850  df-efg 19623  df-frgp 19624

This theorem is referenced by:  frgpgrp  19676  frgpinv  19678  frgpmhm  19679