Theorem frgp0 19778

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 2737	. . 3 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o))
3		frgp0.r	. . 3 ⊢ ∼ = ( ~FG ‘𝐼)
4	1, 2, 3	frgpval 19776	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
5		2on 8520	. . . . 5 ⊢ 2o ∈ On
6		xpexg 7770	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
7	5, 6	mpan2 691	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × 2o) ∈ V)
8		eqid 2737	. . . . 5 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o)))
9	2, 8	frmdbas 18865	. . . 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 2743	. 2 ⊢ (𝐼 ∈ 𝑉 → Word (𝐼 × 2o) = (Base‘(freeMnd‘(𝐼 × 2o))))
12		eqidd 2738	. 2 ⊢ (𝐼 ∈ 𝑉 → (+g‘(freeMnd‘(𝐼 × 2o))) = (+g‘(freeMnd‘(𝐼 × 2o))))
13		eqid 2737	. . . 4 ⊢ ( I ‘Word (𝐼 × 2o)) = ( I ‘Word (𝐼 × 2o))
14	13, 3	efger 19736	. . 3 ⊢ ∼ Er ( I ‘Word (𝐼 × 2o))
15		wrdexg 14562	. . . . 5 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
16		fvi 6985	. . . . 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 8753	. . . 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 6921	. 2 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2o)) ∈ V)
22		eqid 2737	. . . 4 ⊢ (+g‘(freeMnd‘(𝐼 × 2o))) = (+g‘(freeMnd‘(𝐼 × 2o)))
23	1, 2, 3, 22	frgpcpbl 19777	. . 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 18872	. . . . . 6 ⊢ ((𝐼 × 2o) ∈ V → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
26	7, 25	syl 17	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
27	26	3ad2ant1 1134	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
28		simp2 1138	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
29	11	3ad2ant1 1134	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → Word (𝐼 × 2o) = (Base‘(freeMnd‘(𝐼 × 2o))))
30	28, 29	eleqtrd 2843	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
31		simp3 1139	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑦 ∈ Word (𝐼 × 2o))
32	31, 29	eleqtrd 2843	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
33	8, 22	mndcl 18755	. . . 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 2844	. 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 2843	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
42	8, 22	mndcl 18755	. . . . . 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 2844	. . . 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 8765	. . 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 18756	. . . 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 5169	. 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 14577	. . 3 ⊢ ∅ ∈ Word (𝐼 × 2o)
52	51	a1i 11	. 2 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ Word (𝐼 × 2o))
53	51, 11	eleqtrid 2847	. . . . . 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 2827	. . . . . 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 18868	. . . . 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 14624	. . . . 5 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (∅ ++ 𝑥) = 𝑥)
60	59	adantl 481	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅ ++ 𝑥) = 𝑥)
61	58, 60	eqtrd 2777	. . 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 8765	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → 𝑥 ∼ 𝑥)
65	61, 64	eqbrtrd 5165	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) ∼ 𝑥)
66		revcl 14799	. . . 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 2737	. . . . 5 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
69	68	efgmf 19731	. . . 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 14870	. . 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 2843	. . . 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 18868	. . . 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 2827	. . . . 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 2737	. . . . 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 19746	. . . 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 5165	. 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 19076	1 ⊢ (𝐼 ∈ 𝑉 → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∖ cdif 3948  ∅c0 4333  ⟨cop 4632  ⟨cotp 4634   class class class wbr 5143   ↦ cmpt 5225   I cid 5577   × cxp 5683   ∘ ccom 5689  Oncon0 6384  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  1_oc1o 8499  2_oc2o 8500   Er wer 8742  [cec 8743  0cc0 11155  ...cfz 13547  ♯chash 14369  Word cword 14552   ++ cconcat 14608   splice csplice 14787  reversecreverse 14796  ⟨“cs2 14880  Basecbs 17247  +_gcplusg 17297  0_gc0g 17484  Mndcmnd 18747  freeMndcfrmd 18860  Grpcgrp 18951   ~_FG cefg 19724  freeGrpcfrgp 19725

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-lsw 14601  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-splice 14788  df-reverse 14797  df-s2 14887  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-0g 17486  df-imas 17553  df-qus 17554  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-frmd 18862  df-grp 18954  df-efg 19727  df-frgp 19728

This theorem is referenced by:  frgpgrp  19780  frgpinv  19782  frgpmhm  19783