Theorem frgp0 19697

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 2730	. . 3 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o))
3		frgp0.r	. . 3 ⊢ ∼ = ( ~FG ‘𝐼)
4	1, 2, 3	frgpval 19695	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
5		2on 8450	. . . . 5 ⊢ 2o ∈ On
6		xpexg 7729	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
7	5, 6	mpan2 691	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × 2o) ∈ V)
8		eqid 2730	. . . . 5 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o)))
9	2, 8	frmdbas 18786	. . . 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 2736	. 2 ⊢ (𝐼 ∈ 𝑉 → Word (𝐼 × 2o) = (Base‘(freeMnd‘(𝐼 × 2o))))
12		eqidd 2731	. 2 ⊢ (𝐼 ∈ 𝑉 → (+g‘(freeMnd‘(𝐼 × 2o))) = (+g‘(freeMnd‘(𝐼 × 2o))))
13		eqid 2730	. . . 4 ⊢ ( I ‘Word (𝐼 × 2o)) = ( I ‘Word (𝐼 × 2o))
14	13, 3	efger 19655	. . 3 ⊢ ∼ Er ( I ‘Word (𝐼 × 2o))
15		wrdexg 14496	. . . . 5 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
16		fvi 6940	. . . . 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 8682	. . . 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 6876	. 2 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2o)) ∈ V)
22		eqid 2730	. . . 4 ⊢ (+g‘(freeMnd‘(𝐼 × 2o))) = (+g‘(freeMnd‘(𝐼 × 2o)))
23	1, 2, 3, 22	frgpcpbl 19696	. . 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 18793	. . . . . 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 2831	. . . 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 2831	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
33	8, 22	mndcl 18676	. . . 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 2832	. 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 2831	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
42	8, 22	mndcl 18676	. . . . . 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 2832	. . . 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 8694	. . 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 18677	. . . 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 5136	. 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 14511	. . 3 ⊢ ∅ ∈ Word (𝐼 × 2o)
52	51	a1i 11	. 2 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ Word (𝐼 × 2o))
53	51, 11	eleqtrid 2835	. . . . . 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 2815	. . . . . 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 18789	. . . . 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 14558	. . . . 5 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (∅ ++ 𝑥) = 𝑥)
60	59	adantl 481	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅ ++ 𝑥) = 𝑥)
61	58, 60	eqtrd 2765	. . 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 8694	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → 𝑥 ∼ 𝑥)
65	61, 64	eqbrtrd 5132	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) ∼ 𝑥)
66		revcl 14733	. . . 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 2730	. . . . 5 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
69	68	efgmf 19650	. . . 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 14804	. . 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 2831	. . . 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 18789	. . . 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 2815	. . . . 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 2730	. . . . 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 19665	. . . 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 5132	. 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 18997	1 ⊢ (𝐼 ∈ 𝑉 → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∖ cdif 3914  ∅c0 4299  ⟨cop 4598  ⟨cotp 4600   class class class wbr 5110   ↦ cmpt 5191   I cid 5535   × cxp 5639   ∘ ccom 5645  Oncon0 6335  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  1_oc1o 8430  2_oc2o 8431   Er wer 8671  [cec 8672  0cc0 11075  ...cfz 13475  ♯chash 14302  Word cword 14485   ++ cconcat 14542   splice csplice 14721  reversecreverse 14730  ⟨“cs2 14814  Basecbs 17186  +_gcplusg 17227  0_gc0g 17409  Mndcmnd 18668  freeMndcfrmd 18781  Grpcgrp 18872   ~_FG cefg 19643  freeGrpcfrgp 19644

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-ec 8676  df-qs 8680  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-xnn0 12523  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-fzo 13623  df-hash 14303  df-word 14486  df-lsw 14535  df-concat 14543  df-s1 14568  df-substr 14613  df-pfx 14643  df-splice 14722  df-reverse 14731  df-s2 14821  df-struct 17124  df-slot 17159  df-ndx 17171  df-base 17187  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-0g 17411  df-imas 17478  df-qus 17479  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-frmd 18783  df-grp 18875  df-efg 19646  df-frgp 19647

This theorem is referenced by:  frgpgrp  19699  frgpinv  19701  frgpmhm  19702