Theorem frgp0 18613

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 2794	. . 3 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o))
3		frgp0.r	. . 3 ⊢ ∼ = ( ~FG ‘𝐼)
4	1, 2, 3	frgpval 18611	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
5		2on 7965	. . . . 5 ⊢ 2o ∈ On
6		xpexg 7333	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
7	5, 6	mpan2 687	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × 2o) ∈ V)
8		eqid 2794	. . . . 5 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o)))
9	2, 8	frmdbas 17828	. . . 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 2800	. 2 ⊢ (𝐼 ∈ 𝑉 → Word (𝐼 × 2o) = (Base‘(freeMnd‘(𝐼 × 2o))))
12		eqidd 2795	. 2 ⊢ (𝐼 ∈ 𝑉 → (+g‘(freeMnd‘(𝐼 × 2o))) = (+g‘(freeMnd‘(𝐼 × 2o))))
13		eqid 2794	. . . 4 ⊢ ( I ‘Word (𝐼 × 2o)) = ( I ‘Word (𝐼 × 2o))
14	13, 3	efger 18571	. . 3 ⊢ ∼ Er ( I ‘Word (𝐼 × 2o))
15		wrdexg 13717	. . . . 5 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
16		fvi 6610	. . . . 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 8150	. . . 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 234	. 2 ⊢ (𝐼 ∈ 𝑉 → ∼ Er Word (𝐼 × 2o))
21		fvexd 6556	. 2 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2o)) ∈ V)
22		eqid 2794	. . . 4 ⊢ (+g‘(freeMnd‘(𝐼 × 2o))) = (+g‘(freeMnd‘(𝐼 × 2o)))
23	1, 2, 3, 22	frgpcpbl 18612	. . 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 17835	. . . . . 6 ⊢ ((𝐼 × 2o) ∈ V → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
26	7, 25	syl 17	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
27	26	3ad2ant1 1126	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
28		simp2 1130	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
29	11	3ad2ant1 1126	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → Word (𝐼 × 2o) = (Base‘(freeMnd‘(𝐼 × 2o))))
30	28, 29	eleqtrd 2884	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
31		simp3 1131	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑦 ∈ Word (𝐼 × 2o))
32	31, 29	eleqtrd 2884	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
33	8, 22	mndcl 17740	. . . 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 1364	. . 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 2885	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → (𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦) ∈ Word (𝐼 × 2o))
36	20	adantr 481	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → ∼ Er Word (𝐼 × 2o))
37	26	adantr 481	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
38	34	3adant3r3 1177	. . . . . 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 1189	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → 𝑧 ∈ Word (𝐼 × 2o))
40	11	adantr 481	. . . . . . 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 2884	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
42	8, 22	mndcl 17740	. . . . . 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 1364	. . . . 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 2885	. . . 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 8162	. . 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 1177	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
47	32	3adant3r3 1177	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
48	8, 22	mndass 17741	. . . 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 1365	. . 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 4990	. 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 13735	. . 3 ⊢ ∅ ∈ Word (𝐼 × 2o)
52	51	a1i 11	. 2 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ Word (𝐼 × 2o))
53	51, 11	syl5eleq 2888	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
54	53	adantr 481	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
55	11	eleq2d 2867	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ Word (𝐼 × 2o) ↔ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o)))))
56	55	biimpa 477	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
57	2, 8, 22	frmdadd 17831	. . . . 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 13784	. . . . 5 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (∅ ++ 𝑥) = 𝑥)
60	59	adantl 482	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅ ++ 𝑥) = 𝑥)
61	58, 60	eqtrd 2830	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) = 𝑥)
62	20	adantr 481	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → ∼ Er Word (𝐼 × 2o))
63		simpr 485	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
64	62, 63	erref 8162	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → 𝑥 ∼ 𝑥)
65	61, 64	eqbrtrd 4986	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) ∼ 𝑥)
66		revcl 13959	. . . 4 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (reverse‘𝑥) ∈ Word (𝐼 × 2o))
67	66	adantl 482	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (reverse‘𝑥) ∈ Word (𝐼 × 2o))
68		eqid 2794	. . . . 5 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
69	68	efgmf 18566	. . . 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 14029	. . 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 481	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → Word (𝐼 × 2o) = (Base‘(freeMnd‘(𝐼 × 2o))))
74	72, 73	eleqtrd 2884	. . . 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 17831	. . . 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 2867	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ ( I ‘Word (𝐼 × 2o)) ↔ 𝑥 ∈ Word (𝐼 × 2o)))
78	77	biimpar 478	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ ( I ‘Word (𝐼 × 2o)))
79		eqid 2794	. . . . 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 18581	. . . 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 4986	. 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 17974	1 ⊢ (𝐼 ∈ 𝑉 → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2080  Vcvv 3436   ∖ cdif 3858  ∅c0 4213  ⟨cop 4480  ⟨cotp 4482   class class class wbr 4964   ↦ cmpt 5043   I cid 5350   × cxp 5444   ∘ ccom 5450  Oncon0 6069  ⟶wf 6224  ‘cfv 6228  (class class class)co 7019   ∈ cmpo 7021  1_oc1o 7949  2_oc2o 7950   Er wer 8139  [cec 8140  0cc0 10386  ...cfz 12742  ♯chash 13540  Word cword 13707   ++ cconcat 13768   splice csplice 13947  reversecreverse 13956  ⟨“cs2 14039  Basecbs 16312  +_gcplusg 16394  0_gc0g 16542  Mndcmnd 17733  freeMndcfrmd 17823  Grpcgrp 17861   ~_FG cefg 18559  freeGrpcfrgp 18560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322  ax-cnex 10442  ax-resscn 10443  ax-1cn 10444  ax-icn 10445  ax-addcl 10446  ax-addrcl 10447  ax-mulcl 10448  ax-mulrcl 10449  ax-mulcom 10450  ax-addass 10451  ax-mulass 10452  ax-distr 10453  ax-i2m1 10454  ax-1ne0 10455  ax-1rid 10456  ax-rnegex 10457  ax-rrecex 10458  ax-cnre 10459  ax-pre-lttri 10460  ax-pre-lttrn 10461  ax-pre-ltadd 10462  ax-pre-mulgt0 10463

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-nel 3090  df-ral 3109  df-rex 3110  df-reu 3111  df-rmo 3112  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-ot 4483  df-uni 4748  df-int 4785  df-iun 4829  df-iin 4830  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-riota 6980  df-ov 7022  df-oprab 7023  df-mpo 7024  df-om 7440  df-1st 7548  df-2nd 7549  df-wrecs 7801  df-recs 7863  df-rdg 7901  df-1o 7956  df-2o 7957  df-oadd 7960  df-er 8142  df-ec 8144  df-qs 8148  df-map 8261  df-en 8361  df-dom 8362  df-sdom 8363  df-fin 8364  df-sup 8755  df-inf 8756  df-card 9217  df-pnf 10526  df-mnf 10527  df-xr 10528  df-ltxr 10529  df-le 10530  df-sub 10721  df-neg 10722  df-nn 11489  df-2 11550  df-3 11551  df-4 11552  df-5 11553  df-6 11554  df-7 11555  df-8 11556  df-9 11557  df-n0 11748  df-xnn0 11818  df-z 11832  df-dec 11949  df-uz 12094  df-fz 12743  df-fzo 12884  df-hash 13541  df-word 13708  df-lsw 13761  df-concat 13769  df-s1 13794  df-substr 13839  df-pfx 13869  df-splice 13948  df-reverse 13957  df-s2 14046  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-plusg 16407  df-mulr 16408  df-sca 16410  df-vsca 16411  df-ip 16412  df-tset 16413  df-ple 16414  df-ds 16416  df-0g 16544  df-imas 16610  df-qus 16611  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-frmd 17825  df-grp 17864  df-efg 18562  df-frgp 18563

This theorem is referenced by:  frgpgrp  18615  frgpinv  18617  frgpmhm  18618