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Theorem frgp0 19622

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 ∧ [∅] ∼ = (0_g‘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 2732	. . 3 ⊢ (freeMnd‘(𝐼 × 2_o)) = (freeMnd‘(𝐼 × 2_o))
3		frgp0.r	. . 3 ⊢ ∼ = ( ~_FG ‘𝐼)
4	1, 2, 3	frgpval 19620	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2_o)) /_s ∼ ))
5		2on 8476	. . . . 5 ⊢ 2_o ∈ On
6		xpexg 7733	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 2_o ∈ On) → (𝐼 × 2_o) ∈ V)
7	5, 6	mpan2 689	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × 2_o) ∈ V)
8		eqid 2732	. . . . 5 ⊢ (Base‘(freeMnd‘(𝐼 × 2_o))) = (Base‘(freeMnd‘(𝐼 × 2_o)))
9	2, 8	frmdbas 18729	. . . 4 ⊢ ((𝐼 × 2_o) ∈ V → (Base‘(freeMnd‘(𝐼 × 2_o))) = Word (𝐼 × 2_o))
10	7, 9	syl 17	. . 3 ⊢ (𝐼 ∈ 𝑉 → (Base‘(freeMnd‘(𝐼 × 2_o))) = Word (𝐼 × 2_o))
11	10	eqcomd 2738	. 2 ⊢ (𝐼 ∈ 𝑉 → Word (𝐼 × 2_o) = (Base‘(freeMnd‘(𝐼 × 2_o))))
12		eqidd 2733	. 2 ⊢ (𝐼 ∈ 𝑉 → (+_g‘(freeMnd‘(𝐼 × 2_o))) = (+_g‘(freeMnd‘(𝐼 × 2_o))))
13		eqid 2732	. . . 4 ⊢ ( I ‘Word (𝐼 × 2_o)) = ( I ‘Word (𝐼 × 2_o))
14	13, 3	efger 19580	. . 3 ⊢ ∼ Er ( I ‘Word (𝐼 × 2_o))
15		wrdexg 14470	. . . . 5 ⊢ ((𝐼 × 2_o) ∈ V → Word (𝐼 × 2_o) ∈ V)
16		fvi 6964	. . . . 5 ⊢ (Word (𝐼 × 2_o) ∈ V → ( I ‘Word (𝐼 × 2_o)) = Word (𝐼 × 2_o))
17	7, 15, 16	3syl 18	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ( I ‘Word (𝐼 × 2_o)) = Word (𝐼 × 2_o))
18		ereq2 8707	. . . 4 ⊢ (( I ‘Word (𝐼 × 2_o)) = Word (𝐼 × 2_o) → ( ∼ Er ( I ‘Word (𝐼 × 2_o)) ↔ ∼ Er Word (𝐼 × 2_o)))
19	17, 18	syl 17	. . 3 ⊢ (𝐼 ∈ 𝑉 → ( ∼ Er ( I ‘Word (𝐼 × 2_o)) ↔ ∼ Er Word (𝐼 × 2_o)))
20	14, 19	mpbii 232	. 2 ⊢ (𝐼 ∈ 𝑉 → ∼ Er Word (𝐼 × 2_o))
21		fvexd 6903	. 2 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2_o)) ∈ V)
22		eqid 2732	. . . 4 ⊢ (+_g‘(freeMnd‘(𝐼 × 2_o))) = (+_g‘(freeMnd‘(𝐼 × 2_o)))
23	1, 2, 3, 22	frgpcpbl 19621	. . 3 ⊢ ((𝑎 ∼ 𝑏 ∧ 𝑐 ∼ 𝑑) → (𝑎(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑐) ∼ (𝑏(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑑))
24	23	a1i 11	. 2 ⊢ (𝐼 ∈ 𝑉 → ((𝑎 ∼ 𝑏 ∧ 𝑐 ∼ 𝑑) → (𝑎(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑐) ∼ (𝑏(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑑)))
25	2	frmdmnd 18736	. . . . . 6 ⊢ ((𝐼 × 2_o) ∈ V → (freeMnd‘(𝐼 × 2_o)) ∈ Mnd)
26	7, 25	syl 17	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2_o)) ∈ Mnd)
27	26	3ad2ant1 1133	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o)) → (freeMnd‘(𝐼 × 2_o)) ∈ Mnd)
28		simp2 1137	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o)) → 𝑥 ∈ Word (𝐼 × 2_o))
29	11	3ad2ant1 1133	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o)) → Word (𝐼 × 2_o) = (Base‘(freeMnd‘(𝐼 × 2_o))))
30	28, 29	eleqtrd 2835	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))
31		simp3 1138	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o)) → 𝑦 ∈ Word (𝐼 × 2_o))
32	31, 29	eleqtrd 2835	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o)) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))
33	8, 22	mndcl 18629	. . . 4 ⊢ (((freeMnd‘(𝐼 × 2_o)) ∈ Mnd ∧ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2_o))) ∧ 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2_o)))) → (𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))
34	27, 30, 32, 33	syl3anc 1371	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o)) → (𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))
35	34, 29	eleqtrrd 2836	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o)) → (𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑦) ∈ Word (𝐼 × 2_o))
36	20	adantr 481	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o) ∧ 𝑧 ∈ Word (𝐼 × 2_o))) → ∼ Er Word (𝐼 × 2_o))
37	26	adantr 481	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o) ∧ 𝑧 ∈ Word (𝐼 × 2_o))) → (freeMnd‘(𝐼 × 2_o)) ∈ Mnd)
38	34	3adant3r3 1184	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o) ∧ 𝑧 ∈ Word (𝐼 × 2_o))) → (𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))
39		simpr3 1196	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o) ∧ 𝑧 ∈ Word (𝐼 × 2_o))) → 𝑧 ∈ Word (𝐼 × 2_o))
40	11	adantr 481	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o) ∧ 𝑧 ∈ Word (𝐼 × 2_o))) → Word (𝐼 × 2_o) = (Base‘(freeMnd‘(𝐼 × 2_o))))
41	39, 40	eleqtrd 2835	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o) ∧ 𝑧 ∈ Word (𝐼 × 2_o))) → 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))
42	8, 22	mndcl 18629	. . . . . 6 ⊢ (((freeMnd‘(𝐼 × 2_o)) ∈ Mnd ∧ (𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2_o))) ∧ 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2_o)))) → ((𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑦)(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑧) ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))
43	37, 38, 41, 42	syl3anc 1371	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o) ∧ 𝑧 ∈ Word (𝐼 × 2_o))) → ((𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑦)(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑧) ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))
44	43, 40	eleqtrrd 2836	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o) ∧ 𝑧 ∈ Word (𝐼 × 2_o))) → ((𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑦)(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑧) ∈ Word (𝐼 × 2_o))
45	36, 44	erref 8719	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o) ∧ 𝑧 ∈ Word (𝐼 × 2_o))) → ((𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑦)(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑧) ∼ ((𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑦)(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑧))
46	30	3adant3r3 1184	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o) ∧ 𝑧 ∈ Word (𝐼 × 2_o))) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))
47	32	3adant3r3 1184	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o) ∧ 𝑧 ∈ Word (𝐼 × 2_o))) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))
48	8, 22	mndass 18630	. . . 4 ⊢ (((freeMnd‘(𝐼 × 2_o)) ∈ Mnd ∧ (𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2_o))) ∧ 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2_o))) ∧ 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))) → ((𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑦)(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑧) = (𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))(𝑦(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑧)))
49	37, 46, 47, 41, 48	syl13anc 1372	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o) ∧ 𝑧 ∈ Word (𝐼 × 2_o))) → ((𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑦)(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑧) = (𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))(𝑦(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑧)))
50	45, 49	breqtrd 5173	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_o) ∧ 𝑦 ∈ Word (𝐼 × 2_o) ∧ 𝑧 ∈ Word (𝐼 × 2_o))) → ((𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑦)(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑧) ∼ (𝑥(+_g‘(freeMnd‘(𝐼 × 2_o)))(𝑦(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑧)))
51		wrd0 14485	. . 3 ⊢ ∅ ∈ Word (𝐼 × 2_o)
52	51	a1i 11	. 2 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ Word (𝐼 × 2_o))
53	51, 11	eleqtrid 2839	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))
54	53	adantr 481	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))
55	11	eleq2d 2819	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ Word (𝐼 × 2_o) ↔ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2_o)))))
56	55	biimpa 477	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))
57	2, 8, 22	frmdadd 18732	. . . . 5 ⊢ ((∅ ∈ (Base‘(freeMnd‘(𝐼 × 2_o))) ∧ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2_o)))) → (∅(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑥) = (∅ ++ 𝑥))
58	54, 56, 57	syl2anc 584	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → (∅(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑥) = (∅ ++ 𝑥))
59		ccatlid 14532	. . . . 5 ⊢ (𝑥 ∈ Word (𝐼 × 2_o) → (∅ ++ 𝑥) = 𝑥)
60	59	adantl 482	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → (∅ ++ 𝑥) = 𝑥)
61	58, 60	eqtrd 2772	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → (∅(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑥) = 𝑥)
62	20	adantr 481	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → ∼ Er Word (𝐼 × 2_o))
63		simpr 485	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → 𝑥 ∈ Word (𝐼 × 2_o))
64	62, 63	erref 8719	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → 𝑥 ∼ 𝑥)
65	61, 64	eqbrtrd 5169	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → (∅(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑥) ∼ 𝑥)
66		revcl 14707	. . . 4 ⊢ (𝑥 ∈ Word (𝐼 × 2_o) → (reverse‘𝑥) ∈ Word (𝐼 × 2_o))
67	66	adantl 482	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → (reverse‘𝑥) ∈ Word (𝐼 × 2_o))
68		eqid 2732	. . . . 5 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
69	68	efgmf 19575	. . . 4 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩):(𝐼 × 2_o)⟶(𝐼 × 2_o)
70	69	a1i 11	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩):(𝐼 × 2_o)⟶(𝐼 × 2_o))
71		wrdco 14778	. . 3 ⊢ (((reverse‘𝑥) ∈ Word (𝐼 × 2_o) ∧ (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩):(𝐼 × 2_o)⟶(𝐼 × 2_o)) → ((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ∈ Word (𝐼 × 2_o))
72	67, 70, 71	syl2anc 584	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → ((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ∈ Word (𝐼 × 2_o))
73	11	adantr 481	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → Word (𝐼 × 2_o) = (Base‘(freeMnd‘(𝐼 × 2_o))))
74	72, 73	eleqtrd 2835	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → ((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ∈ (Base‘(freeMnd‘(𝐼 × 2_o))))
75	2, 8, 22	frmdadd 18732	. . . 4 ⊢ ((((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ∈ (Base‘(freeMnd‘(𝐼 × 2_o))) ∧ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2_o)))) → (((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ∘ (reverse‘𝑥))(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑥) = (((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥))
76	74, 56, 75	syl2anc 584	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → (((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ∘ (reverse‘𝑥))(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑥) = (((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥))
77	17	eleq2d 2819	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ ( I ‘Word (𝐼 × 2_o)) ↔ 𝑥 ∈ Word (𝐼 × 2_o)))
78	77	biimpar 478	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → 𝑥 ∈ ( I ‘Word (𝐼 × 2_o)))
79		eqid 2732	. . . . 5 ⊢ (𝑣 ∈ ( I ‘Word (𝐼 × 2_o)) ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)‘𝑤)”⟩⟩))) = (𝑣 ∈ ( I ‘Word (𝐼 × 2_o)) ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)‘𝑤)”⟩⟩)))
80	13, 3, 68, 79	efginvrel1 19590	. . . 4 ⊢ (𝑥 ∈ ( I ‘Word (𝐼 × 2_o)) → (((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥) ∼ ∅)
81	78, 80	syl 17	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → (((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥) ∼ ∅)
82	76, 81	eqbrtrd 5169	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_o)) → (((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ∘ (reverse‘𝑥))(+_g‘(freeMnd‘(𝐼 × 2_o)))𝑥) ∼ ∅)
83	4, 11, 12, 20, 21, 24, 35, 50, 52, 65, 72, 82	qusgrp2 18937	1 ⊢ (𝐼 ∈ 𝑉 → (𝐺 ∈ Grp ∧ [∅] ∼ = (0_g‘𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∖ cdif 3944 ∅c0 4321 ⟨cop 4633 ⟨cotp 4635 class class class wbr 5147 ↦ cmpt 5230 I cid 5572 × cxp 5673 ∘ ccom 5679 Oncon0 6361 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 1_oc1o 8455 2_oc2o 8456 Er wer 8696 [cec 8697 0cc0 11106 ...cfz 13480 ♯chash 14286 Word cword 14460 ++ cconcat 14516 splice csplice 14695 reversecreverse 14704 ⟨“cs2 14788 Basecbs 17140 +_gcplusg 17193 0_gc0g 17381 Mndcmnd 18621 freeMndcfrmd 18724 Grpcgrp 18815 ~_FG cefg 19568 freeGrpcfrgp 19569

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-ec 8701 df-qs 8705 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-lsw 14509 df-concat 14517 df-s1 14542 df-substr 14587 df-pfx 14617 df-splice 14696 df-reverse 14705 df-s2 14795 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-0g 17383 df-imas 17450 df-qus 17451 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-frmd 18726 df-grp 18818 df-efg 19571 df-frgp 19572

This theorem is referenced by: frgpgrp 19624 frgpinv 19626 frgpmhm 19627

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