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Theorem frgp0 17996

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 2610	. . 3 ⊢ (freeMnd‘(𝐼 × 2_𝑜)) = (freeMnd‘(𝐼 × 2_𝑜))
3		frgp0.r	. . 3 ⊢ ∼ = ( ~_FG ‘𝐼)
4	1, 2, 3	frgpval 17994	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2_𝑜)) /_s ∼ ))
5		2on 7455	. . . . 5 ⊢ 2_𝑜 ∈ On
6		xpexg 6858	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 2_𝑜 ∈ On) → (𝐼 × 2_𝑜) ∈ V)
7	5, 6	mpan2 703	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × 2_𝑜) ∈ V)
8		eqid 2610	. . . . 5 ⊢ (Base‘(freeMnd‘(𝐼 × 2_𝑜))) = (Base‘(freeMnd‘(𝐼 × 2_𝑜)))
9	2, 8	frmdbas 17212	. . . 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 2616	. 2 ⊢ (𝐼 ∈ 𝑉 → Word (𝐼 × 2_𝑜) = (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
12		eqidd 2611	. 2 ⊢ (𝐼 ∈ 𝑉 → (+_g‘(freeMnd‘(𝐼 × 2_𝑜))) = (+_g‘(freeMnd‘(𝐼 × 2_𝑜))))
13		eqid 2610	. . . 4 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) = ( I ‘Word (𝐼 × 2_𝑜))
14	13, 3	efger 17954	. . 3 ⊢ ∼ Er ( I ‘Word (𝐼 × 2_𝑜))
15		wrdexg 13170	. . . . 5 ⊢ ((𝐼 × 2_𝑜) ∈ V → Word (𝐼 × 2_𝑜) ∈ V)
16		fvi 6165	. . . . 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 7637	. . . 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 222	. 2 ⊢ (𝐼 ∈ 𝑉 → ∼ Er Word (𝐼 × 2_𝑜))
21		fvex 6113	. . 3 ⊢ (freeMnd‘(𝐼 × 2_𝑜)) ∈ V
22	21	a1i 11	. 2 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2_𝑜)) ∈ V)
23		eqid 2610	. . . 4 ⊢ (+_g‘(freeMnd‘(𝐼 × 2_𝑜))) = (+_g‘(freeMnd‘(𝐼 × 2_𝑜)))
24	1, 2, 3, 23	frgpcpbl 17995	. . 3 ⊢ ((𝑎 ∼ 𝑏 ∧ 𝑐 ∼ 𝑑) → (𝑎(+_g‘(freeMnd‘(𝐼 × 2_𝑜)))𝑐) ∼ (𝑏(+_g‘(freeMnd‘(𝐼 × 2_𝑜)))𝑑))
25	24	a1i 11	. 2 ⊢ (𝐼 ∈ 𝑉 → ((𝑎 ∼ 𝑏 ∧ 𝑐 ∼ 𝑑) → (𝑎(+_g‘(freeMnd‘(𝐼 × 2_𝑜)))𝑐) ∼ (𝑏(+_g‘(freeMnd‘(𝐼 × 2_𝑜)))𝑑)))
26	2	frmdmnd 17219	. . . . . 6 ⊢ ((𝐼 × 2_𝑜) ∈ V → (freeMnd‘(𝐼 × 2_𝑜)) ∈ Mnd)
27	7, 26	syl 17	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2_𝑜)) ∈ Mnd)
28	27	3ad2ant1 1075	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2_𝑜)) → (freeMnd‘(𝐼 × 2_𝑜)) ∈ Mnd)
29		simp2 1055	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2_𝑜)) → 𝑥 ∈ Word (𝐼 × 2_𝑜))
30	11	3ad2ant1 1075	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2_𝑜)) → Word (𝐼 × 2_𝑜) = (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
31	29, 30	eleqtrd 2690	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2_𝑜)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
32		simp3 1056	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2_𝑜)) → 𝑦 ∈ Word (𝐼 × 2_𝑜))
33	32, 30	eleqtrd 2690	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2_𝑜)) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
34	8, 23	mndcl 17124	. . . 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))))
35	28, 31, 33, 34	syl3anc 1318	. . 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))))
36	35, 30	eleqtrrd 2691	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2_𝑜)) → (𝑥(+_g‘(freeMnd‘(𝐼 × 2_𝑜)))𝑦) ∈ Word (𝐼 × 2_𝑜))
37	20	adantr 480	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2_𝑜))) → ∼ Er Word (𝐼 × 2_𝑜))
38	27	adantr 480	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2_𝑜))) → (freeMnd‘(𝐼 × 2_𝑜)) ∈ Mnd)
39	35	3adant3r3 1268	. . . . . 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))))
40		simpr3 1062	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2_𝑜))) → 𝑧 ∈ Word (𝐼 × 2_𝑜))
41	11	adantr 480	. . . . . . 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))))
42	40, 41	eleqtrd 2690	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2_𝑜))) → 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
43	8, 23	mndcl 17124	. . . . . 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))))
44	38, 39, 42, 43	syl3anc 1318	. . . . 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))))
45	44, 41	eleqtrrd 2691	. . . 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))
46	37, 45	erref 7649	. . 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))
47	31	3adant3r3 1268	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2_𝑜))) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
48	33	3adant3r3 1268	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2_𝑜))) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
49	8, 23	mndass 17125	. . . 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)))
50	38, 47, 48, 42, 49	syl13anc 1320	. . 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)))
51	46, 50	breqtrd 4609	. 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)))
52		wrd0 13185	. . 3 ⊢ ∅ ∈ Word (𝐼 × 2_𝑜)
53	52	a1i 11	. 2 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ Word (𝐼 × 2_𝑜))
54	52, 11	syl5eleq 2694	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
55	54	adantr 480	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
56	11	eleq2d 2673	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ Word (𝐼 × 2_𝑜) ↔ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2_𝑜)))))
57	56	biimpa 500	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
58	2, 8, 23	frmdadd 17215	. . . . 5 ⊢ ((∅ ∈ (Base‘(freeMnd‘(𝐼 × 2_𝑜))) ∧ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2_𝑜)))) → (∅(+_g‘(freeMnd‘(𝐼 × 2_𝑜)))𝑥) = (∅ ++ 𝑥))
59	55, 57, 58	syl2anc 691	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → (∅(+_g‘(freeMnd‘(𝐼 × 2_𝑜)))𝑥) = (∅ ++ 𝑥))
60		ccatlid 13222	. . . . 5 ⊢ (𝑥 ∈ Word (𝐼 × 2_𝑜) → (∅ ++ 𝑥) = 𝑥)
61	60	adantl 481	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → (∅ ++ 𝑥) = 𝑥)
62	59, 61	eqtrd 2644	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → (∅(+_g‘(freeMnd‘(𝐼 × 2_𝑜)))𝑥) = 𝑥)
63	20	adantr 480	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → ∼ Er Word (𝐼 × 2_𝑜))
64		simpr 476	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → 𝑥 ∈ Word (𝐼 × 2_𝑜))
65	63, 64	erref 7649	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → 𝑥 ∼ 𝑥)
66	62, 65	eqbrtrd 4605	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → (∅(+_g‘(freeMnd‘(𝐼 × 2_𝑜)))𝑥) ∼ 𝑥)
67		revcl 13361	. . . 4 ⊢ (𝑥 ∈ Word (𝐼 × 2_𝑜) → (reverse‘𝑥) ∈ Word (𝐼 × 2_𝑜))
68	67	adantl 481	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → (reverse‘𝑥) ∈ Word (𝐼 × 2_𝑜))
69		eqid 2610	. . . . 5 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩) = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
70	69	efgmf 17949	. . . 4 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩):(𝐼 × 2_𝑜)⟶(𝐼 × 2_𝑜)
71	70	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))
72		wrdco 13428	. . 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))
73	68, 71, 72	syl2anc 691	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → ((𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ∈ Word (𝐼 × 2_𝑜))
74	11	adantr 480	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → Word (𝐼 × 2_𝑜) = (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
75	73, 74	eleqtrd 2690	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → ((𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ∈ (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
76	2, 8, 23	frmdadd 17215	. . . 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))
77	75, 57, 76	syl2anc 691	. . 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))
78	17	eleq2d 2673	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ ( I ‘Word (𝐼 × 2_𝑜)) ↔ 𝑥 ∈ Word (𝐼 × 2_𝑜)))
79	78	biimpar 501	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → 𝑥 ∈ ( I ‘Word (𝐼 × 2_𝑜)))
80		eqid 2610	. . . . 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)”⟩⟩)))
81	13, 3, 69, 80	efginvrel1 17964	. . . 4 ⊢ (𝑥 ∈ ( I ‘Word (𝐼 × 2_𝑜)) → (((𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥) ∼ ∅)
82	79, 81	syl 17	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2_𝑜)) → (((𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥) ∼ ∅)
83	77, 82	eqbrtrd 4605	. 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) ∼ ∅)
84	4, 11, 12, 20, 22, 25, 36, 51, 53, 66, 73, 83	qusgrp2 17356	1 ⊢ (𝐼 ∈ 𝑉 → (𝐺 ∈ Grp ∧ [∅] ∼ = (0_g‘𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 ⟨cop 4131 ⟨cotp 4133 class class class wbr 4583 ↦ cmpt 4643 I cid 4948 × cxp 5036 ∘ ccom 5042 Oncon0 5640 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1_𝑜c1o 7440 2_𝑜c2o 7441 Er wer 7626 [cec 7627 0cc0 9815 ...cfz 12197 #chash 12979 Word cword 13146 ++ cconcat 13148 splice csplice 13151 reversecreverse 13152 ⟨“cs2 13437 Basecbs 15695 +_gcplusg 15768 0_gc0g 15923 Mndcmnd 17117 freeMndcfrmd 17207 Grpcgrp 17245 ~_FG cefg 17942 freeGrpcfrgp 17943

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-splice 13159 df-reverse 13160 df-s2 13444 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-0g 15925 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-frmd 17209 df-grp 17248 df-efg 17945 df-frgp 17946

This theorem is referenced by: frgpgrp 17998 frgpinv 18000 frgpmhm 18001

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