Theorem frgp0 18173

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 2622	. . 3 ⊢ (freeMnd‘(𝐼 × 2𝑜)) = (freeMnd‘(𝐼 × 2𝑜))
3		frgp0.r	. . 3 ⊢ ∼ = ( ~FG ‘𝐼)
4	1, 2, 3	frgpval 18171	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2𝑜)) /s ∼ ))
5		2on 7568	. . . . 5 ⊢ 2𝑜 ∈ On
6		xpexg 6960	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
7	5, 6	mpan2 707	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × 2𝑜) ∈ V)
8		eqid 2622	. . . . 5 ⊢ (Base‘(freeMnd‘(𝐼 × 2𝑜))) = (Base‘(freeMnd‘(𝐼 × 2𝑜)))
9	2, 8	frmdbas 17389	. . . 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 2628	. 2 ⊢ (𝐼 ∈ 𝑉 → Word (𝐼 × 2𝑜) = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
12		eqidd 2623	. 2 ⊢ (𝐼 ∈ 𝑉 → (+g‘(freeMnd‘(𝐼 × 2𝑜))) = (+g‘(freeMnd‘(𝐼 × 2𝑜))))
13		eqid 2622	. . . 4 ⊢ ( I ‘Word (𝐼 × 2𝑜)) = ( I ‘Word (𝐼 × 2𝑜))
14	13, 3	efger 18131	. . 3 ⊢ ∼ Er ( I ‘Word (𝐼 × 2𝑜))
15		wrdexg 13315	. . . . 5 ⊢ ((𝐼 × 2𝑜) ∈ V → Word (𝐼 × 2𝑜) ∈ V)
16		fvi 6255	. . . . 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 7750	. . . 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 223	. 2 ⊢ (𝐼 ∈ 𝑉 → ∼ Er Word (𝐼 × 2𝑜))
21		fvexd 6203	. 2 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2𝑜)) ∈ V)
22		eqid 2622	. . . 4 ⊢ (+g‘(freeMnd‘(𝐼 × 2𝑜))) = (+g‘(freeMnd‘(𝐼 × 2𝑜)))
23	1, 2, 3, 22	frgpcpbl 18172	. . 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 17396	. . . . . 6 ⊢ ((𝐼 × 2𝑜) ∈ V → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
26	7, 25	syl 17	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
27	26	3ad2ant1 1082	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
28		simp2 1062	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ Word (𝐼 × 2𝑜))
29	11	3ad2ant1 1082	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → Word (𝐼 × 2𝑜) = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
30	28, 29	eleqtrd 2703	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
31		simp3 1063	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑦 ∈ Word (𝐼 × 2𝑜))
32	31, 29	eleqtrd 2703	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
33	8, 22	mndcl 17301	. . . 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 1326	. . 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 2704	. 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 1276	. . . . . 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 1069	. . . . . . 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 2703	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
42	8, 22	mndcl 17301	. . . . . 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 1326	. . . . 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 2704	. . . 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 7762	. . 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 1276	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
47	32	3adant3r3 1276	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
48	8, 22	mndass 17302	. . . 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 1328	. . 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 4679	. 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 13330	. . 3 ⊢ ∅ ∈ Word (𝐼 × 2𝑜)
52	51	a1i 11	. 2 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ Word (𝐼 × 2𝑜))
53	51, 11	syl5eleq 2707	. . . . . 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 2687	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ Word (𝐼 × 2𝑜) ↔ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜)))))
56	55	biimpa 501	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
57	2, 8, 22	frmdadd 17392	. . . . 5 ⊢ ((∅ ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))) ∧ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜)))) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) = (∅ ++ 𝑥))
58	54, 56, 57	syl2anc 693	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) = (∅ ++ 𝑥))
59		ccatlid 13369	. . . . 5 ⊢ (𝑥 ∈ Word (𝐼 × 2𝑜) → (∅ ++ 𝑥) = 𝑥)
60	59	adantl 482	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅ ++ 𝑥) = 𝑥)
61	58, 60	eqtrd 2656	. . 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 477	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ Word (𝐼 × 2𝑜))
64	62, 63	erref 7762	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∼ 𝑥)
65	61, 64	eqbrtrd 4675	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) ∼ 𝑥)
66		revcl 13510	. . . 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 2622	. . . . 5 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜 ∖ 𝑧)⟩) = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜 ∖ 𝑧)⟩)
69	68	efgmf 18126	. . . 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 13577	. . 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 693	. 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 2703	. . . 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 17392	. . . 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 693	. . 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 2687	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↔ 𝑥 ∈ Word (𝐼 × 2𝑜)))
78	77	biimpar 502	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ ( I ‘Word (𝐼 × 2𝑜)))
79		eqid 2622	. . . . 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 18141	. . . 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 4675	. 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 17533	1 ⊢ (𝐼 ∈ 𝑉 → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571  ∅c0 3915  ⟨cop 4183  ⟨cotp 4185   class class class wbr 4653   ↦ cmpt 4729   I cid 5023   × cxp 5112   ∘ ccom 5118  Oncon0 5723  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1_𝑜c1o 7553  2_𝑜c2o 7554   Er wer 7739  [cec 7740  0cc0 9936  ...cfz 12326  #chash 13117  Word cword 13291   ++ cconcat 13293   splice csplice 13296  reversecreverse 13297  ⟨“cs2 13586  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100  Mndcmnd 17294  freeMndcfrmd 17384  Grpcgrp 17422   ~_FG cefg 18119  freeGrpcfrgp 18120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-reverse 13305  df-s2 13593  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-0g 16102  df-imas 16168  df-qus 16169  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-frmd 17386  df-grp 17425  df-efg 18122  df-frgp 18123

This theorem is referenced by:  frgpgrp  18175  frgpinv  18177  frgpmhm  18178