Theorem frgp0 19720

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 2727	. . 3 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o))
3		frgp0.r	. . 3 ⊢ ∼ = ( ~FG ‘𝐼)
4	1, 2, 3	frgpval 19718	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ ))
5		2on 8505	. . . . 5 ⊢ 2o ∈ On
6		xpexg 7756	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
7	5, 6	mpan2 689	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × 2o) ∈ V)
8		eqid 2727	. . . . 5 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o)))
9	2, 8	frmdbas 18809	. . . 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 2733	. 2 ⊢ (𝐼 ∈ 𝑉 → Word (𝐼 × 2o) = (Base‘(freeMnd‘(𝐼 × 2o))))
12		eqidd 2728	. 2 ⊢ (𝐼 ∈ 𝑉 → (+g‘(freeMnd‘(𝐼 × 2o))) = (+g‘(freeMnd‘(𝐼 × 2o))))
13		eqid 2727	. . . 4 ⊢ ( I ‘Word (𝐼 × 2o)) = ( I ‘Word (𝐼 × 2o))
14	13, 3	efger 19678	. . 3 ⊢ ∼ Er ( I ‘Word (𝐼 × 2o))
15		wrdexg 14512	. . . . 5 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V)
16		fvi 6977	. . . . 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 8737	. . . 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 232	. 2 ⊢ (𝐼 ∈ 𝑉 → ∼ Er Word (𝐼 × 2o))
21		fvexd 6915	. 2 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2o)) ∈ V)
22		eqid 2727	. . . 4 ⊢ (+g‘(freeMnd‘(𝐼 × 2o))) = (+g‘(freeMnd‘(𝐼 × 2o)))
23	1, 2, 3, 22	frgpcpbl 19719	. . 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 18816	. . . . . 6 ⊢ ((𝐼 × 2o) ∈ V → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
26	7, 25	syl 17	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
27	26	3ad2ant1 1130	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
28		simp2 1134	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
29	11	3ad2ant1 1130	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → Word (𝐼 × 2o) = (Base‘(freeMnd‘(𝐼 × 2o))))
30	28, 29	eleqtrd 2830	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
31		simp3 1135	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑦 ∈ Word (𝐼 × 2o))
32	31, 29	eleqtrd 2830	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
33	8, 22	mndcl 18707	. . . 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 1368	. . 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 2831	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o)) → (𝑥(+g‘(freeMnd‘(𝐼 × 2o)))𝑦) ∈ Word (𝐼 × 2o))
36	20	adantr 479	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → ∼ Er Word (𝐼 × 2o))
37	26	adantr 479	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → (freeMnd‘(𝐼 × 2o)) ∈ Mnd)
38	34	3adant3r3 1181	. . . . . 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 1193	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → 𝑧 ∈ Word (𝐼 × 2o))
40	11	adantr 479	. . . . . . 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 2830	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
42	8, 22	mndcl 18707	. . . . . 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 1368	. . . . 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 2831	. . . 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 8749	. . 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 1181	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
47	32	3adant3r3 1181	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑦 ∈ Word (𝐼 × 2o) ∧ 𝑧 ∈ Word (𝐼 × 2o))) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
48	8, 22	mndass 18708	. . . 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 1369	. . 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 5176	. 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 14527	. . 3 ⊢ ∅ ∈ Word (𝐼 × 2o)
52	51	a1i 11	. 2 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ Word (𝐼 × 2o))
53	51, 11	eleqtrid 2834	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
54	53	adantr 479	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
55	11	eleq2d 2814	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ Word (𝐼 × 2o) ↔ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o)))))
56	55	biimpa 475	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o))))
57	2, 8, 22	frmdadd 18812	. . . . 5 ⊢ ((∅ ∈ (Base‘(freeMnd‘(𝐼 × 2o))) ∧ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2o)))) → (∅(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) = (∅ ++ 𝑥))
58	54, 56, 57	syl2anc 582	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) = (∅ ++ 𝑥))
59		ccatlid 14574	. . . . 5 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (∅ ++ 𝑥) = 𝑥)
60	59	adantl 480	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅ ++ 𝑥) = 𝑥)
61	58, 60	eqtrd 2767	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) = 𝑥)
62	20	adantr 479	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → ∼ Er Word (𝐼 × 2o))
63		simpr 483	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
64	62, 63	erref 8749	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → 𝑥 ∼ 𝑥)
65	61, 64	eqbrtrd 5172	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (∅(+g‘(freeMnd‘(𝐼 × 2o)))𝑥) ∼ 𝑥)
66		revcl 14749	. . . 4 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (reverse‘𝑥) ∈ Word (𝐼 × 2o))
67	66	adantl 480	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → (reverse‘𝑥) ∈ Word (𝐼 × 2o))
68		eqid 2727	. . . . 5 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
69	68	efgmf 19673	. . . 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 14820	. . 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 582	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → ((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) ∘ (reverse‘𝑥)) ∈ Word (𝐼 × 2o))
73	11	adantr 479	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → Word (𝐼 × 2o) = (Base‘(freeMnd‘(𝐼 × 2o))))
74	72, 73	eleqtrd 2830	. . . 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 18812	. . . 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 582	. . 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 2814	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ ( I ‘Word (𝐼 × 2o)) ↔ 𝑥 ∈ Word (𝐼 × 2o)))
78	77	biimpar 476	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word (𝐼 × 2o)) → 𝑥 ∈ ( I ‘Word (𝐼 × 2o)))
79		eqid 2727	. . . . 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 19688	. . . 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 5172	. 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 19019	1 ⊢ (𝐼 ∈ 𝑉 → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3471   ∖ cdif 3944  ∅c0 4324  ⟨cop 4636  ⟨cotp 4638   class class class wbr 5150   ↦ cmpt 5233   I cid 5577   × cxp 5678   ∘ ccom 5684  Oncon0 6372  ⟶wf 6547  ‘cfv 6551  (class class class)co 7424   ∈ cmpo 7426  1_oc1o 8484  2_oc2o 8485   Er wer 8726  [cec 8727  0cc0 11144  ...cfz 13522  ♯chash 14327  Word cword 14502   ++ cconcat 14558   splice csplice 14737  reversecreverse 14746  ⟨“cs2 14830  Basecbs 17185  +_gcplusg 17238  0_gc0g 17426  Mndcmnd 18699  freeMndcfrmd 18804  Grpcgrp 18895   ~_FG cefg 19666  freeGrpcfrgp 19667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-ot 4639  df-uni 4911  df-int 4952  df-iun 5000  df-iin 5001  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-1o 8491  df-2o 8492  df-er 8729  df-ec 8731  df-qs 8735  df-map 8851  df-en 8969  df-dom 8970  df-sdom 8971  df-fin 8972  df-sup 9471  df-inf 9472  df-card 9968  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12509  df-xnn0 12581  df-z 12595  df-dec 12714  df-uz 12859  df-fz 13523  df-fzo 13666  df-hash 14328  df-word 14503  df-lsw 14551  df-concat 14559  df-s1 14584  df-substr 14629  df-pfx 14659  df-splice 14738  df-reverse 14747  df-s2 14837  df-struct 17121  df-slot 17156  df-ndx 17168  df-base 17186  df-plusg 17251  df-mulr 17252  df-sca 17254  df-vsca 17255  df-ip 17256  df-tset 17257  df-ple 17258  df-ds 17260  df-0g 17428  df-imas 17495  df-qus 17496  df-mgm 18605  df-sgrp 18684  df-mnd 18700  df-frmd 18806  df-grp 18898  df-efg 19669  df-frgp 19670

This theorem is referenced by:  frgpgrp  19722  frgpinv  19724  frgpmhm  19725

Otomatik - 173.255.232.114 CloudFlare DNS Türk Telekom DNS Google DNS Open DNS

OSZAR »