Theorem frgpcyg 19741

Description: A free group is cyclic iff it has zero or one generator. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 18-Apr-2021.)

Hypothesis

Ref	Expression
frgpcyg.g	⊢ 𝐺 = (freeGrp‘𝐼)

Assertion

Ref	Expression
frgpcyg	⊢ (𝐼 ≼ 1𝑜 ↔ 𝐺 ∈ CycGrp)

Proof of Theorem frgpcyg

Dummy variables 𝑓 𝑔 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdom2 7871	. . 3 ⊢ (𝐼 ≼ 1𝑜 ↔ (𝐼 ≺ 1𝑜 ∨ 𝐼 ≈ 1𝑜))
2		sdom1 8045	. . . . 5 ⊢ (𝐼 ≺ 1𝑜 ↔ 𝐼 = ∅)
3		frgpcyg.g	. . . . . . 7 ⊢ 𝐺 = (freeGrp‘𝐼)
4		fveq2 6103	. . . . . . 7 ⊢ (𝐼 = ∅ → (freeGrp‘𝐼) = (freeGrp‘∅))
5	3, 4	syl5eq 2656	. . . . . 6 ⊢ (𝐼 = ∅ → 𝐺 = (freeGrp‘∅))
6		0ex 4718	. . . . . . . 8 ⊢ ∅ ∈ V
7		eqid 2610	. . . . . . . . 9 ⊢ (freeGrp‘∅) = (freeGrp‘∅)
8	7	frgpgrp 17998	. . . . . . . 8 ⊢ (∅ ∈ V → (freeGrp‘∅) ∈ Grp)
9	6, 8	ax-mp 5	. . . . . . 7 ⊢ (freeGrp‘∅) ∈ Grp
10		eqid 2610	. . . . . . . 8 ⊢ (Base‘(freeGrp‘∅)) = (Base‘(freeGrp‘∅))
11	7, 10	0frgp 18015	. . . . . . 7 ⊢ (Base‘(freeGrp‘∅)) ≈ 1𝑜
12	10	0cyg 18117	. . . . . . 7 ⊢ (((freeGrp‘∅) ∈ Grp ∧ (Base‘(freeGrp‘∅)) ≈ 1𝑜) → (freeGrp‘∅) ∈ CycGrp)
13	9, 11, 12	mp2an 704	. . . . . 6 ⊢ (freeGrp‘∅) ∈ CycGrp
14	5, 13	syl6eqel 2696	. . . . 5 ⊢ (𝐼 = ∅ → 𝐺 ∈ CycGrp)
15	2, 14	sylbi 206	. . . 4 ⊢ (𝐼 ≺ 1𝑜 → 𝐺 ∈ CycGrp)
16		eqid 2610	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
17		eqid 2610	. . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺)
18		relen 7846	. . . . . . 7 ⊢ Rel ≈
19	18	brrelexi 5082	. . . . . 6 ⊢ (𝐼 ≈ 1𝑜 → 𝐼 ∈ V)
20	3	frgpgrp 17998	. . . . . 6 ⊢ (𝐼 ∈ V → 𝐺 ∈ Grp)
21	19, 20	syl 17	. . . . 5 ⊢ (𝐼 ≈ 1𝑜 → 𝐺 ∈ Grp)
22		eqid 2610	. . . . . . . 8 ⊢ ( ~FG ‘𝐼) = ( ~FG ‘𝐼)
23		eqid 2610	. . . . . . . 8 ⊢ (varFGrp‘𝐼) = (varFGrp‘𝐼)
24	22, 23, 3, 16	vrgpf 18004	. . . . . . 7 ⊢ (𝐼 ∈ V → (varFGrp‘𝐼):𝐼⟶(Base‘𝐺))
25	19, 24	syl 17	. . . . . 6 ⊢ (𝐼 ≈ 1𝑜 → (varFGrp‘𝐼):𝐼⟶(Base‘𝐺))
26		en1uniel 7914	. . . . . 6 ⊢ (𝐼 ≈ 1𝑜 → ∪ 𝐼 ∈ 𝐼)
27	25, 26	ffvelrnd 6268	. . . . 5 ⊢ (𝐼 ≈ 1𝑜 → ((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺))
28		zringgrp 19642	. . . . . . . . 9 ⊢ ℤring ∈ Grp
29		uniexg 6853	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ V → ∪ 𝐼 ∈ V)
30	19, 29	syl 17	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1𝑜 → ∪ 𝐼 ∈ V)
31		1zzd 11285	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1𝑜 → 1 ∈ ℤ)
32	30, 31	fsnd 6091	. . . . . . . . . 10 ⊢ (𝐼 ≈ 1𝑜 → {⟨∪ 𝐼, 1⟩}:{∪ 𝐼}⟶ℤ)
33		en1b 7910	. . . . . . . . . . . 12 ⊢ (𝐼 ≈ 1𝑜 ↔ 𝐼 = {∪ 𝐼})
34	33	biimpi 205	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1𝑜 → 𝐼 = {∪ 𝐼})
35	34	feq2d 5944	. . . . . . . . . 10 ⊢ (𝐼 ≈ 1𝑜 → ({⟨∪ 𝐼, 1⟩}:𝐼⟶ℤ ↔ {⟨∪ 𝐼, 1⟩}:{∪ 𝐼}⟶ℤ))
36	32, 35	mpbird 246	. . . . . . . . 9 ⊢ (𝐼 ≈ 1𝑜 → {⟨∪ 𝐼, 1⟩}:𝐼⟶ℤ)
37		zringbas 19643	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤring)
38	3, 37, 23	frgpup3 18014	. . . . . . . . 9 ⊢ ((ℤring ∈ Grp ∧ 𝐼 ∈ V ∧ {⟨∪ 𝐼, 1⟩}:𝐼⟶ℤ) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
39	28, 19, 36, 38	mp3an2i 1421	. . . . . . . 8 ⊢ (𝐼 ≈ 1𝑜 → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
40	39	adantr 480	. . . . . . 7 ⊢ ((𝐼 ≈ 1𝑜 ∧ 𝑥 ∈ (Base‘𝐺)) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
41		reurex 3137	. . . . . . 7 ⊢ (∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → ∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
42	40, 41	syl 17	. . . . . 6 ⊢ ((𝐼 ≈ 1𝑜 ∧ 𝑥 ∈ (Base‘𝐺)) → ∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
43		fveq1 6102	. . . . . . . . . 10 ⊢ ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → ((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼))
44		fvco3 6185	. . . . . . . . . . . 12 ⊢ (((varFGrp‘𝐼):𝐼⟶(Base‘𝐺) ∧ ∪ 𝐼 ∈ 𝐼) → ((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)))
45	25, 26, 44	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1𝑜 → ((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)))
46		1z 11284	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
47		fvsng 6352	. . . . . . . . . . . 12 ⊢ ((∪ 𝐼 ∈ V ∧ 1 ∈ ℤ) → ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼) = 1)
48	30, 46, 47	sylancl 693	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1𝑜 → ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼) = 1)
49	45, 48	eqeq12d 2625	. . . . . . . . . 10 ⊢ (𝐼 ≈ 1𝑜 → (((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼) ↔ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1))
50	43, 49	syl5ib 233	. . . . . . . . 9 ⊢ (𝐼 ≈ 1𝑜 → ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1))
51	50	ad2antrr 758	. . . . . . . 8 ⊢ (((𝐼 ≈ 1𝑜 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1))
52	16, 37	ghmf 17487	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐺 GrpHom ℤring) → 𝑓:(Base‘𝐺)⟶ℤ)
53	52	ad2antrl 760	. . . . . . . . . . . 12 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓:(Base‘𝐺)⟶ℤ)
54	53	ffvelrnda 6267	. . . . . . . . . . 11 ⊢ (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓‘𝑥) ∈ ℤ)
55	54	an32s 842	. . . . . . . . . 10 ⊢ (((𝐼 ≈ 1𝑜 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑓‘𝑥) ∈ ℤ)
56		mptresid 5375	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = ( I ↾ (Base‘𝐺))
57	3, 16, 23	frgpup3 18014	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝐼 ∈ V ∧ (varFGrp‘𝐼):𝐼⟶(Base‘𝐺)) → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
58	21, 19, 25, 57	syl3anc 1318	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ≈ 1𝑜 → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
59		reurmo 3138	. . . . . . . . . . . . . . . . 17 ⊢ (∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼) → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
60	58, 59	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ≈ 1𝑜 → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
61	60	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
62	21	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝐺 ∈ Grp)
63	16	idghm 17498	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → ( I ↾ (Base‘𝐺)) ∈ (𝐺 GrpHom 𝐺))
64	62, 63	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ( I ↾ (Base‘𝐺)) ∈ (𝐺 GrpHom 𝐺))
65	25	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (varFGrp‘𝐼):𝐼⟶(Base‘𝐺))
66		fcoi2 5992	. . . . . . . . . . . . . . . 16 ⊢ ((varFGrp‘𝐼):𝐼⟶(Base‘𝐺) → (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
67	65, 66	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
68	53	feqmptd 6159	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓 = (𝑥 ∈ (Base‘𝐺) ↦ (𝑓‘𝑥)))
69		eqidd 2611	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
70		oveq1 6556	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑓‘𝑥) → (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
71	54, 68, 69, 70	fmptco 6303	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∘ 𝑓) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
72	27	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺))
73		eqid 2610	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
74	17, 73, 16	mulgghm2 19664	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ ((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺)) → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∈ (ℤring GrpHom 𝐺))
75	62, 72, 74	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∈ (ℤring GrpHom 𝐺))
76		simprl 790	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓 ∈ (𝐺 GrpHom ℤring))
77		ghmco 17503	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∈ (ℤring GrpHom 𝐺) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∘ 𝑓) ∈ (𝐺 GrpHom 𝐺))
78	75, 76, 77	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∘ 𝑓) ∈ (𝐺 GrpHom 𝐺))
79	71, 78	eqeltrrd 2689	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∈ (𝐺 GrpHom 𝐺))
80	34	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝐼 = {∪ 𝐼})
81	80	eleq2d 2673	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑦 ∈ 𝐼 ↔ 𝑦 ∈ {∪ 𝐼}))
82		simprr 792	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)
83	82	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ((𝑓‘((varFGrp‘𝐼)‘∪ 𝐼))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = (1(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
84	16, 17	mulg1 17371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺) → (1(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((varFGrp‘𝐼)‘∪ 𝐼))
85	72, 84	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (1(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((varFGrp‘𝐼)‘∪ 𝐼))
86	83, 85	eqtrd 2644	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ((𝑓‘((varFGrp‘𝐼)‘∪ 𝐼))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((varFGrp‘𝐼)‘∪ 𝐼))
87		elsni 4142	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ {∪ 𝐼} → 𝑦 = ∪ 𝐼)
88	87	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ {∪ 𝐼} → ((varFGrp‘𝐼)‘𝑦) = ((varFGrp‘𝐼)‘∪ 𝐼))
89	88	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ {∪ 𝐼} → (𝑓‘((varFGrp‘𝐼)‘𝑦)) = (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)))
90	89	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ {∪ 𝐼} → ((𝑓‘((varFGrp‘𝐼)‘𝑦))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((𝑓‘((varFGrp‘𝐼)‘∪ 𝐼))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
91	90, 88	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ {∪ 𝐼} → (((𝑓‘((varFGrp‘𝐼)‘𝑦))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((varFGrp‘𝐼)‘𝑦) ↔ ((𝑓‘((varFGrp‘𝐼)‘∪ 𝐼))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((varFGrp‘𝐼)‘∪ 𝐼)))
92	86, 91	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑦 ∈ {∪ 𝐼} → ((𝑓‘((varFGrp‘𝐼)‘𝑦))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((varFGrp‘𝐼)‘𝑦)))
93	81, 92	sylbid 229	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑦 ∈ 𝐼 → ((𝑓‘((varFGrp‘𝐼)‘𝑦))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((varFGrp‘𝐼)‘𝑦)))
94	93	imp 444	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) ∧ 𝑦 ∈ 𝐼) → ((𝑓‘((varFGrp‘𝐼)‘𝑦))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((varFGrp‘𝐼)‘𝑦))
95	94	mpteq2dva 4672	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑦 ∈ 𝐼 ↦ ((𝑓‘((varFGrp‘𝐼)‘𝑦))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) = (𝑦 ∈ 𝐼 ↦ ((varFGrp‘𝐼)‘𝑦)))
96	65	ffvelrnda 6267	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) ∧ 𝑦 ∈ 𝐼) → ((varFGrp‘𝐼)‘𝑦) ∈ (Base‘𝐺))
97	65	feqmptd 6159	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (varFGrp‘𝐼) = (𝑦 ∈ 𝐼 ↦ ((varFGrp‘𝐼)‘𝑦)))
98		eqidd 2611	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
99		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ((varFGrp‘𝐼)‘𝑦) → (𝑓‘𝑥) = (𝑓‘((varFGrp‘𝐼)‘𝑦)))
100	99	oveq1d 6564	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ((varFGrp‘𝐼)‘𝑦) → ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((𝑓‘((varFGrp‘𝐼)‘𝑦))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
101	96, 97, 98, 100	fmptco 6303	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∘ (varFGrp‘𝐼)) = (𝑦 ∈ 𝐼 ↦ ((𝑓‘((varFGrp‘𝐼)‘𝑦))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
102	95, 101, 97	3eqtr4d 2654	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
103		coeq1 5201	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ( I ↾ (Base‘𝐺)) → (𝑔 ∘ (varFGrp‘𝐼)) = (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)))
104	103	eqeq1d 2612	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = ( I ↾ (Base‘𝐺)) → ((𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼) ↔ (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼)))
105		coeq1 5201	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) → (𝑔 ∘ (varFGrp‘𝐼)) = ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∘ (varFGrp‘𝐼)))
106	105	eqeq1d 2612	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) → ((𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼)))
107	104, 106	rmoi 3496	. . . . . . . . . . . . . . 15 ⊢ ((∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼) ∧ (( I ↾ (Base‘𝐺)) ∈ (𝐺 GrpHom 𝐺) ∧ (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼)) ∧ ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∈ (𝐺 GrpHom 𝐺) ∧ ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))) → ( I ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
108	61, 64, 67, 79, 102, 107	syl122anc 1327	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ( I ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
109	56, 108	syl5eq 2656	. . . . . . . . . . . . 13 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
110		mpteqb 6207	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (Base‘𝐺)𝑥 ∈ (Base‘𝐺) → ((𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ↔ ∀𝑥 ∈ (Base‘𝐺)𝑥 = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
111		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (Base‘𝐺) → 𝑥 ∈ (Base‘𝐺))
112	110, 111	mprg 2910	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ↔ ∀𝑥 ∈ (Base‘𝐺)𝑥 = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
113	109, 112	sylib 207	. . . . . . . . . . . 12 ⊢ ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ∀𝑥 ∈ (Base‘𝐺)𝑥 = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
114	113	r19.21bi 2916	. . . . . . . . . . 11 ⊢ (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
115	114	an32s 842	. . . . . . . . . 10 ⊢ (((𝐼 ≈ 1𝑜 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑥 = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
116	70	eqeq2d 2620	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑓‘𝑥) → (𝑥 = (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) ↔ 𝑥 = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
117	116	rspcev 3282	. . . . . . . . . 10 ⊢ (((𝑓‘𝑥) ∈ ℤ ∧ 𝑥 = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
118	55, 115, 117	syl2anc 691	. . . . . . . . 9 ⊢ (((𝐼 ≈ 1𝑜 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
119	118	expr 641	. . . . . . . 8 ⊢ (((𝐼 ≈ 1𝑜 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1 → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
120	51, 119	syld 46	. . . . . . 7 ⊢ (((𝐼 ≈ 1𝑜 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
121	120	rexlimdva 3013	. . . . . 6 ⊢ ((𝐼 ≈ 1𝑜 ∧ 𝑥 ∈ (Base‘𝐺)) → (∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
122	42, 121	mpd 15	. . . . 5 ⊢ ((𝐼 ≈ 1𝑜 ∧ 𝑥 ∈ (Base‘𝐺)) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
123	16, 17, 21, 27, 122	iscygd 18112	. . . 4 ⊢ (𝐼 ≈ 1𝑜 → 𝐺 ∈ CycGrp)
124	15, 123	jaoi 393	. . 3 ⊢ ((𝐼 ≺ 1𝑜 ∨ 𝐼 ≈ 1𝑜) → 𝐺 ∈ CycGrp)
125	1, 124	sylbi 206	. 2 ⊢ (𝐼 ≼ 1𝑜 → 𝐺 ∈ CycGrp)
126		cygabl 18115	. . 3 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
127	3	frgpnabl 18101	. . . . 5 ⊢ (1𝑜 ≺ 𝐼 → ¬ 𝐺 ∈ Abel)
128	127	con2i 133	. . . 4 ⊢ (𝐺 ∈ Abel → ¬ 1𝑜 ≺ 𝐼)
129		ablgrp 18021	. . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
130		eqid 2610	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
131	16, 130	grpidcl 17273	. . . . . 6 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
132	3, 16	elbasfv 15748	. . . . . 6 ⊢ ((0g‘𝐺) ∈ (Base‘𝐺) → 𝐼 ∈ V)
133	129, 131, 132	3syl 18	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐼 ∈ V)
134		1onn 7606	. . . . . 6 ⊢ 1𝑜 ∈ ω
135		nnfi 8038	. . . . . 6 ⊢ (1𝑜 ∈ ω → 1𝑜 ∈ Fin)
136	134, 135	ax-mp 5	. . . . 5 ⊢ 1𝑜 ∈ Fin
137		fidomtri2 8703	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 1𝑜 ∈ Fin) → (𝐼 ≼ 1𝑜 ↔ ¬ 1𝑜 ≺ 𝐼))
138	133, 136, 137	sylancl 693	. . . 4 ⊢ (𝐺 ∈ Abel → (𝐼 ≼ 1𝑜 ↔ ¬ 1𝑜 ≺ 𝐼))
139	128, 138	mpbird 246	. . 3 ⊢ (𝐺 ∈ Abel → 𝐼 ≼ 1𝑜)
140	126, 139	syl 17	. 2 ⊢ (𝐺 ∈ CycGrp → 𝐼 ≼ 1𝑜)
141	125, 140	impbii 198	1 ⊢ (𝐼 ≼ 1𝑜 ↔ 𝐺 ∈ CycGrp)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  ∃*wrmo 2899  Vcvv 3173  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   ↾ cres 5040   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ωcom 6957  1_𝑜c1o 7440   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  Fincfn 7841  1c1 9816  ℤcz 11254  Basecbs 15695  0_gc0g 15923  Grpcgrp 17245  ._gcmg 17363   GrpHom cghm 17480   ~_FG cefg 17942  freeGrpcfrgp 17943  var_FGrpcvrgp 17944  Abelcabl 18017  CycGrpccyg 18102  ℤ_ringzring 19637

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-inf2 8421  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  ax-addf 9894  ax-mulf 9895

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-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  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-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-0g 15925  df-gsum 15926  df-imas 15991  df-qus 15992  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-frmd 17209  df-vrmd 17210  df-grp 17248  df-minusg 17249  df-mulg 17364  df-subg 17414  df-ghm 17481  df-efg 17945  df-frgp 17946  df-vrgp 17947  df-cmn 18018  df-abl 18019  df-cyg 18103  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-subrg 18601  df-cnfld 19568  df-zring 19638

This theorem is referenced by: (None)