Theorem frgpcyg 21880

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 9010	. . 3 ⊢ (𝐼 ≼ 1o ↔ (𝐼 ≺ 1o ∨ 𝐼 ≈ 1o))
2		sdom1 9242	. . . . 5 ⊢ (𝐼 ≺ 1o ↔ 𝐼 = ∅)
3		frgpcyg.g	. . . . . . 7 ⊢ 𝐺 = (freeGrp‘𝐼)
4		fveq2 6885	. . . . . . 7 ⊢ (𝐼 = ∅ → (freeGrp‘𝐼) = (freeGrp‘∅))
5	3, 4	eqtrid 2807	. . . . . 6 ⊢ (𝐼 = ∅ → 𝐺 = (freeGrp‘∅))
6		0ex 5260	. . . . . . . 8 ⊢ ∅ ∈ V
7		eqid 2760	. . . . . . . . 9 ⊢ (freeGrp‘∅) = (freeGrp‘∅)
8	7	frgpgrp 19977	. . . . . . . 8 ⊢ (∅ ∈ V → (freeGrp‘∅) ∈ Grp)
9	6, 8	ax-mp 5	. . . . . . 7 ⊢ (freeGrp‘∅) ∈ Grp
10		eqid 2760	. . . . . . . 8 ⊢ (Base‘(freeGrp‘∅)) = (Base‘(freeGrp‘∅))
11	7, 10	0frgp 19994	. . . . . . 7 ⊢ (Base‘(freeGrp‘∅)) ≈ 1o
12	10	0cyg 20108	. . . . . . 7 ⊢ (((freeGrp‘∅) ∈ Grp ∧ (Base‘(freeGrp‘∅)) ≈ 1o) → (freeGrp‘∅) ∈ CycGrp)
13	9, 11, 12	mp2an 705	. . . . . 6 ⊢ (freeGrp‘∅) ∈ CycGrp
14	5, 13	eqeltrdi 2868	. . . . 5 ⊢ (𝐼 = ∅ → 𝐺 ∈ CycGrp)
15	2, 14	sylbi 220	. . . 4 ⊢ (𝐼 ≺ 1o → 𝐺 ∈ CycGrp)
16		eqid 2760	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
17		eqid 2760	. . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺)
18		relen 8979	. . . . . . 7 ⊢ Rel ≈
19	18	brrelex1i 5706	. . . . . 6 ⊢ (𝐼 ≈ 1o → 𝐼 ∈ V)
20	3	frgpgrp 19977	. . . . . 6 ⊢ (𝐼 ∈ V → 𝐺 ∈ Grp)
21	19, 20	syl 18	. . . . 5 ⊢ (𝐼 ≈ 1o → 𝐺 ∈ Grp)
22		eqid 2760	. . . . . . . 8 ⊢ ( ~FG ‘𝐼) = ( ~FG ‘𝐼)
23		eqid 2760	. . . . . . . 8 ⊢ (varFGrp‘𝐼) = (varFGrp‘𝐼)
24	22, 23, 3, 16	vrgpf 19983	. . . . . . 7 ⊢ (𝐼 ∈ V → (varFGrp‘𝐼):𝐼⟶(Base‘𝐺))
25	19, 24	syl 18	. . . . . 6 ⊢ (𝐼 ≈ 1o → (varFGrp‘𝐼):𝐼⟶(Base‘𝐺))
26		en1uniel 9058	. . . . . 6 ⊢ (𝐼 ≈ 1o → ∪ 𝐼 ∈ 𝐼)
27	25, 26	ffvelcdmd 7085	. . . . 5 ⊢ (𝐼 ≈ 1o → ((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺))
28		zringgrp 21759	. . . . . . . . 9 ⊢ ℤring ∈ Grp
29	19	uniexd 7759	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → ∪ 𝐼 ∈ V)
30		1zzd 12728	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → 1 ∈ ℤ)
31	29, 30	fsnd 6869	. . . . . . . . . 10 ⊢ (𝐼 ≈ 1o → {⟨∪ 𝐼, 1⟩}:{∪ 𝐼}⟶ℤ)
32		en1b 9053	. . . . . . . . . . . 12 ⊢ (𝐼 ≈ 1o ↔ 𝐼 = {∪ 𝐼})
33	32	biimpi 219	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → 𝐼 = {∪ 𝐼})
34	33	feq2d 6693	. . . . . . . . . 10 ⊢ (𝐼 ≈ 1o → ({⟨∪ 𝐼, 1⟩}:𝐼⟶ℤ ↔ {⟨∪ 𝐼, 1⟩}:{∪ 𝐼}⟶ℤ))
35	31, 34	mpbird 260	. . . . . . . . 9 ⊢ (𝐼 ≈ 1o → {⟨∪ 𝐼, 1⟩}:𝐼⟶ℤ)
36		zringbas 21760	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤring)
37	3, 36, 23	frgpup3 19993	. . . . . . . . 9 ⊢ ((ℤring ∈ Grp ∧ 𝐼 ∈ V ∧ {⟨∪ 𝐼, 1⟩}:𝐼⟶ℤ) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
38	28, 19, 35, 37	mp3an2i 1495	. . . . . . . 8 ⊢ (𝐼 ≈ 1o → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
39	38	adantr 486	. . . . . . 7 ⊢ ((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
40		reurex 3369	. . . . . . 7 ⊢ (∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → ∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
41	39, 40	syl 18	. . . . . 6 ⊢ ((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) → ∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
42		fveq1 6884	. . . . . . . . . 10 ⊢ ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → ((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼))
43	25, 26	fvco3d 6986	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → ((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)))
44		1z 12727	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
45		fvsng 7185	. . . . . . . . . . . 12 ⊢ ((∪ 𝐼 ∈ V ∧ 1 ∈ ℤ) → ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼) = 1)
46	29, 44, 45	sylancl 598	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼) = 1)
47	43, 46	eqeq12d 2776	. . . . . . . . . 10 ⊢ (𝐼 ≈ 1o → (((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼) ↔ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1))
48	42, 47	imbitrid 247	. . . . . . . . 9 ⊢ (𝐼 ≈ 1o → ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1))
49	48	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1))
50	16, 36	ghmf 19435	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐺 GrpHom ℤring) → 𝑓:(Base‘𝐺)⟶ℤ)
51	50	ad2antrl 741	. . . . . . . . . . . 12 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓:(Base‘𝐺)⟶ℤ)
52	51	ffvelcdmda 7084	. . . . . . . . . . 11 ⊢ (((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓‘𝑥) ∈ ℤ)
53	52	an32s 665	. . . . . . . . . 10 ⊢ (((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑓‘𝑥) ∈ ℤ)
54		mptresid 6042	. . . . . . . . . . . . . 14 ⊢ ( I ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ 𝑥)
55	3, 16, 23	frgpup3 19993	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝐼 ∈ V ∧ (varFGrp‘𝐼):𝐼⟶(Base‘𝐺)) → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
56	21, 19, 25, 55	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ≈ 1o → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
57		reurmo 3368	. . . . . . . . . . . . . . . . 17 ⊢ (∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼) → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
58	56, 57	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ≈ 1o → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
59	58	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
60	21	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝐺 ∈ Grp)
61	16	idghm 19446	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → ( I ↾ (Base‘𝐺)) ∈ (𝐺 GrpHom 𝐺))
62	60, 61	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ( I ↾ (Base‘𝐺)) ∈ (𝐺 GrpHom 𝐺))
63	25	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (varFGrp‘𝐼):𝐼⟶(Base‘𝐺))
64		fcoi2 6757	. . . . . . . . . . . . . . . 16 ⊢ ((varFGrp‘𝐼):𝐼⟶(Base‘𝐺) → (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
65	63, 64	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
66	51	feqmptd 6953	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓 = (𝑥 ∈ (Base‘𝐺) ↦ (𝑓‘𝑥)))
67		eqidd 2761	. . . . . . . . . . . . . . . . 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))))
68		oveq1 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑓‘𝑥) → (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
69	52, 66, 67, 68	fmptco 7130	. . . . . . . . . . . . . . . 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))))
70	27	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺))
71		eqid 2760	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
72	17, 71, 16	mulgghm2 21783	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ ((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺)) → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∈ (ℤring GrpHom 𝐺))
73	60, 70, 72	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∈ (ℤring GrpHom 𝐺))
74		simprl 783	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓 ∈ (𝐺 GrpHom ℤring))
75		ghmco 19451	. . . . . . . . . . . . . . . . 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))
76	73, 74, 75	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∘ 𝑓) ∈ (𝐺 GrpHom 𝐺))
77	69, 76	eqeltrrd 2861	. . . . . . . . . . . . . . 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))
78	33	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝐼 = {∪ 𝐼})
79	78	eleq2d 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑦 ∈ 𝐼 ↔ 𝑦 ∈ {∪ 𝐼}))
80		simprr 785	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)
81	80	oveq1d 7435	. . . . . . . . . . . . . . . . . . . . 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)))
82	16, 17	mulg1 19292	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺) → (1(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((varFGrp‘𝐼)‘∪ 𝐼))
83	70, 82	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (1(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((varFGrp‘𝐼)‘∪ 𝐼))
84	81, 83	eqtrd 2795	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ((𝑓‘((varFGrp‘𝐼)‘∪ 𝐼))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((varFGrp‘𝐼)‘∪ 𝐼))
85		elsni 4600	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ {∪ 𝐼} → 𝑦 = ∪ 𝐼)
86	85	fveq2d 6889	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ {∪ 𝐼} → ((varFGrp‘𝐼)‘𝑦) = ((varFGrp‘𝐼)‘∪ 𝐼))
87	86	fveq2d 6889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ {∪ 𝐼} → (𝑓‘((varFGrp‘𝐼)‘𝑦)) = (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)))
88	87	oveq1d 7435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ {∪ 𝐼} → ((𝑓‘((varFGrp‘𝐼)‘𝑦))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((𝑓‘((varFGrp‘𝐼)‘∪ 𝐼))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
89	88, 86	eqeq12d 2776	. . . . . . . . . . . . . . . . . . . 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)))
90	84, 89	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . 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)))
91	79, 90	sylbid 243	. . . . . . . . . . . . . . . . . 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)))
92	91	imp 412	. . . . . . . . . . . . . . . . 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))
93	92	mpteq2dva 5197	. . . . . . . . . . . . . . . 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)))
94	63	ffvelcdmda 7084	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) ∧ 𝑦 ∈ 𝐼) → ((varFGrp‘𝐼)‘𝑦) ∈ (Base‘𝐺))
95	63	feqmptd 6953	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (varFGrp‘𝐼) = (𝑦 ∈ 𝐼 ↦ ((varFGrp‘𝐼)‘𝑦)))
96		eqidd 2761	. . . . . . . . . . . . . . . . 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))))
97		fveq2 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ((varFGrp‘𝐼)‘𝑦) → (𝑓‘𝑥) = (𝑓‘((varFGrp‘𝐼)‘𝑦)))
98	97	oveq1d 7435	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ((varFGrp‘𝐼)‘𝑦) → ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((𝑓‘((varFGrp‘𝐼)‘𝑦))(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
99	94, 95, 96, 98	fmptco 7130	. . . . . . . . . . . . . . . 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))))
100	93, 99, 95	3eqtr4d 2805	. . . . . . . . . . . . . . 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))
101		coeq1 5834	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ( I ↾ (Base‘𝐺)) → (𝑔 ∘ (varFGrp‘𝐼)) = (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)))
102	101	eqeq1d 2762	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = ( I ↾ (Base‘𝐺)) → ((𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼) ↔ (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼)))
103		coeq1 5834	. . . . . . . . . . . . . . . . 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)))
104	103	eqeq1d 2762	. . . . . . . . . . . . . . . 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)))
105	102, 104	rmoi 3837	. . . . . . . . . . . . . . 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))))
106	59, 62, 65, 77, 100, 105	syl122anc 1406	. . . . . . . . . . . . . 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))))
107	54, 106	eqtr3id 2809	. . . . . . . . . . . . 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))))
108		mpteqb 7013	. . . . . . . . . . . . . 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))))
109		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (Base‘𝐺) → 𝑥 ∈ (Base‘𝐺))
110	108, 109	mprg 3082	. . . . . . . . . . . . 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)))
111	107, 110	sylib 221	. . . . . . . . . . . 12 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ∀𝑥 ∈ (Base‘𝐺)𝑥 = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
112	111	r19.21bi 3254	. . . . . . . . . . 11 ⊢ (((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
113	112	an32s 665	. . . . . . . . . 10 ⊢ (((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑥 = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
114	68	rspceeqv 3598	. . . . . . . . . 10 ⊢ (((𝑓‘𝑥) ∈ ℤ ∧ 𝑥 = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
115	53, 113, 114	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
116	115	expr 462	. . . . . . . 8 ⊢ (((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1 → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
117	49, 116	syld 48	. . . . . . 7 ⊢ (((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
118	117	rexlimdva 3163	. . . . . 6 ⊢ ((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) → (∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))))
119	41, 118	mpd 16	. . . . 5 ⊢ ((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
120	16, 17, 21, 27, 119	iscygd 20102	. . . 4 ⊢ (𝐼 ≈ 1o → 𝐺 ∈ CycGrp)
121	15, 120	jaoi 871	. . 3 ⊢ ((𝐼 ≺ 1o ∨ 𝐼 ≈ 1o) → 𝐺 ∈ CycGrp)
122	1, 121	sylbi 220	. 2 ⊢ (𝐼 ≼ 1o → 𝐺 ∈ CycGrp)
123		cygabl 20106	. . 3 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
124	3	frgpnabl 20090	. . . . 5 ⊢ (1o ≺ 𝐼 → ¬ 𝐺 ∈ Abel)
125	124	con2i 140	. . . 4 ⊢ (𝐺 ∈ Abel → ¬ 1o ≺ 𝐼)
126		ablgrp 20000	. . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
127		eqid 2760	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
128	16, 127	grpidcl 19177	. . . . . 6 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
129	3, 16	elbasfv 17394	. . . . . 6 ⊢ ((0g‘𝐺) ∈ (Base‘𝐺) → 𝐼 ∈ V)
130	126, 128, 129	3syl 19	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐼 ∈ V)
131		1onn 8650	. . . . . 6 ⊢ 1o ∈ ω
132		nnfi 9184	. . . . . 6 ⊢ (1o ∈ ω → 1o ∈ Fin)
133	131, 132	ax-mp 5	. . . . 5 ⊢ 1o ∈ Fin
134		fidomtri2 10076	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 1o ∈ Fin) → (𝐼 ≼ 1o ↔ ¬ 1o ≺ 𝐼))
135	130, 133, 134	sylancl 598	. . . 4 ⊢ (𝐺 ∈ Abel → (𝐼 ≼ 1o ↔ ¬ 1o ≺ 𝐼))
136	125, 135	mpbird 260	. . 3 ⊢ (𝐺 ∈ Abel → 𝐼 ≼ 1o)
137	123, 136	syl 18	. 2 ⊢ (𝐺 ∈ CycGrp → 𝐼 ≼ 1o)
138	122, 137	impbii 212	1 ⊢ (𝐼 ≼ 1o ↔ 𝐺 ∈ CycGrp)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363  ∃*wrmo 3364  Vcvv 3450  ∅c0 4278  {csn 4583  ⟨cop 4589  ∪ cuni 4866   class class class wbr 5102   ↦ cmpt 5185   I cid 5544   ↾ cres 5652   ∘ ccom 5654  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  ωcom 7877  1_oc1o 8470   ≈ cen 8971   ≼ cdom 8972   ≺ csdm 8973  Fincfn 8974  1c1 11202  ℤcz 12694  Basecbs 17388  0_gc0g 17611  Grpcgrp 19145  ._gcmg 19278   GrpHom cghm 19428   ~_FG cefg 19921  freeGrpcfrgp 19922  var_FGrpcvrgp 19923  Abelcabl 19996  CycGrpccyg 20092  ℤ_ringczring 21753

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11257  ax-resscn 11258  ax-1cn 11259  ax-icn 11260  ax-addcl 11261  ax-addrcl 11262  ax-mulcl 11263  ax-mulrcl 11264  ax-mulcom 11265  ax-addass 11266  ax-mulass 11267  ax-distr 11268  ax-i2m1 11269  ax-1ne0 11270  ax-1rid 11271  ax-rnegex 11272  ax-rrecex 11273  ax-cnre 11274  ax-pre-lttri 11275  ax-pre-lttrn 11276  ax-pre-ltadd 11277  ax-pre-mulgt0 11278  ax-addf 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-2o 8478  df-er 8718  df-ec 8720  df-qs 8724  df-map 8850  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9435  df-inf 9436  df-card 10021  df-pnf 11346  df-mnf 11347  df-xr 11348  df-ltxr 11349  df-le 11350  df-sub 11544  df-neg 11545  df-nn 12337  df-2 12406  df-3 12407  df-4 12408  df-5 12409  df-6 12410  df-7 12411  df-8 12412  df-9 12413  df-n0 12608  df-xnn0 12681  df-z 12695  df-dec 12816  df-uz 12967  df-rp 13122  df-fz 13641  df-fzo 13790  df-seq 14146  df-hash 14476  df-word 14660  df-lsw 14709  df-concat 14717  df-s1 14744  df-substr 14790  df-pfx 14822  df-splice 14900  df-reverse 14909  df-s2 15000  df-struct 17326  df-sets 17343  df-slot 17361  df-ndx 17373  df-base 17389  df-ress 17410  df-plusg 17442  df-mulr 17443  df-starv 17444  df-sca 17445  df-vsca 17446  df-ip 17447  df-tset 17448  df-ple 17449  df-ds 17451  df-unif 17452  df-0g 17613  df-gsum 17614  df-imas 17681  df-qus 17682  df-mgm 18817  df-sgrp 18909  df-mnd 18925  df-mhm 18979  df-submnd 18980  df-frmd 19046  df-vrmd 19047  df-grp 19148  df-minusg 19149  df-mulg 19279  df-subg 19334  df-ghm 19429  df-efg 19924  df-frgp 19925  df-vrgp 19926  df-cmn 19997  df-abl 19998  df-cyg 20093  df-mgp 20362  df-rng 20376  df-ur 20409  df-ring 20462  df-cring 20463  df-subrng 20799  df-subrg 20823  df-cnfld 21680  df-zring 21754

This theorem is used by: (None)