Theorem frgpcyg 21348

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 8980	. . 3 ⊢ (𝐼 ≼ 1o ↔ (𝐼 ≺ 1o ∨ 𝐼 ≈ 1o))
2		sdom1 9244	. . . . 5 ⊢ (𝐼 ≺ 1o ↔ 𝐼 = ∅)
3		frgpcyg.g	. . . . . . 7 ⊢ 𝐺 = (freeGrp‘𝐼)
4		fveq2 6890	. . . . . . 7 ⊢ (𝐼 = ∅ → (freeGrp‘𝐼) = (freeGrp‘∅))
5	3, 4	eqtrid 2782	. . . . . 6 ⊢ (𝐼 = ∅ → 𝐺 = (freeGrp‘∅))
6		0ex 5306	. . . . . . . 8 ⊢ ∅ ∈ V
7		eqid 2730	. . . . . . . . 9 ⊢ (freeGrp‘∅) = (freeGrp‘∅)
8	7	frgpgrp 19671	. . . . . . . 8 ⊢ (∅ ∈ V → (freeGrp‘∅) ∈ Grp)
9	6, 8	ax-mp 5	. . . . . . 7 ⊢ (freeGrp‘∅) ∈ Grp
10		eqid 2730	. . . . . . . 8 ⊢ (Base‘(freeGrp‘∅)) = (Base‘(freeGrp‘∅))
11	7, 10	0frgp 19688	. . . . . . 7 ⊢ (Base‘(freeGrp‘∅)) ≈ 1o
12	10	0cyg 19802	. . . . . . 7 ⊢ (((freeGrp‘∅) ∈ Grp ∧ (Base‘(freeGrp‘∅)) ≈ 1o) → (freeGrp‘∅) ∈ CycGrp)
13	9, 11, 12	mp2an 688	. . . . . 6 ⊢ (freeGrp‘∅) ∈ CycGrp
14	5, 13	eqeltrdi 2839	. . . . 5 ⊢ (𝐼 = ∅ → 𝐺 ∈ CycGrp)
15	2, 14	sylbi 216	. . . 4 ⊢ (𝐼 ≺ 1o → 𝐺 ∈ CycGrp)
16		eqid 2730	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
17		eqid 2730	. . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺)
18		relen 8946	. . . . . . 7 ⊢ Rel ≈
19	18	brrelex1i 5731	. . . . . 6 ⊢ (𝐼 ≈ 1o → 𝐼 ∈ V)
20	3	frgpgrp 19671	. . . . . 6 ⊢ (𝐼 ∈ V → 𝐺 ∈ Grp)
21	19, 20	syl 17	. . . . 5 ⊢ (𝐼 ≈ 1o → 𝐺 ∈ Grp)
22		eqid 2730	. . . . . . . 8 ⊢ ( ~FG ‘𝐼) = ( ~FG ‘𝐼)
23		eqid 2730	. . . . . . . 8 ⊢ (varFGrp‘𝐼) = (varFGrp‘𝐼)
24	22, 23, 3, 16	vrgpf 19677	. . . . . . 7 ⊢ (𝐼 ∈ V → (varFGrp‘𝐼):𝐼⟶(Base‘𝐺))
25	19, 24	syl 17	. . . . . 6 ⊢ (𝐼 ≈ 1o → (varFGrp‘𝐼):𝐼⟶(Base‘𝐺))
26		en1uniel 9030	. . . . . 6 ⊢ (𝐼 ≈ 1o → ∪ 𝐼 ∈ 𝐼)
27	25, 26	ffvelcdmd 7086	. . . . 5 ⊢ (𝐼 ≈ 1o → ((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺))
28		zringgrp 21223	. . . . . . . . 9 ⊢ ℤring ∈ Grp
29	19	uniexd 7734	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → ∪ 𝐼 ∈ V)
30		1zzd 12597	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → 1 ∈ ℤ)
31	29, 30	fsnd 6875	. . . . . . . . . 10 ⊢ (𝐼 ≈ 1o → {⟨∪ 𝐼, 1⟩}:{∪ 𝐼}⟶ℤ)
32		en1b 9025	. . . . . . . . . . . 12 ⊢ (𝐼 ≈ 1o ↔ 𝐼 = {∪ 𝐼})
33	32	biimpi 215	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → 𝐼 = {∪ 𝐼})
34	33	feq2d 6702	. . . . . . . . . 10 ⊢ (𝐼 ≈ 1o → ({⟨∪ 𝐼, 1⟩}:𝐼⟶ℤ ↔ {⟨∪ 𝐼, 1⟩}:{∪ 𝐼}⟶ℤ))
35	31, 34	mpbird 256	. . . . . . . . 9 ⊢ (𝐼 ≈ 1o → {⟨∪ 𝐼, 1⟩}:𝐼⟶ℤ)
36		zringbas 21224	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤring)
37	3, 36, 23	frgpup3 19687	. . . . . . . . 9 ⊢ ((ℤring ∈ Grp ∧ 𝐼 ∈ V ∧ {⟨∪ 𝐼, 1⟩}:𝐼⟶ℤ) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
38	28, 19, 35, 37	mp3an2i 1464	. . . . . . . 8 ⊢ (𝐼 ≈ 1o → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
39	38	adantr 479	. . . . . . 7 ⊢ ((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
40		reurex 3378	. . . . . . 7 ⊢ (∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → ∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
41	39, 40	syl 17	. . . . . 6 ⊢ ((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) → ∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
42		fveq1 6889	. . . . . . . . . 10 ⊢ ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → ((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼))
43	25, 26	fvco3d 6990	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → ((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)))
44		1z 12596	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
45		fvsng 7179	. . . . . . . . . . . 12 ⊢ ((∪ 𝐼 ∈ V ∧ 1 ∈ ℤ) → ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼) = 1)
46	29, 44, 45	sylancl 584	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼) = 1)
47	43, 46	eqeq12d 2746	. . . . . . . . . 10 ⊢ (𝐼 ≈ 1o → (((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼) ↔ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1))
48	42, 47	imbitrid 243	. . . . . . . . 9 ⊢ (𝐼 ≈ 1o → ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1))
49	48	ad2antrr 722	. . . . . . . 8 ⊢ (((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1))
50	16, 36	ghmf 19134	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐺 GrpHom ℤring) → 𝑓:(Base‘𝐺)⟶ℤ)
51	50	ad2antrl 724	. . . . . . . . . . . 12 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓:(Base‘𝐺)⟶ℤ)
52	51	ffvelcdmda 7085	. . . . . . . . . . 11 ⊢ (((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓‘𝑥) ∈ ℤ)
53	52	an32s 648	. . . . . . . . . 10 ⊢ (((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑓‘𝑥) ∈ ℤ)
54		mptresid 6049	. . . . . . . . . . . . . 14 ⊢ ( I ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ 𝑥)
55	3, 16, 23	frgpup3 19687	. . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ≈ 1o → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
57		reurmo 3377	. . . . . . . . . . . . . . . . 17 ⊢ (∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼) → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
58	56, 57	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ≈ 1o → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
59	58	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
60	21	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝐺 ∈ Grp)
61	16	idghm 19145	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → ( I ↾ (Base‘𝐺)) ∈ (𝐺 GrpHom 𝐺))
62	60, 61	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ( I ↾ (Base‘𝐺)) ∈ (𝐺 GrpHom 𝐺))
63	25	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (varFGrp‘𝐼):𝐼⟶(Base‘𝐺))
64		fcoi2 6765	. . . . . . . . . . . . . . . 16 ⊢ ((varFGrp‘𝐼):𝐼⟶(Base‘𝐺) → (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
65	63, 64	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
66	51	feqmptd 6959	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓 = (𝑥 ∈ (Base‘𝐺) ↦ (𝑓‘𝑥)))
67		eqidd 2731	. . . . . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑓‘𝑥) → (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
69	52, 66, 67, 68	fmptco 7128	. . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺))
71		eqid 2730	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
72	17, 71, 16	mulgghm2 21247	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ ((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺)) → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∈ (ℤring GrpHom 𝐺))
73	60, 70, 72	syl2anc 582	. . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓 ∈ (𝐺 GrpHom ℤring))
75		ghmco 19150	. . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . 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 2832	. . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝐼 = {∪ 𝐼})
79	78	eleq2d 2817	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑦 ∈ 𝐼 ↔ 𝑦 ∈ {∪ 𝐼}))
80		simprr 769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)
81	80	oveq1d 7426	. . . . . . . . . . . . . . . . . . . . 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 18997	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺) → (1(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((varFGrp‘𝐼)‘∪ 𝐼))
83	70, 82	syl 17	. . . . . . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . . . . . 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 4644	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ {∪ 𝐼} → 𝑦 = ∪ 𝐼)
86	85	fveq2d 6894	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ {∪ 𝐼} → ((varFGrp‘𝐼)‘𝑦) = ((varFGrp‘𝐼)‘∪ 𝐼))
87	86	fveq2d 6894	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ {∪ 𝐼} → (𝑓‘((varFGrp‘𝐼)‘𝑦)) = (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)))
88	87	oveq1d 7426	. . . . . . . . . . . . . . . . . . . . 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 2746	. . . . . . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . . . 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 5247	. . . . . . . . . . . . . . . 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 7085	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) ∧ 𝑦 ∈ 𝐼) → ((varFGrp‘𝐼)‘𝑦) ∈ (Base‘𝐺))
95	63	feqmptd 6959	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (varFGrp‘𝐼) = (𝑦 ∈ 𝐼 ↦ ((varFGrp‘𝐼)‘𝑦)))
96		eqidd 2731	. . . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ((varFGrp‘𝐼)‘𝑦) → (𝑓‘𝑥) = (𝑓‘((varFGrp‘𝐼)‘𝑦)))
98	97	oveq1d 7426	. . . . . . . . . . . . . . . . 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 7128	. . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . 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 5856	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ( I ↾ (Base‘𝐺)) → (𝑔 ∘ (varFGrp‘𝐼)) = (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)))
102	101	eqeq1d 2732	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = ( I ↾ (Base‘𝐺)) → ((𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼) ↔ (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼)))
103		coeq1 5856	. . . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . 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 3884	. . . . . . . . . . . . . . 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 1377	. . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . 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 7016	. . . . . . . . . . . . . 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 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (Base‘𝐺) → 𝑥 ∈ (Base‘𝐺))
110	108, 109	mprg 3065	. . . . . . . . . . . . 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 217	. . . . . . . . . . . 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 3246	. . . . . . . . . . 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 648	. . . . . . . . . 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 3632	. . . . . . . . . 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 582	. . . . . . . . 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 455	. . . . . . . 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 47	. . . . . . 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 3153	. . . . . 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 15	. . . . 5 ⊢ ((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
120	16, 17, 21, 27, 119	iscygd 19796	. . . 4 ⊢ (𝐼 ≈ 1o → 𝐺 ∈ CycGrp)
121	15, 120	jaoi 853	. . 3 ⊢ ((𝐼 ≺ 1o ∨ 𝐼 ≈ 1o) → 𝐺 ∈ CycGrp)
122	1, 121	sylbi 216	. 2 ⊢ (𝐼 ≼ 1o → 𝐺 ∈ CycGrp)
123		cygabl 19800	. . 3 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
124	3	frgpnabl 19784	. . . . 5 ⊢ (1o ≺ 𝐼 → ¬ 𝐺 ∈ Abel)
125	124	con2i 139	. . . 4 ⊢ (𝐺 ∈ Abel → ¬ 1o ≺ 𝐼)
126		ablgrp 19694	. . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
127		eqid 2730	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
128	16, 127	grpidcl 18886	. . . . . 6 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
129	3, 16	elbasfv 17154	. . . . . 6 ⊢ ((0g‘𝐺) ∈ (Base‘𝐺) → 𝐼 ∈ V)
130	126, 128, 129	3syl 18	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐼 ∈ V)
131		1onn 8641	. . . . . 6 ⊢ 1o ∈ ω
132		nnfi 9169	. . . . . 6 ⊢ (1o ∈ ω → 1o ∈ Fin)
133	131, 132	ax-mp 5	. . . . 5 ⊢ 1o ∈ Fin
134		fidomtri2 9991	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 1o ∈ Fin) → (𝐼 ≼ 1o ↔ ¬ 1o ≺ 𝐼))
135	130, 133, 134	sylancl 584	. . . 4 ⊢ (𝐺 ∈ Abel → (𝐼 ≼ 1o ↔ ¬ 1o ≺ 𝐼))
136	125, 135	mpbird 256	. . 3 ⊢ (𝐺 ∈ Abel → 𝐼 ≼ 1o)
137	123, 136	syl 17	. 2 ⊢ (𝐺 ∈ CycGrp → 𝐼 ≼ 1o)
138	122, 137	impbii 208	1 ⊢ (𝐼 ≼ 1o ↔ 𝐺 ∈ CycGrp)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104  ∀wral 3059  ∃wrex 3068  ∃!wreu 3372  ∃*wrmo 3373  Vcvv 3472  ∅c0 4321  {csn 4627  ⟨cop 4633  ∪ cuni 4907   class class class wbr 5147   ↦ cmpt 5230   I cid 5572   ↾ cres 5677   ∘ ccom 5679  ⟶wf 6538  ‘cfv 6542  (class class class)co 7411  ωcom 7857  1_oc1o 8461   ≈ cen 8938   ≼ cdom 8939   ≺ csdm 8940  Fincfn 8941  1c1 11113  ℤcz 12562  Basecbs 17148  0_gc0g 17389  Grpcgrp 18855  ._gcmg 18986   GrpHom cghm 19127   ~_FG cefg 19615  freeGrpcfrgp 19616  var_FGrpcvrgp 19617  Abelcabl 19690  CycGrpccyg 19786  ℤ_ringczring 21217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-addf 11191  ax-mulf 11192

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-ot 4636  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-ec 8707  df-qs 8711  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-xnn0 12549  df-z 12563  df-dec 12682  df-uz 12827  df-rp 12979  df-fz 13489  df-fzo 13632  df-seq 13971  df-hash 14295  df-word 14469  df-lsw 14517  df-concat 14525  df-s1 14550  df-substr 14595  df-pfx 14625  df-splice 14704  df-reverse 14713  df-s2 14803  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-starv 17216  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ds 17223  df-unif 17224  df-0g 17391  df-gsum 17392  df-imas 17458  df-qus 17459  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18705  df-submnd 18706  df-frmd 18766  df-vrmd 18767  df-grp 18858  df-minusg 18859  df-mulg 18987  df-subg 19039  df-ghm 19128  df-efg 19618  df-frgp 19619  df-vrgp 19620  df-cmn 19691  df-abl 19692  df-cyg 19787  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-cring 20130  df-subrng 20434  df-subrg 20459  df-cnfld 21145  df-zring 21218

This theorem is referenced by: (None)