Theorem frgpcyg 21543

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 8931	. . 3 ⊢ (𝐼 ≼ 1o ↔ (𝐼 ≺ 1o ∨ 𝐼 ≈ 1o))
2		sdom1 9162	. . . . 5 ⊢ (𝐼 ≺ 1o ↔ 𝐼 = ∅)
3		frgpcyg.g	. . . . . . 7 ⊢ 𝐺 = (freeGrp‘𝐼)
4		fveq2 6842	. . . . . . 7 ⊢ (𝐼 = ∅ → (freeGrp‘𝐼) = (freeGrp‘∅))
5	3, 4	eqtrid 2784	. . . . . 6 ⊢ (𝐼 = ∅ → 𝐺 = (freeGrp‘∅))
6		0ex 5254	. . . . . . . 8 ⊢ ∅ ∈ V
7		eqid 2737	. . . . . . . . 9 ⊢ (freeGrp‘∅) = (freeGrp‘∅)
8	7	frgpgrp 19706	. . . . . . . 8 ⊢ (∅ ∈ V → (freeGrp‘∅) ∈ Grp)
9	6, 8	ax-mp 5	. . . . . . 7 ⊢ (freeGrp‘∅) ∈ Grp
10		eqid 2737	. . . . . . . 8 ⊢ (Base‘(freeGrp‘∅)) = (Base‘(freeGrp‘∅))
11	7, 10	0frgp 19723	. . . . . . 7 ⊢ (Base‘(freeGrp‘∅)) ≈ 1o
12	10	0cyg 19837	. . . . . . 7 ⊢ (((freeGrp‘∅) ∈ Grp ∧ (Base‘(freeGrp‘∅)) ≈ 1o) → (freeGrp‘∅) ∈ CycGrp)
13	9, 11, 12	mp2an 693	. . . . . 6 ⊢ (freeGrp‘∅) ∈ CycGrp
14	5, 13	eqeltrdi 2845	. . . . 5 ⊢ (𝐼 = ∅ → 𝐺 ∈ CycGrp)
15	2, 14	sylbi 217	. . . 4 ⊢ (𝐼 ≺ 1o → 𝐺 ∈ CycGrp)
16		eqid 2737	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
17		eqid 2737	. . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺)
18		relen 8900	. . . . . . 7 ⊢ Rel ≈
19	18	brrelex1i 5688	. . . . . 6 ⊢ (𝐼 ≈ 1o → 𝐼 ∈ V)
20	3	frgpgrp 19706	. . . . . 6 ⊢ (𝐼 ∈ V → 𝐺 ∈ Grp)
21	19, 20	syl 17	. . . . 5 ⊢ (𝐼 ≈ 1o → 𝐺 ∈ Grp)
22		eqid 2737	. . . . . . . 8 ⊢ ( ~FG ‘𝐼) = ( ~FG ‘𝐼)
23		eqid 2737	. . . . . . . 8 ⊢ (varFGrp‘𝐼) = (varFGrp‘𝐼)
24	22, 23, 3, 16	vrgpf 19712	. . . . . . 7 ⊢ (𝐼 ∈ V → (varFGrp‘𝐼):𝐼⟶(Base‘𝐺))
25	19, 24	syl 17	. . . . . 6 ⊢ (𝐼 ≈ 1o → (varFGrp‘𝐼):𝐼⟶(Base‘𝐺))
26		en1uniel 8978	. . . . . 6 ⊢ (𝐼 ≈ 1o → ∪ 𝐼 ∈ 𝐼)
27	25, 26	ffvelcdmd 7039	. . . . 5 ⊢ (𝐼 ≈ 1o → ((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺))
28		zringgrp 21422	. . . . . . . . 9 ⊢ ℤring ∈ Grp
29	19	uniexd 7697	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → ∪ 𝐼 ∈ V)
30		1zzd 12534	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → 1 ∈ ℤ)
31	29, 30	fsnd 6826	. . . . . . . . . 10 ⊢ (𝐼 ≈ 1o → {⟨∪ 𝐼, 1⟩}:{∪ 𝐼}⟶ℤ)
32		en1b 8974	. . . . . . . . . . . 12 ⊢ (𝐼 ≈ 1o ↔ 𝐼 = {∪ 𝐼})
33	32	biimpi 216	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → 𝐼 = {∪ 𝐼})
34	33	feq2d 6654	. . . . . . . . . 10 ⊢ (𝐼 ≈ 1o → ({⟨∪ 𝐼, 1⟩}:𝐼⟶ℤ ↔ {⟨∪ 𝐼, 1⟩}:{∪ 𝐼}⟶ℤ))
35	31, 34	mpbird 257	. . . . . . . . 9 ⊢ (𝐼 ≈ 1o → {⟨∪ 𝐼, 1⟩}:𝐼⟶ℤ)
36		zringbas 21423	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤring)
37	3, 36, 23	frgpup3 19722	. . . . . . . . 9 ⊢ ((ℤring ∈ Grp ∧ 𝐼 ∈ V ∧ {⟨∪ 𝐼, 1⟩}:𝐼⟶ℤ) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
38	28, 19, 35, 37	mp3an2i 1469	. . . . . . . 8 ⊢ (𝐼 ≈ 1o → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
39	38	adantr 480	. . . . . . 7 ⊢ ((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
40		reurex 3356	. . . . . . 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 6841	. . . . . . . . . 10 ⊢ ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → ((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼))
43	25, 26	fvco3d 6942	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → ((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)))
44		1z 12533	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
45		fvsng 7136	. . . . . . . . . . . 12 ⊢ ((∪ 𝐼 ∈ V ∧ 1 ∈ ℤ) → ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼) = 1)
46	29, 44, 45	sylancl 587	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼) = 1)
47	43, 46	eqeq12d 2753	. . . . . . . . . 10 ⊢ (𝐼 ≈ 1o → (((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼) ↔ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1))
48	42, 47	imbitrid 244	. . . . . . . . 9 ⊢ (𝐼 ≈ 1o → ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1))
49	48	ad2antrr 727	. . . . . . . 8 ⊢ (((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1))
50	16, 36	ghmf 19164	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐺 GrpHom ℤring) → 𝑓:(Base‘𝐺)⟶ℤ)
51	50	ad2antrl 729	. . . . . . . . . . . 12 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓:(Base‘𝐺)⟶ℤ)
52	51	ffvelcdmda 7038	. . . . . . . . . . 11 ⊢ (((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓‘𝑥) ∈ ℤ)
53	52	an32s 653	. . . . . . . . . 10 ⊢ (((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑓‘𝑥) ∈ ℤ)
54		mptresid 6018	. . . . . . . . . . . . . 14 ⊢ ( I ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ 𝑥)
55	3, 16, 23	frgpup3 19722	. . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ≈ 1o → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
57		reurmo 3355	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
60	21	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝐺 ∈ Grp)
61	16	idghm 19175	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (varFGrp‘𝐼):𝐼⟶(Base‘𝐺))
64		fcoi2 6717	. . . . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓 = (𝑥 ∈ (Base‘𝐺) ↦ (𝑓‘𝑥)))
67		eqidd 2738	. . . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑓‘𝑥) → (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
69	52, 66, 67, 68	fmptco 7084	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → ((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺))
71		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
72	17, 71, 16	mulgghm2 21446	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ ((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺)) → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∈ (ℤring GrpHom 𝐺))
73	60, 70, 72	syl2anc 585	. . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓 ∈ (𝐺 GrpHom ℤring))
75		ghmco 19180	. . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝐼 = {∪ 𝐼})
79	78	eleq2d 2823	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑦 ∈ 𝐼 ↔ 𝑦 ∈ {∪ 𝐼}))
80		simprr 773	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)
81	80	oveq1d 7383	. . . . . . . . . . . . . . . . . . . . 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 19026	. . . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . 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 4599	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ {∪ 𝐼} → 𝑦 = ∪ 𝐼)
86	85	fveq2d 6846	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ {∪ 𝐼} → ((varFGrp‘𝐼)‘𝑦) = ((varFGrp‘𝐼)‘∪ 𝐼))
87	86	fveq2d 6846	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ {∪ 𝐼} → (𝑓‘((varFGrp‘𝐼)‘𝑦)) = (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)))
88	87	oveq1d 7383	. . . . . . . . . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . 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 5193	. . . . . . . . . . . . . . . 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 7038	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) ∧ 𝑦 ∈ 𝐼) → ((varFGrp‘𝐼)‘𝑦) ∈ (Base‘𝐺))
95	63	feqmptd 6910	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (varFGrp‘𝐼) = (𝑦 ∈ 𝐼 ↦ ((varFGrp‘𝐼)‘𝑦)))
96		eqidd 2738	. . . . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ((varFGrp‘𝐼)‘𝑦) → (𝑓‘𝑥) = (𝑓‘((varFGrp‘𝐼)‘𝑦)))
98	97	oveq1d 7383	. . . . . . . . . . . . . . . . 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 7084	. . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . 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 5814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ( I ↾ (Base‘𝐺)) → (𝑔 ∘ (varFGrp‘𝐼)) = (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)))
102	101	eqeq1d 2739	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = ( I ↾ (Base‘𝐺)) → ((𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼) ↔ (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼)))
103		coeq1 5814	. . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . 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 3843	. . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . 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 6969	. . . . . . . . . . . . . 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 3058	. . . . . . . . . . . . 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 218	. . . . . . . . . . . 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 3230	. . . . . . . . . . 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 653	. . . . . . . . . 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 3601	. . . . . . . . . 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 585	. . . . . . . . 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 456	. . . . . . . 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 3139	. . . . . 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 19831	. . . 4 ⊢ (𝐼 ≈ 1o → 𝐺 ∈ CycGrp)
121	15, 120	jaoi 858	. . 3 ⊢ ((𝐼 ≺ 1o ∨ 𝐼 ≈ 1o) → 𝐺 ∈ CycGrp)
122	1, 121	sylbi 217	. 2 ⊢ (𝐼 ≼ 1o → 𝐺 ∈ CycGrp)
123		cygabl 19835	. . 3 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
124	3	frgpnabl 19819	. . . . 5 ⊢ (1o ≺ 𝐼 → ¬ 𝐺 ∈ Abel)
125	124	con2i 139	. . . 4 ⊢ (𝐺 ∈ Abel → ¬ 1o ≺ 𝐼)
126		ablgrp 19729	. . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
127		eqid 2737	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
128	16, 127	grpidcl 18910	. . . . . 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 8578	. . . . . 6 ⊢ 1o ∈ ω
132		nnfi 9104	. . . . . 6 ⊢ (1o ∈ ω → 1o ∈ Fin)
133	131, 132	ax-mp 5	. . . . 5 ⊢ 1o ∈ Fin
134		fidomtri2 9918	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 1o ∈ Fin) → (𝐼 ≼ 1o ↔ ¬ 1o ≺ 𝐼))
135	130, 133, 134	sylancl 587	. . . 4 ⊢ (𝐺 ∈ Abel → (𝐼 ≼ 1o ↔ ¬ 1o ≺ 𝐼))
136	125, 135	mpbird 257	. . 3 ⊢ (𝐺 ∈ Abel → 𝐼 ≼ 1o)
137	123, 136	syl 17	. 2 ⊢ (𝐺 ∈ CycGrp → 𝐼 ≼ 1o)
138	122, 137	impbii 209	1 ⊢ (𝐼 ≼ 1o ↔ 𝐺 ∈ CycGrp)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ∃!wreu 3350  ∃*wrmo 3351  Vcvv 3442  ∅c0 4287  {csn 4582  ⟨cop 4588  ∪ cuni 4865   class class class wbr 5100   ↦ cmpt 5181   I cid 5526   ↾ cres 5634   ∘ ccom 5636  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  ωcom 7818  1_oc1o 8400   ≈ cen 8892   ≼ cdom 8893   ≺ csdm 8894  Fincfn 8895  1c1 11039  ℤcz 12500  Basecbs 17148  0_gc0g 17371  Grpcgrp 18878  ._gcmg 19012   GrpHom cghm 19156   ~_FG cefg 19650  freeGrpcfrgp 19651  var_FGrpcvrgp 19652  Abelcabl 19725  CycGrpccyg 19821  ℤ_ringczring 21416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-addf 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-ec 8647  df-qs 8651  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-inf 9358  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-xnn0 12487  df-z 12501  df-dec 12620  df-uz 12764  df-rp 12918  df-fz 13436  df-fzo 13583  df-seq 13937  df-hash 14266  df-word 14449  df-lsw 14498  df-concat 14506  df-s1 14532  df-substr 14577  df-pfx 14607  df-splice 14685  df-reverse 14694  df-s2 14783  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-starv 17204  df-sca 17205  df-vsca 17206  df-ip 17207  df-tset 17208  df-ple 17209  df-ds 17211  df-unif 17212  df-0g 17373  df-gsum 17374  df-imas 17441  df-qus 17442  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-mhm 18720  df-submnd 18721  df-frmd 18786  df-vrmd 18787  df-grp 18881  df-minusg 18882  df-mulg 19013  df-subg 19068  df-ghm 19157  df-efg 19653  df-frgp 19654  df-vrgp 19655  df-cmn 19726  df-abl 19727  df-cyg 19822  df-mgp 20091  df-rng 20103  df-ur 20132  df-ring 20185  df-cring 20186  df-subrng 20494  df-subrg 20518  df-cnfld 21325  df-zring 21417

This theorem is referenced by: (None)