Theorem frgpcyg 21534

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 8996	. . 3 ⊢ (𝐼 ≼ 1o ↔ (𝐼 ≺ 1o ∨ 𝐼 ≈ 1o))
2		sdom1 9250	. . . . 5 ⊢ (𝐼 ≺ 1o ↔ 𝐼 = ∅)
3		frgpcyg.g	. . . . . . 7 ⊢ 𝐺 = (freeGrp‘𝐼)
4		fveq2 6876	. . . . . . 7 ⊢ (𝐼 = ∅ → (freeGrp‘𝐼) = (freeGrp‘∅))
5	3, 4	eqtrid 2782	. . . . . 6 ⊢ (𝐼 = ∅ → 𝐺 = (freeGrp‘∅))
6		0ex 5277	. . . . . . . 8 ⊢ ∅ ∈ V
7		eqid 2735	. . . . . . . . 9 ⊢ (freeGrp‘∅) = (freeGrp‘∅)
8	7	frgpgrp 19743	. . . . . . . 8 ⊢ (∅ ∈ V → (freeGrp‘∅) ∈ Grp)
9	6, 8	ax-mp 5	. . . . . . 7 ⊢ (freeGrp‘∅) ∈ Grp
10		eqid 2735	. . . . . . . 8 ⊢ (Base‘(freeGrp‘∅)) = (Base‘(freeGrp‘∅))
11	7, 10	0frgp 19760	. . . . . . 7 ⊢ (Base‘(freeGrp‘∅)) ≈ 1o
12	10	0cyg 19874	. . . . . . 7 ⊢ (((freeGrp‘∅) ∈ Grp ∧ (Base‘(freeGrp‘∅)) ≈ 1o) → (freeGrp‘∅) ∈ CycGrp)
13	9, 11, 12	mp2an 692	. . . . . 6 ⊢ (freeGrp‘∅) ∈ CycGrp
14	5, 13	eqeltrdi 2842	. . . . 5 ⊢ (𝐼 = ∅ → 𝐺 ∈ CycGrp)
15	2, 14	sylbi 217	. . . 4 ⊢ (𝐼 ≺ 1o → 𝐺 ∈ CycGrp)
16		eqid 2735	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
17		eqid 2735	. . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺)
18		relen 8964	. . . . . . 7 ⊢ Rel ≈
19	18	brrelex1i 5710	. . . . . 6 ⊢ (𝐼 ≈ 1o → 𝐼 ∈ V)
20	3	frgpgrp 19743	. . . . . 6 ⊢ (𝐼 ∈ V → 𝐺 ∈ Grp)
21	19, 20	syl 17	. . . . 5 ⊢ (𝐼 ≈ 1o → 𝐺 ∈ Grp)
22		eqid 2735	. . . . . . . 8 ⊢ ( ~FG ‘𝐼) = ( ~FG ‘𝐼)
23		eqid 2735	. . . . . . . 8 ⊢ (varFGrp‘𝐼) = (varFGrp‘𝐼)
24	22, 23, 3, 16	vrgpf 19749	. . . . . . 7 ⊢ (𝐼 ∈ V → (varFGrp‘𝐼):𝐼⟶(Base‘𝐺))
25	19, 24	syl 17	. . . . . 6 ⊢ (𝐼 ≈ 1o → (varFGrp‘𝐼):𝐼⟶(Base‘𝐺))
26		en1uniel 9043	. . . . . 6 ⊢ (𝐼 ≈ 1o → ∪ 𝐼 ∈ 𝐼)
27	25, 26	ffvelcdmd 7075	. . . . 5 ⊢ (𝐼 ≈ 1o → ((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺))
28		zringgrp 21413	. . . . . . . . 9 ⊢ ℤring ∈ Grp
29	19	uniexd 7736	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → ∪ 𝐼 ∈ V)
30		1zzd 12623	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → 1 ∈ ℤ)
31	29, 30	fsnd 6861	. . . . . . . . . 10 ⊢ (𝐼 ≈ 1o → {⟨∪ 𝐼, 1⟩}:{∪ 𝐼}⟶ℤ)
32		en1b 9039	. . . . . . . . . . . 12 ⊢ (𝐼 ≈ 1o ↔ 𝐼 = {∪ 𝐼})
33	32	biimpi 216	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → 𝐼 = {∪ 𝐼})
34	33	feq2d 6692	. . . . . . . . . 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 21414	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤring)
37	3, 36, 23	frgpup3 19759	. . . . . . . . 9 ⊢ ((ℤring ∈ Grp ∧ 𝐼 ∈ V ∧ {⟨∪ 𝐼, 1⟩}:𝐼⟶ℤ) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩})
38	28, 19, 35, 37	mp3an2i 1468	. . . . . . . 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 3363	. . . . . . 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 6875	. . . . . . . . . 10 ⊢ ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → ((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼))
43	25, 26	fvco3d 6979	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → ((𝑓 ∘ (varFGrp‘𝐼))‘∪ 𝐼) = (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)))
44		1z 12622	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
45		fvsng 7172	. . . . . . . . . . . 12 ⊢ ((∪ 𝐼 ∈ V ∧ 1 ∈ ℤ) → ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼) = 1)
46	29, 44, 45	sylancl 586	. . . . . . . . . . 11 ⊢ (𝐼 ≈ 1o → ({⟨∪ 𝐼, 1⟩}‘∪ 𝐼) = 1)
47	43, 46	eqeq12d 2751	. . . . . . . . . 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 726	. . . . . . . 8 ⊢ (((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp‘𝐼)) = {⟨∪ 𝐼, 1⟩} → (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1))
50	16, 36	ghmf 19203	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐺 GrpHom ℤring) → 𝑓:(Base‘𝐺)⟶ℤ)
51	50	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓:(Base‘𝐺)⟶ℤ)
52	51	ffvelcdmda 7074	. . . . . . . . . . 11 ⊢ (((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓‘𝑥) ∈ ℤ)
53	52	an32s 652	. . . . . . . . . 10 ⊢ (((𝐼 ≈ 1o ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑓‘𝑥) ∈ ℤ)
54		mptresid 6038	. . . . . . . . . . . . . 14 ⊢ ( I ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ 𝑥)
55	3, 16, 23	frgpup3 19759	. . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ≈ 1o → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼))
57		reurmo 3362	. . . . . . . . . . . . . . . . 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 19214	. . . . . . . . . . . . . . . 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 6753	. . . . . . . . . . . . . . . 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 6947	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓 = (𝑥 ∈ (Base‘𝐺) ↦ (𝑓‘𝑥)))
67		eqidd 2736	. . . . . . . . . . . . . . . . 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 7412	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑓‘𝑥) → (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)) = ((𝑓‘𝑥)(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
69	52, 66, 67, 68	fmptco 7119	. . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼)))
72	17, 71, 16	mulgghm2 21437	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ ((varFGrp‘𝐼)‘∪ 𝐼) ∈ (Base‘𝐺)) → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)((varFGrp‘𝐼)‘∪ 𝐼))) ∈ (ℤring GrpHom 𝐺))
73	60, 70, 72	syl2anc 584	. . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → 𝑓 ∈ (𝐺 GrpHom ℤring))
75		ghmco 19219	. . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑦 ∈ 𝐼 ↔ 𝑦 ∈ {∪ 𝐼}))
80		simprr 772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)
81	80	oveq1d 7420	. . . . . . . . . . . . . . . . . . . . 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 19064	. . . . . . . . . . . . . . . . . . . . . 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 4618	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ {∪ 𝐼} → 𝑦 = ∪ 𝐼)
86	85	fveq2d 6880	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ {∪ 𝐼} → ((varFGrp‘𝐼)‘𝑦) = ((varFGrp‘𝐼)‘∪ 𝐼))
87	86	fveq2d 6880	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ {∪ 𝐼} → (𝑓‘((varFGrp‘𝐼)‘𝑦)) = (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)))
88	87	oveq1d 7420	. . . . . . . . . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . . . . . . . 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 5214	. . . . . . . . . . . . . . . 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 7074	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) ∧ 𝑦 ∈ 𝐼) → ((varFGrp‘𝐼)‘𝑦) ∈ (Base‘𝐺))
95	63	feqmptd 6947	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp‘𝐼)‘∪ 𝐼)) = 1)) → (varFGrp‘𝐼) = (𝑦 ∈ 𝐼 ↦ ((varFGrp‘𝐼)‘𝑦)))
96		eqidd 2736	. . . . . . . . . . . . . . . . 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 6876	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ((varFGrp‘𝐼)‘𝑦) → (𝑓‘𝑥) = (𝑓‘((varFGrp‘𝐼)‘𝑦)))
98	97	oveq1d 7420	. . . . . . . . . . . . . . . . 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 7119	. . . . . . . . . . . . . . . 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 5837	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ( I ↾ (Base‘𝐺)) → (𝑔 ∘ (varFGrp‘𝐼)) = (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)))
102	101	eqeq1d 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = ( I ↾ (Base‘𝐺)) → ((𝑔 ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼) ↔ (( I ↾ (Base‘𝐺)) ∘ (varFGrp‘𝐼)) = (varFGrp‘𝐼)))
103		coeq1 5837	. . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . 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 3866	. . . . . . . . . . . . . . 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 1381	. . . . . . . . . . . . . 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 7005	. . . . . . . . . . . . . 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 3057	. . . . . . . . . . . . 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 3234	. . . . . . . . . . 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 652	. . . . . . . . . 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 3624	. . . . . . . . . 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 584	. . . . . . . . 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 3141	. . . . . 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 19868	. . . 4 ⊢ (𝐼 ≈ 1o → 𝐺 ∈ CycGrp)
121	15, 120	jaoi 857	. . 3 ⊢ ((𝐼 ≺ 1o ∨ 𝐼 ≈ 1o) → 𝐺 ∈ CycGrp)
122	1, 121	sylbi 217	. 2 ⊢ (𝐼 ≼ 1o → 𝐺 ∈ CycGrp)
123		cygabl 19872	. . 3 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
124	3	frgpnabl 19856	. . . . 5 ⊢ (1o ≺ 𝐼 → ¬ 𝐺 ∈ Abel)
125	124	con2i 139	. . . 4 ⊢ (𝐺 ∈ Abel → ¬ 1o ≺ 𝐼)
126		ablgrp 19766	. . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
127		eqid 2735	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
128	16, 127	grpidcl 18948	. . . . . 6 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
129	3, 16	elbasfv 17234	. . . . . 6 ⊢ ((0g‘𝐺) ∈ (Base‘𝐺) → 𝐼 ∈ V)
130	126, 128, 129	3syl 18	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐼 ∈ V)
131		1onn 8652	. . . . . 6 ⊢ 1o ∈ ω
132		nnfi 9181	. . . . . 6 ⊢ (1o ∈ ω → 1o ∈ Fin)
133	131, 132	ax-mp 5	. . . . 5 ⊢ 1o ∈ Fin
134		fidomtri2 10008	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 1o ∈ Fin) → (𝐼 ≼ 1o ↔ ¬ 1o ≺ 𝐼))
135	130, 133, 134	sylancl 586	. . . 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 847   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  ∃!wreu 3357  ∃*wrmo 3358  Vcvv 3459  ∅c0 4308  {csn 4601  ⟨cop 4607  ∪ cuni 4883   class class class wbr 5119   ↦ cmpt 5201   I cid 5547   ↾ cres 5656   ∘ ccom 5658  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  ωcom 7861  1_oc1o 8473   ≈ cen 8956   ≼ cdom 8957   ≺ csdm 8958  Fincfn 8959  1c1 11130  ℤcz 12588  Basecbs 17228  0_gc0g 17453  Grpcgrp 18916  ._gcmg 19050   GrpHom cghm 19195   ~_FG cefg 19687  freeGrpcfrgp 19688  var_FGrpcvrgp 19689  Abelcabl 19762  CycGrpccyg 19858  ℤ_ringczring 21407

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-addf 11208

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-ot 4610  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-ec 8721  df-qs 8725  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-sup 9454  df-inf 9455  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-xnn0 12575  df-z 12589  df-dec 12709  df-uz 12853  df-rp 13009  df-fz 13525  df-fzo 13672  df-seq 14020  df-hash 14349  df-word 14532  df-lsw 14581  df-concat 14589  df-s1 14614  df-substr 14659  df-pfx 14689  df-splice 14768  df-reverse 14777  df-s2 14867  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-starv 17286  df-sca 17287  df-vsca 17288  df-ip 17289  df-tset 17290  df-ple 17291  df-ds 17293  df-unif 17294  df-0g 17455  df-gsum 17456  df-imas 17522  df-qus 17523  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-mhm 18761  df-submnd 18762  df-frmd 18827  df-vrmd 18828  df-grp 18919  df-minusg 18920  df-mulg 19051  df-subg 19106  df-ghm 19196  df-efg 19690  df-frgp 19691  df-vrgp 19692  df-cmn 19763  df-abl 19764  df-cyg 19859  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-cring 20196  df-subrng 20506  df-subrg 20530  df-cnfld 21316  df-zring 21408

This theorem is referenced by: (None)