Theorem unitprodclb 33474

Description: A finite product is a unit iff all factors are units. (Contributed by Thierry Arnoux, 27-May-2025.)

Hypotheses

Ref	Expression
unitprodclb.1	⊢ 𝐵 = (Base‘𝑅)
unitprodclb.u	⊢ 𝑈 = (Unit‘𝑅)
unitprodclb.m	⊢ 𝑀 = (mulGrp‘𝑅)
unitprodclb.r	⊢ (𝜑 → 𝑅 ∈ CRing)
unitprodclb.f	⊢ (𝜑 → 𝐹 ∈ Word 𝐵)

Assertion

Ref	Expression
unitprodclb	⊢ (𝜑 → ((𝑀 Σg 𝐹) ∈ 𝑈 ↔ ran 𝐹 ⊆ 𝑈))

Proof of Theorem unitprodclb

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

Step	Hyp	Ref	Expression
1		unitprodclb.f	. 2 ⊢ (𝜑 → 𝐹 ∈ Word 𝐵)
2		unitprodclb.r	. 2 ⊢ (𝜑 → 𝑅 ∈ CRing)
3		oveq2 7367	. . . . . 6 ⊢ (𝑔 = ∅ → (𝑀 Σg 𝑔) = (𝑀 Σg ∅))
4	3	eleq1d 2826	. . . . 5 ⊢ (𝑔 = ∅ → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg ∅) ∈ 𝑈))
5		rneq 5884	. . . . . 6 ⊢ (𝑔 = ∅ → ran 𝑔 = ran ∅)
6	5	sseq1d 3947	. . . . 5 ⊢ (𝑔 = ∅ → (ran 𝑔 ⊆ 𝑈 ↔ ran ∅ ⊆ 𝑈))
7	4, 6	bibi12d 347	. . . 4 ⊢ (𝑔 = ∅ → (((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈) ↔ ((𝑀 Σg ∅) ∈ 𝑈 ↔ ran ∅ ⊆ 𝑈)))
8	7	imbi2d 342	. . 3 ⊢ (𝑔 = ∅ → ((𝑅 ∈ CRing → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈)) ↔ (𝑅 ∈ CRing → ((𝑀 Σg ∅) ∈ 𝑈 ↔ ran ∅ ⊆ 𝑈))))
9		oveq2 7367	. . . . . 6 ⊢ (𝑔 = 𝑓 → (𝑀 Σg 𝑔) = (𝑀 Σg 𝑓))
10	9	eleq1d 2826	. . . . 5 ⊢ (𝑔 = 𝑓 → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg 𝑓) ∈ 𝑈))
11		rneq 5884	. . . . . 6 ⊢ (𝑔 = 𝑓 → ran 𝑔 = ran 𝑓)
12	11	sseq1d 3947	. . . . 5 ⊢ (𝑔 = 𝑓 → (ran 𝑔 ⊆ 𝑈 ↔ ran 𝑓 ⊆ 𝑈))
13	10, 12	bibi12d 347	. . . 4 ⊢ (𝑔 = 𝑓 → (((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈) ↔ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)))
14	13	imbi2d 342	. . 3 ⊢ (𝑔 = 𝑓 → ((𝑅 ∈ CRing → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈)) ↔ (𝑅 ∈ CRing → ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈))))
15		oveq2 7367	. . . . . 6 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → (𝑀 Σg 𝑔) = (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)))
16	15	eleq1d 2826	. . . . 5 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) ∈ 𝑈))
17		rneq 5884	. . . . . 6 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → ran 𝑔 = ran (𝑓 ++ ⟨“𝑥”⟩))
18	17	sseq1d 3947	. . . . 5 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → (ran 𝑔 ⊆ 𝑈 ↔ ran (𝑓 ++ ⟨“𝑥”⟩) ⊆ 𝑈))
19	16, 18	bibi12d 347	. . . 4 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → (((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈) ↔ ((𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) ∈ 𝑈 ↔ ran (𝑓 ++ ⟨“𝑥”⟩) ⊆ 𝑈)))
20	19	imbi2d 342	. . 3 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → ((𝑅 ∈ CRing → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈)) ↔ (𝑅 ∈ CRing → ((𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) ∈ 𝑈 ↔ ran (𝑓 ++ ⟨“𝑥”⟩) ⊆ 𝑈))))
21		oveq2 7367	. . . . . 6 ⊢ (𝑔 = 𝐹 → (𝑀 Σg 𝑔) = (𝑀 Σg 𝐹))
22	21	eleq1d 2826	. . . . 5 ⊢ (𝑔 = 𝐹 → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg 𝐹) ∈ 𝑈))
23		rneq 5884	. . . . . 6 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
24	23	sseq1d 3947	. . . . 5 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ⊆ 𝑈 ↔ ran 𝐹 ⊆ 𝑈))
25	22, 24	bibi12d 347	. . . 4 ⊢ (𝑔 = 𝐹 → (((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈) ↔ ((𝑀 Σg 𝐹) ∈ 𝑈 ↔ ran 𝐹 ⊆ 𝑈)))
26	25	imbi2d 342	. . 3 ⊢ (𝑔 = 𝐹 → ((𝑅 ∈ CRing → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈)) ↔ (𝑅 ∈ CRing → ((𝑀 Σg 𝐹) ∈ 𝑈 ↔ ran 𝐹 ⊆ 𝑈))))
27		unitprodclb.m	. . . . . . 7 ⊢ 𝑀 = (mulGrp‘𝑅)
28		eqid 2741	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
29	27, 28	ringidval 20158	. . . . . 6 ⊢ (1r‘𝑅) = (0g‘𝑀)
30	29	gsum0 18647	. . . . 5 ⊢ (𝑀 Σg ∅) = (1r‘𝑅)
31		crngring 20220	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
32		unitprodclb.u	. . . . . . 7 ⊢ 𝑈 = (Unit‘𝑅)
33	32, 28	1unit 20348	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝑈)
34	31, 33	syl 17	. . . . 5 ⊢ (𝑅 ∈ CRing → (1r‘𝑅) ∈ 𝑈)
35	30, 34	eqeltrid 2845	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑀 Σg ∅) ∈ 𝑈)
36		rn0 5874	. . . . . 6 ⊢ ran ∅ = ∅
37		0ss 4330	. . . . . 6 ⊢ ∅ ⊆ 𝑈
38	36, 37	eqsstri 3962	. . . . 5 ⊢ ran ∅ ⊆ 𝑈
39	38	a1i 11	. . . 4 ⊢ (𝑅 ∈ CRing → ran ∅ ⊆ 𝑈)
40	35, 39	2thd 267	. . 3 ⊢ (𝑅 ∈ CRing → ((𝑀 Σg ∅) ∈ 𝑈 ↔ ran ∅ ⊆ 𝑈))
41		simplr 775	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑅 ∈ CRing)
42		unitprodclb.1	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅)
43	27, 42	mgpbas 20120	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
44	27	crngmgp 20216	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ CMnd)
45	44	ad2antlr 734	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑀 ∈ CMnd)
46		ovexd 7394	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (0..^(♯‘𝑓)) ∈ V)
47		wrdf 14475	. . . . . . . . . 10 ⊢ (𝑓 ∈ Word 𝐵 → 𝑓:(0..^(♯‘𝑓))⟶𝐵)
48	47	ad3antrrr 737	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑓:(0..^(♯‘𝑓))⟶𝐵)
49		fvexd 6845	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (1r‘𝑅) ∈ V)
50		simplll 781	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑓 ∈ Word 𝐵)
51	49, 50	wrdfsupp 33018	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑓 finSupp (1r‘𝑅))
52	43, 29, 45, 46, 48, 51	gsumcl 19884	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (𝑀 Σg 𝑓) ∈ 𝐵)
53		simpllr 782	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑥 ∈ 𝐵)
54		eqid 2741	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
55	32, 54, 42	unitmulclb 20355	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ (𝑀 Σg 𝑓) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (((𝑀 Σg 𝑓)(.r‘𝑅)𝑥) ∈ 𝑈 ↔ ((𝑀 Σg 𝑓) ∈ 𝑈 ∧ 𝑥 ∈ 𝑈)))
56	41, 52, 53, 55	syl3anc 1380	. . . . . . 7 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (((𝑀 Σg 𝑓)(.r‘𝑅)𝑥) ∈ 𝑈 ↔ ((𝑀 Σg 𝑓) ∈ 𝑈 ∧ 𝑥 ∈ 𝑈)))
57		simpr 486	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈))
58		vex 3437	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
59	58	snss 4718	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑈 ↔ {𝑥} ⊆ 𝑈)
60		s1rn 14557	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → ran ⟨“𝑥”⟩ = {𝑥})
61	60	sseq1d 3947	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (ran ⟨“𝑥”⟩ ⊆ 𝑈 ↔ {𝑥} ⊆ 𝑈))
62	59, 61	bitr4id 292	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝑈 ↔ ran ⟨“𝑥”⟩ ⊆ 𝑈))
63	53, 62	syl 17	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (𝑥 ∈ 𝑈 ↔ ran ⟨“𝑥”⟩ ⊆ 𝑈))
64	57, 63	anbi12d 639	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (((𝑀 Σg 𝑓) ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ↔ (ran 𝑓 ⊆ 𝑈 ∧ ran ⟨“𝑥”⟩ ⊆ 𝑈)))
65		unss 4121	. . . . . . . 8 ⊢ ((ran 𝑓 ⊆ 𝑈 ∧ ran ⟨“𝑥”⟩ ⊆ 𝑈) ↔ (ran 𝑓 ∪ ran ⟨“𝑥”⟩) ⊆ 𝑈)
66	64, 65	bitrdi 289	. . . . . . 7 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (((𝑀 Σg 𝑓) ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ↔ (ran 𝑓 ∪ ran ⟨“𝑥”⟩) ⊆ 𝑈))
67	56, 66	bitrd 281	. . . . . 6 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (((𝑀 Σg 𝑓)(.r‘𝑅)𝑥) ∈ 𝑈 ↔ (ran 𝑓 ∪ ran ⟨“𝑥”⟩) ⊆ 𝑈))
68	27	ringmgp 20214	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
69	31, 68	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ Mnd)
70	69	ad2antlr 734	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑀 ∈ Mnd)
71	27, 54	mgpplusg 20119	. . . . . . . . 9 ⊢ (.r‘𝑅) = (+g‘𝑀)
72	43, 71	gsumccatsn 18806	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ 𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) = ((𝑀 Σg 𝑓)(.r‘𝑅)𝑥))
73	70, 50, 53, 72	syl3anc 1380	. . . . . . 7 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) = ((𝑀 Σg 𝑓)(.r‘𝑅)𝑥))
74	73	eleq1d 2826	. . . . . 6 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → ((𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) ∈ 𝑈 ↔ ((𝑀 Σg 𝑓)(.r‘𝑅)𝑥) ∈ 𝑈))
75	53	s1cld 14561	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → ⟨“𝑥”⟩ ∈ Word 𝐵)
76		ccatrn 14547	. . . . . . . 8 ⊢ ((𝑓 ∈ Word 𝐵 ∧ ⟨“𝑥”⟩ ∈ Word 𝐵) → ran (𝑓 ++ ⟨“𝑥”⟩) = (ran 𝑓 ∪ ran ⟨“𝑥”⟩))
77	50, 75, 76	syl2anc 591	. . . . . . 7 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → ran (𝑓 ++ ⟨“𝑥”⟩) = (ran 𝑓 ∪ ran ⟨“𝑥”⟩))
78	77	sseq1d 3947	. . . . . 6 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (ran (𝑓 ++ ⟨“𝑥”⟩) ⊆ 𝑈 ↔ (ran 𝑓 ∪ ran ⟨“𝑥”⟩) ⊆ 𝑈))
79	67, 74, 78	3bitr4d 313	. . . . 5 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → ((𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) ∈ 𝑈 ↔ ran (𝑓 ++ ⟨“𝑥”⟩) ⊆ 𝑈))
80	79	exp31 421	. . . 4 ⊢ ((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑅 ∈ CRing → (((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈) → ((𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) ∈ 𝑈 ↔ ran (𝑓 ++ ⟨“𝑥”⟩) ⊆ 𝑈))))
81	80	a2d 29	. . 3 ⊢ ((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑅 ∈ CRing → ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (𝑅 ∈ CRing → ((𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) ∈ 𝑈 ↔ ran (𝑓 ++ ⟨“𝑥”⟩) ⊆ 𝑈))))
82	8, 14, 20, 26, 40, 81	wrdind 14679	. 2 ⊢ (𝐹 ∈ Word 𝐵 → (𝑅 ∈ CRing → ((𝑀 Σg 𝐹) ∈ 𝑈 ↔ ran 𝐹 ⊆ 𝑈)))
83	1, 2, 82	sylc 65	1 ⊢ (𝜑 → ((𝑀 Σg 𝐹) ∈ 𝑈 ↔ ran 𝐹 ⊆ 𝑈))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  Vcvv 3433   ∪ cun 3882   ⊆ wss 3884  ∅c0 4263  {csn 4557  ran crn 5621  ⟶wf 6484  ‘cfv 6488  (class class class)co 7359  0cc0 11034  ..^cfzo 13603  ♯chash 14287  Word cword 14470   ++ cconcat 14527  ⟨“cs1 14553  Basecbs 17174  ._rcmulr 17216   Σ_g cgsu 17398  Mndcmnd 18697  CMndccmn 19749  mulGrpcmgp 20115  1_rcur 20156  Ringcrg 20208  CRingccrg 20209  Unitcui 20329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681  ax-cnex 11090  ax-resscn 11091  ax-1cn 11092  ax-icn 11093  ax-addcl 11094  ax-addrcl 11095  ax-mulcl 11096  ax-mulrcl 11097  ax-mulcom 11098  ax-addass 11099  ax-mulass 11100  ax-distr 11101  ax-i2m1 11102  ax-1ne0 11103  ax-1rid 11104  ax-rnegex 11105  ax-rrecex 11106  ax-cnre 11107  ax-pre-lttri 11108  ax-pre-lttrn 11109  ax-pre-ltadd 11110  ax-pre-mulgt0 11111

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-isom 6497  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-om 7810  df-1st 7933  df-2nd 7934  df-supp 8103  df-tpos 8168  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-oi 9419  df-card 9858  df-pnf 11177  df-mnf 11178  df-xr 11179  df-ltxr 11180  df-le 11181  df-sub 11375  df-neg 11376  df-nn 12170  df-2 12239  df-3 12240  df-n0 12433  df-xnn0 12506  df-z 12520  df-uz 12784  df-fz 13457  df-fzo 13604  df-seq 13959  df-hash 14288  df-word 14471  df-lsw 14520  df-concat 14528  df-s1 14554  df-substr 14599  df-pfx 14629  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-mulr 17229  df-0g 17399  df-gsum 17400  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-submnd 18747  df-grp 18907  df-minusg 18908  df-cntz 19286  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-ring 20210  df-cring 20211  df-oppr 20311  df-dvdsr 20331  df-unit 20332

This theorem is referenced by:  1arithidom  33630