Theorem unitprodclb 33618

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 7408	. . . . . 6 ⊢ (𝑔 = ∅ → (𝑀 Σg 𝑔) = (𝑀 Σg ∅))
4	3	eleq1d 2850	. . . . 5 ⊢ (𝑔 = ∅ → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg ∅) ∈ 𝑈))
5		rneq 5917	. . . . . 6 ⊢ (𝑔 = ∅ → ran 𝑔 = ran ∅)
6	5	sseq1d 3970	. . . . 5 ⊢ (𝑔 = ∅ → (ran 𝑔 ⊆ 𝑈 ↔ ran ∅ ⊆ 𝑈))
7	4, 6	bibi12d 348	. . . 4 ⊢ (𝑔 = ∅ → (((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈) ↔ ((𝑀 Σg ∅) ∈ 𝑈 ↔ ran ∅ ⊆ 𝑈)))
8	7	imbi2d 343	. . 3 ⊢ (𝑔 = ∅ → ((𝑅 ∈ CRing → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈)) ↔ (𝑅 ∈ CRing → ((𝑀 Σg ∅) ∈ 𝑈 ↔ ran ∅ ⊆ 𝑈))))
9		oveq2 7408	. . . . . 6 ⊢ (𝑔 = 𝑓 → (𝑀 Σg 𝑔) = (𝑀 Σg 𝑓))
10	9	eleq1d 2850	. . . . 5 ⊢ (𝑔 = 𝑓 → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg 𝑓) ∈ 𝑈))
11		rneq 5917	. . . . . 6 ⊢ (𝑔 = 𝑓 → ran 𝑔 = ran 𝑓)
12	11	sseq1d 3970	. . . . 5 ⊢ (𝑔 = 𝑓 → (ran 𝑔 ⊆ 𝑈 ↔ ran 𝑓 ⊆ 𝑈))
13	10, 12	bibi12d 348	. . . 4 ⊢ (𝑔 = 𝑓 → (((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈) ↔ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)))
14	13	imbi2d 343	. . 3 ⊢ (𝑔 = 𝑓 → ((𝑅 ∈ CRing → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈)) ↔ (𝑅 ∈ CRing → ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈))))
15		oveq2 7408	. . . . . 6 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → (𝑀 Σg 𝑔) = (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)))
16	15	eleq1d 2850	. . . . 5 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) ∈ 𝑈))
17		rneq 5917	. . . . . 6 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → ran 𝑔 = ran (𝑓 ++ ⟨“𝑥”⟩))
18	17	sseq1d 3970	. . . . 5 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → (ran 𝑔 ⊆ 𝑈 ↔ ran (𝑓 ++ ⟨“𝑥”⟩) ⊆ 𝑈))
19	16, 18	bibi12d 348	. . . 4 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → (((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈) ↔ ((𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) ∈ 𝑈 ↔ ran (𝑓 ++ ⟨“𝑥”⟩) ⊆ 𝑈)))
20	19	imbi2d 343	. . 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 7408	. . . . . 6 ⊢ (𝑔 = 𝐹 → (𝑀 Σg 𝑔) = (𝑀 Σg 𝐹))
22	21	eleq1d 2850	. . . . 5 ⊢ (𝑔 = 𝐹 → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg 𝐹) ∈ 𝑈))
23		rneq 5917	. . . . . 6 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
24	23	sseq1d 3970	. . . . 5 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ⊆ 𝑈 ↔ ran 𝐹 ⊆ 𝑈))
25	22, 24	bibi12d 348	. . . 4 ⊢ (𝑔 = 𝐹 → (((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈) ↔ ((𝑀 Σg 𝐹) ∈ 𝑈 ↔ ran 𝐹 ⊆ 𝑈)))
26	25	imbi2d 343	. . 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 2765	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
29	27, 28	ringidval 20256	. . . . . 6 ⊢ (1r‘𝑅) = (0g‘𝑀)
30	29	gsum0 18732	. . . . 5 ⊢ (𝑀 Σg ∅) = (1r‘𝑅)
31		crngring 20318	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
32		unitprodclb.u	. . . . . . 7 ⊢ 𝑈 = (Unit‘𝑅)
33	32, 28	1unit 20447	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝑈)
34	31, 33	syl 18	. . . . 5 ⊢ (𝑅 ∈ CRing → (1r‘𝑅) ∈ 𝑈)
35	30, 34	eqeltrid 2869	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑀 Σg ∅) ∈ 𝑈)
36		rn0 5907	. . . . . 6 ⊢ ran ∅ = ∅
37		0ss 4357	. . . . . 6 ⊢ ∅ ⊆ 𝑈
38	36, 37	eqsstri 3985	. . . . 5 ⊢ ran ∅ ⊆ 𝑈
39	38	a1i 11	. . . 4 ⊢ (𝑅 ∈ CRing → ran ∅ ⊆ 𝑈)
40	35, 39	2thd 268	. . 3 ⊢ (𝑅 ∈ CRing → ((𝑀 Σg ∅) ∈ 𝑈 ↔ ran ∅ ⊆ 𝑈))
41		simplr 780	. . . . . . . 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 20212	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
44	27	crngmgp 20314	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ CMnd)
45	44	ad2antlr 739	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑀 ∈ CMnd)
46		ovexd 7435	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (0..^(♯‘𝑓)) ∈ V)
47		wrdf 14545	. . . . . . . . . 10 ⊢ (𝑓 ∈ Word 𝐵 → 𝑓:(0..^(♯‘𝑓))⟶𝐵)
48	47	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑓:(0..^(♯‘𝑓))⟶𝐵)
49		fvexd 6886	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (1r‘𝑅) ∈ V)
50		simplll 786	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑓 ∈ Word 𝐵)
51	49, 50	wrdfsupp 33170	. . . . . . . . 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 19976	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (𝑀 Σg 𝑓) ∈ 𝐵)
53		simpllr 787	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑥 ∈ 𝐵)
54		eqid 2765	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
55	32, 54, 42	unitmulclb 20454	. . . . . . . 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 1394	. . . . . . 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 489	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈))
58		vex 3461	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
59	58	snss 4746	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑈 ↔ {𝑥} ⊆ 𝑈)
60		s1rn 14627	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → ran ⟨“𝑥”⟩ = {𝑥})
61	60	sseq1d 3970	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (ran ⟨“𝑥”⟩ ⊆ 𝑈 ↔ {𝑥} ⊆ 𝑈))
62	59, 61	bitr4id 293	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝑈 ↔ ran ⟨“𝑥”⟩ ⊆ 𝑈))
63	53, 62	syl 18	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (𝑥 ∈ 𝑈 ↔ ran ⟨“𝑥”⟩ ⊆ 𝑈))
64	57, 63	anbi12d 643	. . . . . . . 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 4145	. . . . . . . 8 ⊢ ((ran 𝑓 ⊆ 𝑈 ∧ ran ⟨“𝑥”⟩ ⊆ 𝑈) ↔ (ran 𝑓 ∪ ran ⟨“𝑥”⟩) ⊆ 𝑈)
66	64, 65	bitrdi 290	. . . . . . 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 282	. . . . . 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 20312	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
69	31, 68	syl 18	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ Mnd)
70	69	ad2antlr 739	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑀 ∈ Mnd)
71	27, 54	mgpplusg 20211	. . . . . . . . 9 ⊢ (.r‘𝑅) = (+g‘𝑀)
72	43, 71	gsumccatsn 18892	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ 𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) = ((𝑀 Σg 𝑓)(.r‘𝑅)𝑥))
73	70, 50, 53, 72	syl3anc 1394	. . . . . . 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 2850	. . . . . 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 14631	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → ⟨“𝑥”⟩ ∈ Word 𝐵)
76		ccatrn 14617	. . . . . . . 8 ⊢ ((𝑓 ∈ Word 𝐵 ∧ ⟨“𝑥”⟩ ∈ Word 𝐵) → ran (𝑓 ++ ⟨“𝑥”⟩) = (ran 𝑓 ∪ ran ⟨“𝑥”⟩))
77	50, 75, 76	syl2anc 595	. . . . . . 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 3970	. . . . . 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 314	. . . . 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 424	. . . 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 30	. . 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 14749	. 2 ⊢ (𝐹 ∈ Word 𝐵 → (𝑅 ∈ CRing → ((𝑀 Σg 𝐹) ∈ 𝑈 ↔ ran 𝐹 ⊆ 𝑈)))
83	1, 2, 82	sylc 66	1 ⊢ (𝜑 → ((𝑀 Σg 𝐹) ∈ 𝑈 ↔ ran 𝐹 ⊆ 𝑈))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∪ cun 3905   ⊆ wss 3907  ∅c0 4288  {csn 4585  ran crn 5653  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  0cc0 11088  ..^cfzo 13673  ♯chash 14357  Word cword 14540   ++ cconcat 14597  ⟨“cs1 14623  Basecbs 17259  ._rcmulr 17301   Σ_g cgsu 17483  Mndcmnd 18782  CMndccmn 19841  mulGrpcmgp 20207  1_rcur 20254  Ringcrg 20306  CRingccrg 20307  Unitcui 20428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-tpos 8210  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-n0 12496  df-xnn0 12569  df-z 12583  df-uz 12854  df-fz 13527  df-fzo 13674  df-seq 14029  df-hash 14358  df-word 14541  df-lsw 14590  df-concat 14598  df-s1 14624  df-substr 14669  df-pfx 14699  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-mulr 17314  df-0g 17484  df-gsum 17485  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-submnd 18832  df-grp 18993  df-minusg 18994  df-cntz 19378  df-cmn 19843  df-abl 19844  df-mgp 20208  df-rng 20222  df-ur 20255  df-ring 20308  df-cring 20309  df-oppr 20410  df-dvdsr 20430  df-unit 20431

This theorem is referenced by:  1arithidom  33744