Theorem unitprodclb 33449

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 7375	. . . . . 6 ⊢ (𝑔 = ∅ → (𝑀 Σg 𝑔) = (𝑀 Σg ∅))
4	3	eleq1d 2821	. . . . 5 ⊢ (𝑔 = ∅ → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg ∅) ∈ 𝑈))
5		rneq 5891	. . . . . 6 ⊢ (𝑔 = ∅ → ran 𝑔 = ran ∅)
6	5	sseq1d 3953	. . . . 5 ⊢ (𝑔 = ∅ → (ran 𝑔 ⊆ 𝑈 ↔ ran ∅ ⊆ 𝑈))
7	4, 6	bibi12d 345	. . . 4 ⊢ (𝑔 = ∅ → (((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈) ↔ ((𝑀 Σg ∅) ∈ 𝑈 ↔ ran ∅ ⊆ 𝑈)))
8	7	imbi2d 340	. . 3 ⊢ (𝑔 = ∅ → ((𝑅 ∈ CRing → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈)) ↔ (𝑅 ∈ CRing → ((𝑀 Σg ∅) ∈ 𝑈 ↔ ran ∅ ⊆ 𝑈))))
9		oveq2 7375	. . . . . 6 ⊢ (𝑔 = 𝑓 → (𝑀 Σg 𝑔) = (𝑀 Σg 𝑓))
10	9	eleq1d 2821	. . . . 5 ⊢ (𝑔 = 𝑓 → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg 𝑓) ∈ 𝑈))
11		rneq 5891	. . . . . 6 ⊢ (𝑔 = 𝑓 → ran 𝑔 = ran 𝑓)
12	11	sseq1d 3953	. . . . 5 ⊢ (𝑔 = 𝑓 → (ran 𝑔 ⊆ 𝑈 ↔ ran 𝑓 ⊆ 𝑈))
13	10, 12	bibi12d 345	. . . 4 ⊢ (𝑔 = 𝑓 → (((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈) ↔ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)))
14	13	imbi2d 340	. . 3 ⊢ (𝑔 = 𝑓 → ((𝑅 ∈ CRing → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈)) ↔ (𝑅 ∈ CRing → ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈))))
15		oveq2 7375	. . . . . 6 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → (𝑀 Σg 𝑔) = (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)))
16	15	eleq1d 2821	. . . . 5 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) ∈ 𝑈))
17		rneq 5891	. . . . . 6 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → ran 𝑔 = ran (𝑓 ++ ⟨“𝑥”⟩))
18	17	sseq1d 3953	. . . . 5 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → (ran 𝑔 ⊆ 𝑈 ↔ ran (𝑓 ++ ⟨“𝑥”⟩) ⊆ 𝑈))
19	16, 18	bibi12d 345	. . . 4 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → (((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈) ↔ ((𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) ∈ 𝑈 ↔ ran (𝑓 ++ ⟨“𝑥”⟩) ⊆ 𝑈)))
20	19	imbi2d 340	. . 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 7375	. . . . . 6 ⊢ (𝑔 = 𝐹 → (𝑀 Σg 𝑔) = (𝑀 Σg 𝐹))
22	21	eleq1d 2821	. . . . 5 ⊢ (𝑔 = 𝐹 → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg 𝐹) ∈ 𝑈))
23		rneq 5891	. . . . . 6 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
24	23	sseq1d 3953	. . . . 5 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ⊆ 𝑈 ↔ ran 𝐹 ⊆ 𝑈))
25	22, 24	bibi12d 345	. . . 4 ⊢ (𝑔 = 𝐹 → (((𝑀 Σg 𝑔) ∈ 𝑈 ↔ ran 𝑔 ⊆ 𝑈) ↔ ((𝑀 Σg 𝐹) ∈ 𝑈 ↔ ran 𝐹 ⊆ 𝑈)))
26	25	imbi2d 340	. . 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 2736	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
29	27, 28	ringidval 20164	. . . . . 6 ⊢ (1r‘𝑅) = (0g‘𝑀)
30	29	gsum0 18652	. . . . 5 ⊢ (𝑀 Σg ∅) = (1r‘𝑅)
31		crngring 20226	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
32		unitprodclb.u	. . . . . . 7 ⊢ 𝑈 = (Unit‘𝑅)
33	32, 28	1unit 20354	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝑈)
34	31, 33	syl 17	. . . . 5 ⊢ (𝑅 ∈ CRing → (1r‘𝑅) ∈ 𝑈)
35	30, 34	eqeltrid 2840	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑀 Σg ∅) ∈ 𝑈)
36		rn0 5881	. . . . . 6 ⊢ ran ∅ = ∅
37		0ss 4340	. . . . . 6 ⊢ ∅ ⊆ 𝑈
38	36, 37	eqsstri 3968	. . . . 5 ⊢ ran ∅ ⊆ 𝑈
39	38	a1i 11	. . . 4 ⊢ (𝑅 ∈ CRing → ran ∅ ⊆ 𝑈)
40	35, 39	2thd 265	. . 3 ⊢ (𝑅 ∈ CRing → ((𝑀 Σg ∅) ∈ 𝑈 ↔ ran ∅ ⊆ 𝑈))
41		simplr 769	. . . . . . . 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 20126	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
44	27	crngmgp 20222	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ CMnd)
45	44	ad2antlr 728	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑀 ∈ CMnd)
46		ovexd 7402	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (0..^(♯‘𝑓)) ∈ V)
47		wrdf 14480	. . . . . . . . . 10 ⊢ (𝑓 ∈ Word 𝐵 → 𝑓:(0..^(♯‘𝑓))⟶𝐵)
48	47	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑓:(0..^(♯‘𝑓))⟶𝐵)
49		fvexd 6855	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (1r‘𝑅) ∈ V)
50		simplll 775	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑓 ∈ Word 𝐵)
51	49, 50	wrdfsupp 32997	. . . . . . . . 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 19890	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (𝑀 Σg 𝑓) ∈ 𝐵)
53		simpllr 776	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑥 ∈ 𝐵)
54		eqid 2736	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
55	32, 54, 42	unitmulclb 20361	. . . . . . . 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 1374	. . . . . . 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 484	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈))
58		vex 3433	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
59	58	snss 4728	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑈 ↔ {𝑥} ⊆ 𝑈)
60		s1rn 14562	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → ran ⟨“𝑥”⟩ = {𝑥})
61	60	sseq1d 3953	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (ran ⟨“𝑥”⟩ ⊆ 𝑈 ↔ {𝑥} ⊆ 𝑈))
62	59, 61	bitr4id 290	. . . . . . . . . 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 633	. . . . . . . 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 4130	. . . . . . . 8 ⊢ ((ran 𝑓 ⊆ 𝑈 ∧ ran ⟨“𝑥”⟩ ⊆ 𝑈) ↔ (ran 𝑓 ∪ ran ⟨“𝑥”⟩) ⊆ 𝑈)
66	64, 65	bitrdi 287	. . . . . . 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 279	. . . . . 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 20220	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
69	31, 68	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ Mnd)
70	69	ad2antlr 728	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑀 ∈ Mnd)
71	27, 54	mgpplusg 20125	. . . . . . . . 9 ⊢ (.r‘𝑅) = (+g‘𝑀)
72	43, 71	gsumccatsn 18811	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ 𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) = ((𝑀 Σg 𝑓)(.r‘𝑅)𝑥))
73	70, 50, 53, 72	syl3anc 1374	. . . . . . 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 2821	. . . . . 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 14566	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → ⟨“𝑥”⟩ ∈ Word 𝐵)
76		ccatrn 14552	. . . . . . . 8 ⊢ ((𝑓 ∈ Word 𝐵 ∧ ⟨“𝑥”⟩ ∈ Word 𝐵) → ran (𝑓 ++ ⟨“𝑥”⟩) = (ran 𝑓 ∪ ran ⟨“𝑥”⟩))
77	50, 75, 76	syl2anc 585	. . . . . . 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 3953	. . . . . 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 311	. . . . 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 419	. . . 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 14684	. 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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∪ cun 3887   ⊆ wss 3889  ∅c0 4273  {csn 4567  ran crn 5632  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  0cc0 11038  ..^cfzo 13608  ♯chash 14292  Word cword 14475   ++ cconcat 14532  ⟨“cs1 14558  Basecbs 17179  ._rcmulr 17221   Σ_g cgsu 17403  Mndcmnd 18702  CMndccmn 19755  mulGrpcmgp 20121  1_rcur 20162  Ringcrg 20214  CRingccrg 20215  Unitcui 20335

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  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

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  df-word 14476  df-lsw 14525  df-concat 14533  df-s1 14559  df-substr 14604  df-pfx 14634  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-0g 17404  df-gsum 17405  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-submnd 18752  df-grp 18912  df-minusg 18913  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-cring 20217  df-oppr 20317  df-dvdsr 20337  df-unit 20338

This theorem is referenced by:  1arithidom  33597