Theorem unitprodclb 33360

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 7395	. . . . . 6 ⊢ (𝑔 = ∅ → (𝑀 Σg 𝑔) = (𝑀 Σg ∅))
4	3	eleq1d 2813	. . . . 5 ⊢ (𝑔 = ∅ → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg ∅) ∈ 𝑈))
5		rneq 5900	. . . . . 6 ⊢ (𝑔 = ∅ → ran 𝑔 = ran ∅)
6	5	sseq1d 3978	. . . . 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 7395	. . . . . 6 ⊢ (𝑔 = 𝑓 → (𝑀 Σg 𝑔) = (𝑀 Σg 𝑓))
10	9	eleq1d 2813	. . . . 5 ⊢ (𝑔 = 𝑓 → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg 𝑓) ∈ 𝑈))
11		rneq 5900	. . . . . 6 ⊢ (𝑔 = 𝑓 → ran 𝑔 = ran 𝑓)
12	11	sseq1d 3978	. . . . 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 7395	. . . . . 6 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → (𝑀 Σg 𝑔) = (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)))
16	15	eleq1d 2813	. . . . 5 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) ∈ 𝑈))
17		rneq 5900	. . . . . 6 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → ran 𝑔 = ran (𝑓 ++ ⟨“𝑥”⟩))
18	17	sseq1d 3978	. . . . 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 7395	. . . . . 6 ⊢ (𝑔 = 𝐹 → (𝑀 Σg 𝑔) = (𝑀 Σg 𝐹))
22	21	eleq1d 2813	. . . . 5 ⊢ (𝑔 = 𝐹 → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg 𝐹) ∈ 𝑈))
23		rneq 5900	. . . . . 6 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
24	23	sseq1d 3978	. . . . 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 2729	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
29	27, 28	ringidval 20092	. . . . . 6 ⊢ (1r‘𝑅) = (0g‘𝑀)
30	29	gsum0 18611	. . . . 5 ⊢ (𝑀 Σg ∅) = (1r‘𝑅)
31		crngring 20154	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
32		unitprodclb.u	. . . . . . 7 ⊢ 𝑈 = (Unit‘𝑅)
33	32, 28	1unit 20283	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝑈)
34	31, 33	syl 17	. . . . 5 ⊢ (𝑅 ∈ CRing → (1r‘𝑅) ∈ 𝑈)
35	30, 34	eqeltrid 2832	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑀 Σg ∅) ∈ 𝑈)
36		rn0 5889	. . . . . 6 ⊢ ran ∅ = ∅
37		0ss 4363	. . . . . 6 ⊢ ∅ ⊆ 𝑈
38	36, 37	eqsstri 3993	. . . . 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 768	. . . . . . . 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 20054	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
44	27	crngmgp 20150	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ CMnd)
45	44	ad2antlr 727	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑀 ∈ CMnd)
46		ovexd 7422	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (0..^(♯‘𝑓)) ∈ V)
47		wrdf 14483	. . . . . . . . . 10 ⊢ (𝑓 ∈ Word 𝐵 → 𝑓:(0..^(♯‘𝑓))⟶𝐵)
48	47	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑓:(0..^(♯‘𝑓))⟶𝐵)
49		fvexd 6873	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (1r‘𝑅) ∈ V)
50		simplll 774	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑓 ∈ Word 𝐵)
51	49, 50	wrdfsupp 32858	. . . . . . . . 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 19845	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (𝑀 Σg 𝑓) ∈ 𝐵)
53		simpllr 775	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑥 ∈ 𝐵)
54		eqid 2729	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
55	32, 54, 42	unitmulclb 20290	. . . . . . . 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 1373	. . . . . . 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 3451	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
59	58	snss 4749	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑈 ↔ {𝑥} ⊆ 𝑈)
60		s1rn 14564	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → ran ⟨“𝑥”⟩ = {𝑥})
61	60	sseq1d 3978	. . . . . . . . . . 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 632	. . . . . . . 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 4153	. . . . . . . 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 20148	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
69	31, 68	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ Mnd)
70	69	ad2antlr 727	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑀 ∈ Mnd)
71	27, 54	mgpplusg 20053	. . . . . . . . 9 ⊢ (.r‘𝑅) = (+g‘𝑀)
72	43, 71	gsumccatsn 18770	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ 𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) = ((𝑀 Σg 𝑓)(.r‘𝑅)𝑥))
73	70, 50, 53, 72	syl3anc 1373	. . . . . . 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 2813	. . . . . 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 14568	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → ⟨“𝑥”⟩ ∈ Word 𝐵)
76		ccatrn 14554	. . . . . . . 8 ⊢ ((𝑓 ∈ Word 𝐵 ∧ ⟨“𝑥”⟩ ∈ Word 𝐵) → ran (𝑓 ++ ⟨“𝑥”⟩) = (ran 𝑓 ∪ ran ⟨“𝑥”⟩))
77	50, 75, 76	syl2anc 584	. . . . . . 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 3978	. . . . . 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 14687	. 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 1540   ∈ wcel 2109  Vcvv 3447   ∪ cun 3912   ⊆ wss 3914  ∅c0 4296  {csn 4589  ran crn 5639  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  0cc0 11068  ..^cfzo 13615  ♯chash 14295  Word cword 14478   ++ cconcat 14535  ⟨“cs1 14560  Basecbs 17179  ._rcmulr 17221   Σ_g cgsu 17403  Mndcmnd 18661  CMndccmn 19710  mulGrpcmgp 20049  1_rcur 20090  Ringcrg 20142  CRingccrg 20143  Unitcui 20264

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-xnn0 12516  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  df-word 14479  df-lsw 14528  df-concat 14536  df-s1 14561  df-substr 14606  df-pfx 14636  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 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-grp 18868  df-minusg 18869  df-cntz 19249  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-ring 20144  df-cring 20145  df-oppr 20246  df-dvdsr 20266  df-unit 20267

This theorem is referenced by:  1arithidom  33508