Theorem unitprodclb 33926

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 7420	. . . . . 6 ⊢ (𝑔 = ∅ → (𝑀 Σg 𝑔) = (𝑀 Σg ∅))
4	3	eleq1d 2846	. . . . 5 ⊢ (𝑔 = ∅ → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg ∅) ∈ 𝑈))
5		rneq 5918	. . . . . 6 ⊢ (𝑔 = ∅ → ran 𝑔 = ran ∅)
6	5	sseq1d 3962	. . . . 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 7420	. . . . . 6 ⊢ (𝑔 = 𝑓 → (𝑀 Σg 𝑔) = (𝑀 Σg 𝑓))
10	9	eleq1d 2846	. . . . 5 ⊢ (𝑔 = 𝑓 → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg 𝑓) ∈ 𝑈))
11		rneq 5918	. . . . . 6 ⊢ (𝑔 = 𝑓 → ran 𝑔 = ran 𝑓)
12	11	sseq1d 3962	. . . . 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 7420	. . . . . 6 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → (𝑀 Σg 𝑔) = (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)))
16	15	eleq1d 2846	. . . . 5 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) ∈ 𝑈))
17		rneq 5918	. . . . . 6 ⊢ (𝑔 = (𝑓 ++ ⟨“𝑥”⟩) → ran 𝑔 = ran (𝑓 ++ ⟨“𝑥”⟩))
18	17	sseq1d 3962	. . . . 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 7420	. . . . . 6 ⊢ (𝑔 = 𝐹 → (𝑀 Σg 𝑔) = (𝑀 Σg 𝐹))
22	21	eleq1d 2846	. . . . 5 ⊢ (𝑔 = 𝐹 → ((𝑀 Σg 𝑔) ∈ 𝑈 ↔ (𝑀 Σg 𝐹) ∈ 𝑈))
23		rneq 5918	. . . . . 6 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
24	23	sseq1d 3962	. . . . 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 2761	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
29	27, 28	ringidval 20389	. . . . . 6 ⊢ (1r‘𝑅) = (0g‘𝑀)
30	29	gsum0 18853	. . . . 5 ⊢ (𝑀 Σg ∅) = (1r‘𝑅)
31		crngring 20452	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
32		unitprodclb.u	. . . . . . 7 ⊢ 𝑈 = (Unit‘𝑅)
33	32, 28	1unit 20584	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝑈)
34	31, 33	syl 18	. . . . 5 ⊢ (𝑅 ∈ CRing → (1r‘𝑅) ∈ 𝑈)
35	30, 34	eqeltrid 2865	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑀 Σg ∅) ∈ 𝑈)
36		rn0 5908	. . . . . 6 ⊢ ran ∅ = ∅
37		0ss 4350	. . . . . 6 ⊢ ∅ ⊆ 𝑈
38	36, 37	eqsstri 3977	. . . . 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 781	. . . . . . . 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 20345	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
44	27	crngmgp 20447	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ CMnd)
45	44	ad2antlr 740	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑀 ∈ CMnd)
46		ovexd 7447	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (0..^(♯‘𝑓)) ∈ V)
47		wrdf 14643	. . . . . . . . . 10 ⊢ (𝑓 ∈ Word 𝐵 → 𝑓:(0..^(♯‘𝑓))⟶𝐵)
48	47	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑓:(0..^(♯‘𝑓))⟶𝐵)
49		fvexd 6892	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (1r‘𝑅) ∈ V)
50		simplll 787	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑓 ∈ Word 𝐵)
51	49, 50	wrdfsupp 33486	. . . . . . . . 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 20109	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → (𝑀 Σg 𝑓) ∈ 𝐵)
53		simpllr 788	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑥 ∈ 𝐵)
54		eqid 2761	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
55	32, 54, 42	unitmulclb 20591	. . . . . . . 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 1398	. . . . . . 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 490	. . . . . . . . 9 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈))
58		vex 3455	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
59	58	snss 4745	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑈 ↔ {𝑥} ⊆ 𝑈)
60		s1rn 14726	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → ran ⟨“𝑥”⟩ = {𝑥})
61	60	sseq1d 3962	. . . . . . . . . . 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 644	. . . . . . . 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 4136	. . . . . . . 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 20445	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
69	31, 68	syl 18	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑀 ∈ Mnd)
70	69	ad2antlr 740	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → 𝑀 ∈ Mnd)
71	27, 54	mgpplusg 20344	. . . . . . . . 9 ⊢ (.r‘𝑅) = (+g‘𝑀)
72	43, 71	gsumccatsn 19019	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ 𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑀 Σg (𝑓 ++ ⟨“𝑥”⟩)) = ((𝑀 Σg 𝑓)(.r‘𝑅)𝑥))
73	70, 50, 53, 72	syl3anc 1398	. . . . . . 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 2846	. . . . . 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 14730	. . . . . . . 8 ⊢ ((((𝑓 ∈ Word 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑅 ∈ CRing) ∧ ((𝑀 Σg 𝑓) ∈ 𝑈 ↔ ran 𝑓 ⊆ 𝑈)) → ⟨“𝑥”⟩ ∈ Word 𝐵)
76		ccatrn 14715	. . . . . . . 8 ⊢ ((𝑓 ∈ Word 𝐵 ∧ ⟨“𝑥”⟩ ∈ Word 𝐵) → ran (𝑓 ++ ⟨“𝑥”⟩) = (ran 𝑓 ∪ ran ⟨“𝑥”⟩))
77	50, 75, 76	syl2anc 596	. . . . . . 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 3962	. . . . . 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 425	. . . 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 14851	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  {csn 4584  ran crn 5652  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  0cc0 11181  ..^cfzo 13768  ♯chash 14454  Word cword 14638   ++ cconcat 14695  ⟨“cs1 14722  Basecbs 17367  ._rcmulr 17409   Σ_g cgsu 17591  Mndcmnd 18903  CMndccmn 19974  mulGrpcmgp 20340  1_rcur 20387  Ringcrg 20439  CRingccrg 20440  Unitcui 20565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8162  df-tpos 8227  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9338  df-oi 9488  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-xnn0 12661  df-z 12675  df-uz 12947  df-fz 13621  df-fzo 13769  df-seq 14125  df-hash 14455  df-word 14639  df-lsw 14688  df-concat 14696  df-s1 14723  df-substr 14769  df-pfx 14801  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-plusg 17421  df-mulr 17422  df-0g 17592  df-gsum 17593  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-submnd 18959  df-grp 19127  df-minusg 19128  df-cntz 19511  df-cmn 19976  df-abl 19977  df-mgp 20341  df-rng 20355  df-ur 20388  df-ring 20441  df-cring 20442  df-oppr 20547  df-dvdsr 20567  df-unit 20568

This theorem is used by:  1arithidom  34051