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Theorem ablfac1c 18470

Description: The factors of ablfac1b 18469 cover the entire group. (Contributed by Mario Carneiro, 21-Apr-2016.)

**Hypotheses**
Ref	Expression
ablfac1.b	⊢ 𝐵 = (Base‘𝐺)
ablfac1.o	⊢ 𝑂 = (od‘𝐺)
ablfac1.s	⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})
ablfac1.g	⊢ (𝜑 → 𝐺 ∈ Abel)
ablfac1.f	⊢ (𝜑 → 𝐵 ∈ Fin)
ablfac1.1	⊢ (𝜑 → 𝐴 ⊆ ℙ)
ablfac1c.d	⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (#‘𝐵)}
ablfac1.2	⊢ (𝜑 → 𝐷 ⊆ 𝐴)

**Assertion**
Ref	Expression
ablfac1c	⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝐵)

Distinct variable groups: 𝑤,𝑝,𝑥,𝐵 𝐷,𝑝,𝑥 𝜑,𝑝,𝑤,𝑥 𝐴,𝑝,𝑥 𝑂,𝑝,𝑥 𝐺,𝑝,𝑥 Allowed substitution hints: 𝐴(𝑤) 𝐷(𝑤) 𝑆(𝑥,𝑤,𝑝) 𝐺(𝑤) 𝑂(𝑤)

Proof of Theorem ablfac1c Dummy variable 𝑞 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ablfac1.f	. 2 ⊢ (𝜑 → 𝐵 ∈ Fin)
2		ablfac1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	2	dprdssv 18415	. . 3 ⊢ (𝐺 DProd 𝑆) ⊆ 𝐵
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝐵)
5		ssfi 8180	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd 𝑆) ⊆ 𝐵) → (𝐺 DProd 𝑆) ∈ Fin)
6	1, 3, 5	sylancl 694	. . . . 5 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ Fin)
7		hashcl 13147	. . . . 5 ⊢ ((𝐺 DProd 𝑆) ∈ Fin → (#‘(𝐺 DProd 𝑆)) ∈ ℕ₀)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (#‘(𝐺 DProd 𝑆)) ∈ ℕ₀)
9		hashcl 13147	. . . . 5 ⊢ (𝐵 ∈ Fin → (#‘𝐵) ∈ ℕ₀)
10	1, 9	syl 17	. . . 4 ⊢ (𝜑 → (#‘𝐵) ∈ ℕ₀)
11		ablfac1.o	. . . . . . 7 ⊢ 𝑂 = (od‘𝐺)
12		ablfac1.s	. . . . . . 7 ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})
13		ablfac1.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Abel)
14		ablfac1.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
15	2, 11, 12, 13, 1, 14	ablfac1b 18469	. . . . . 6 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
16		dprdsubg 18423	. . . . . 6 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
17	15, 16	syl 17	. . . . 5 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
18	2	lagsubg 17656	. . . . 5 ⊢ (((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) ∧ 𝐵 ∈ Fin) → (#‘(𝐺 DProd 𝑆)) ∥ (#‘𝐵))
19	17, 1, 18	syl2anc 693	. . . 4 ⊢ (𝜑 → (#‘(𝐺 DProd 𝑆)) ∥ (#‘𝐵))
20		breq1 4656	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑞 → (𝑤 ∥ (#‘𝐵) ↔ 𝑞 ∥ (#‘𝐵)))
21		ablfac1c.d	. . . . . . . . . . 11 ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (#‘𝐵)}
22	20, 21	elrab2 3366	. . . . . . . . . 10 ⊢ (𝑞 ∈ 𝐷 ↔ (𝑞 ∈ ℙ ∧ 𝑞 ∥ (#‘𝐵)))
23		ablfac1.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
24	23	sseld 3602	. . . . . . . . . 10 ⊢ (𝜑 → (𝑞 ∈ 𝐷 → 𝑞 ∈ 𝐴))
25	22, 24	syl5bir 233	. . . . . . . . 9 ⊢ (𝜑 → ((𝑞 ∈ ℙ ∧ 𝑞 ∥ (#‘𝐵)) → 𝑞 ∈ 𝐴))
26	25	impl 650	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (#‘𝐵)) → 𝑞 ∈ 𝐴)
27	2, 11, 12, 13, 1, 14	ablfac1a 18468	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (#‘(𝑆‘𝑞)) = (𝑞↑(𝑞 pCnt (#‘𝐵))))
28		fvex 6201	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘𝐺) ∈ V
29	2, 28	eqeltri 2697	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐵 ∈ V
30	29	rabex 4813	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} ∈ V
31	30, 12	dmmpti 6023	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑆 = 𝐴
32	31	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝑆 = 𝐴)
33	15, 32	dprdf2 18406	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
34	33	ffvelrnda 6359	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
35	15	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd 𝑆)
36	31	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → dom 𝑆 = 𝐴)
37		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴)
38	35, 36, 37	dprdub 18424	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ⊆ (𝐺 DProd 𝑆))
39	17	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
40		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ↾_s (𝐺 DProd 𝑆)) = (𝐺 ↾_s (𝐺 DProd 𝑆))
41	40	subsubg 17617	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) → ((𝑆‘𝑞) ∈ (SubGrp‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ↔ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) ∧ (𝑆‘𝑞) ⊆ (𝐺 DProd 𝑆))))
42	39, 41	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑆‘𝑞) ∈ (SubGrp‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ↔ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) ∧ (𝑆‘𝑞) ⊆ (𝐺 DProd 𝑆))))
43	34, 38, 42	mpbir2and 957	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘(𝐺 ↾_s (𝐺 DProd 𝑆))))
44	40	subgbas 17598	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) → (𝐺 DProd 𝑆) = (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))))
45	39, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) = (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))))
46	6	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) ∈ Fin)
47	45, 46	eqeltrrd 2702	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ∈ Fin)
48		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) = (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))
49	48	lagsubg 17656	. . . . . . . . . . . . 13 ⊢ (((𝑆‘𝑞) ∈ (SubGrp‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ∧ (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ∈ Fin) → (#‘(𝑆‘𝑞)) ∥ (#‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
50	43, 47, 49	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (#‘(𝑆‘𝑞)) ∥ (#‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
51	45	fveq2d 6195	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (#‘(𝐺 DProd 𝑆)) = (#‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
52	50, 51	breqtrrd 4681	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (#‘(𝑆‘𝑞)) ∥ (#‘(𝐺 DProd 𝑆)))
53	27, 52	eqbrtrrd 4677	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (#‘𝐵))) ∥ (#‘(𝐺 DProd 𝑆)))
54	14	sselda 3603	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℙ)
55	8	nn0zd 11480	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘(𝐺 DProd 𝑆)) ∈ ℤ)
56	55	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (#‘(𝐺 DProd 𝑆)) ∈ ℤ)
57		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → 𝑞 ∈ ℙ)
58		ablgrp 18198	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
59	2	grpbn0 17451	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
60	13, 58, 59	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≠ ∅)
61		hashnncl 13157	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Fin → ((#‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
62	1, 61	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((#‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
63	60, 62	mpbird 247	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (#‘𝐵) ∈ ℕ)
64	63	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (#‘𝐵) ∈ ℕ)
65	57, 64	pccld 15555	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (#‘𝐵)) ∈ ℕ₀)
66	54, 65	syldan 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (#‘𝐵)) ∈ ℕ₀)
67		pcdvdsb 15573	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ ℙ ∧ (#‘(𝐺 DProd 𝑆)) ∈ ℤ ∧ (𝑞 pCnt (#‘𝐵)) ∈ ℕ₀) → ((𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))) ↔ (𝑞↑(𝑞 pCnt (#‘𝐵))) ∥ (#‘(𝐺 DProd 𝑆))))
68	54, 56, 66, 67	syl3anc 1326	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))) ↔ (𝑞↑(𝑞 pCnt (#‘𝐵))) ∥ (#‘(𝐺 DProd 𝑆))))
69	53, 68	mpbird 247	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
70	69	adantlr 751	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
71	26, 70	syldan 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (#‘𝐵)) → (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
72		pceq0 15575	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℙ ∧ (#‘𝐵) ∈ ℕ) → ((𝑞 pCnt (#‘𝐵)) = 0 ↔ ¬ 𝑞 ∥ (#‘𝐵)))
73	57, 64, 72	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt (#‘𝐵)) = 0 ↔ ¬ 𝑞 ∥ (#‘𝐵)))
74	73	biimpar 502	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (#‘𝐵)) → (𝑞 pCnt (#‘𝐵)) = 0)
75		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
76	75	subg0cl 17602	. . . . . . . . . . . . . 14 ⊢ ((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝐺 DProd 𝑆))
77		ne0i 3921	. . . . . . . . . . . . . 14 ⊢ ((0_g‘𝐺) ∈ (𝐺 DProd 𝑆) → (𝐺 DProd 𝑆) ≠ ∅)
78	17, 76, 77	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 DProd 𝑆) ≠ ∅)
79		hashnncl 13157	. . . . . . . . . . . . . 14 ⊢ ((𝐺 DProd 𝑆) ∈ Fin → ((#‘(𝐺 DProd 𝑆)) ∈ ℕ ↔ (𝐺 DProd 𝑆) ≠ ∅))
80	6, 79	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((#‘(𝐺 DProd 𝑆)) ∈ ℕ ↔ (𝐺 DProd 𝑆) ≠ ∅))
81	78, 80	mpbird 247	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘(𝐺 DProd 𝑆)) ∈ ℕ)
82	81	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (#‘(𝐺 DProd 𝑆)) ∈ ℕ)
83	57, 82	pccld 15555	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (#‘(𝐺 DProd 𝑆))) ∈ ℕ₀)
84	83	nn0ge0d 11354	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → 0 ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
85	84	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (#‘𝐵)) → 0 ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
86	74, 85	eqbrtrd 4675	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (#‘𝐵)) → (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
87	71, 86	pm2.61dan 832	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
88	87	ralrimiva 2966	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ ℙ (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
89	10	nn0zd 11480	. . . . . 6 ⊢ (𝜑 → (#‘𝐵) ∈ ℤ)
90		pc2dvds 15583	. . . . . 6 ⊢ (((#‘𝐵) ∈ ℤ ∧ (#‘(𝐺 DProd 𝑆)) ∈ ℤ) → ((#‘𝐵) ∥ (#‘(𝐺 DProd 𝑆)) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆)))))
91	89, 55, 90	syl2anc 693	. . . . 5 ⊢ (𝜑 → ((#‘𝐵) ∥ (#‘(𝐺 DProd 𝑆)) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆)))))
92	88, 91	mpbird 247	. . . 4 ⊢ (𝜑 → (#‘𝐵) ∥ (#‘(𝐺 DProd 𝑆)))
93		dvdseq 15036	. . . 4 ⊢ ((((#‘(𝐺 DProd 𝑆)) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) ∧ ((#‘(𝐺 DProd 𝑆)) ∥ (#‘𝐵) ∧ (#‘𝐵) ∥ (#‘(𝐺 DProd 𝑆)))) → (#‘(𝐺 DProd 𝑆)) = (#‘𝐵))
94	8, 10, 19, 92, 93	syl22anc 1327	. . 3 ⊢ (𝜑 → (#‘(𝐺 DProd 𝑆)) = (#‘𝐵))
95		hashen 13135	. . . 4 ⊢ (((𝐺 DProd 𝑆) ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘(𝐺 DProd 𝑆)) = (#‘𝐵) ↔ (𝐺 DProd 𝑆) ≈ 𝐵))
96	6, 1, 95	syl2anc 693	. . 3 ⊢ (𝜑 → ((#‘(𝐺 DProd 𝑆)) = (#‘𝐵) ↔ (𝐺 DProd 𝑆) ≈ 𝐵))
97	94, 96	mpbid 222	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ≈ 𝐵)
98		fisseneq 8171	. 2 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd 𝑆) ⊆ 𝐵 ∧ (𝐺 DProd 𝑆) ≈ 𝐵) → (𝐺 DProd 𝑆) = 𝐵)
99	1, 4, 97, 98	syl3anc 1326	1 ⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 ≈ cen 7952 Fincfn 7955 0cc0 9936 ≤ cle 10075 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ↑cexp 12860 #chash 13117 ∥ cdvds 14983 ℙcprime 15385 pCnt cpc 15541 Basecbs 15857 ↾_s cress 15858 0_gc0g 16100 Grpcgrp 17422 SubGrpcsubg 17588 odcod 17944 Abelcabl 18194 DProd cdprd 18392

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-eqg 17593 df-ghm 17658 df-gim 17701 df-ga 17723 df-cntz 17750 df-oppg 17776 df-od 17948 df-lsm 18051 df-pj1 18052 df-cmn 18195 df-abl 18196 df-dprd 18394

This theorem is referenced by: ablfaclem2 18485

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