ablfac1c - Metamath Proof Explorer

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Theorem ablfac1c 20003

Description: The factors of ablfac1b 20002 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 19948	. . 3 ⊢ (𝐺 DProd 𝑆) ⊆ 𝐵
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝐵)
5		ssfi 9137	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd 𝑆) ⊆ 𝐵) → (𝐺 DProd 𝑆) ∈ Fin)
6	1, 3, 5	sylancl 586	. . . . 5 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ Fin)
7		hashcl 14321	. . . . 5 ⊢ ((𝐺 DProd 𝑆) ∈ Fin → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ₀)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ₀)
9		hashcl 14321	. . . . 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 20002	. . . . . 6 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
16		dprdsubg 19956	. . . . . 6 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
17	15, 16	syl 17	. . . . 5 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
18	2	lagsubg 19127	. . . . 5 ⊢ (((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) ∧ 𝐵 ∈ Fin) → (♯‘(𝐺 DProd 𝑆)) ∥ (♯‘𝐵))
19	17, 1, 18	syl2anc 584	. . . 4 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∥ (♯‘𝐵))
20		breq1 5110	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑞 → (𝑤 ∥ (♯‘𝐵) ↔ 𝑞 ∥ (♯‘𝐵)))
21		ablfac1c.d	. . . . . . . . . . 11 ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
22	20, 21	elrab2 3662	. . . . . . . . . 10 ⊢ (𝑞 ∈ 𝐷 ↔ (𝑞 ∈ ℙ ∧ 𝑞 ∥ (♯‘𝐵)))
23		ablfac1.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
24	23	sseld 3945	. . . . . . . . . 10 ⊢ (𝜑 → (𝑞 ∈ 𝐷 → 𝑞 ∈ 𝐴))
25	22, 24	biimtrrid 243	. . . . . . . . 9 ⊢ (𝜑 → ((𝑞 ∈ ℙ ∧ 𝑞 ∥ (♯‘𝐵)) → 𝑞 ∈ 𝐴))
26	25	impl 455	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (♯‘𝐵)) → 𝑞 ∈ 𝐴)
27	2, 11, 12, 13, 1, 14	ablfac1a 20001	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
28	2	fvexi 6872	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐵 ∈ V
29	28	rabex 5294	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
30	29, 12	dmmpti 6662	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑆 = 𝐴
31	30	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝑆 = 𝐴)
32	15, 31	dprdf2 19939	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
33	32	ffvelcdmda 7056	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
34	15	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd 𝑆)
35	30	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → dom 𝑆 = 𝐴)
36		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴)
37	34, 35, 36	dprdub 19957	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ⊆ (𝐺 DProd 𝑆))
38	17	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
39		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ↾_s (𝐺 DProd 𝑆)) = (𝐺 ↾_s (𝐺 DProd 𝑆))
40	39	subsubg 19081	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) → ((𝑆‘𝑞) ∈ (SubGrp‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ↔ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) ∧ (𝑆‘𝑞) ⊆ (𝐺 DProd 𝑆))))
41	38, 40	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑆‘𝑞) ∈ (SubGrp‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ↔ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) ∧ (𝑆‘𝑞) ⊆ (𝐺 DProd 𝑆))))
42	33, 37, 41	mpbir2and 713	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘(𝐺 ↾_s (𝐺 DProd 𝑆))))
43	39	subgbas 19062	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) → (𝐺 DProd 𝑆) = (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))))
44	38, 43	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) = (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))))
45	6	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) ∈ Fin)
46	44, 45	eqeltrrd 2829	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ∈ Fin)
47		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) = (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))
48	47	lagsubg 19127	. . . . . . . . . . . . 13 ⊢ (((𝑆‘𝑞) ∈ (SubGrp‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ∧ (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ∈ Fin) → (♯‘(𝑆‘𝑞)) ∥ (♯‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
49	42, 46, 48	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) ∥ (♯‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
50	44	fveq2d 6862	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd 𝑆)) = (♯‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
51	49, 50	breqtrrd 5135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) ∥ (♯‘(𝐺 DProd 𝑆)))
52	27, 51	eqbrtrrd 5131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘(𝐺 DProd 𝑆)))
53	14	sselda 3946	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℙ)
54	8	nn0zd 12555	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∈ ℤ)
55	54	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd 𝑆)) ∈ ℤ)
56		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → 𝑞 ∈ ℙ)
57		ablgrp 19715	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
58	2	grpbn0 18898	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
59	13, 57, 58	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≠ ∅)
60		hashnncl 14331	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
61	1, 60	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
62	59, 61	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
63	62	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (♯‘𝐵) ∈ ℕ)
64	56, 63	pccld 16821	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ₀)
65	53, 64	syldan 591	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ₀)
66		pcdvdsb 16840	. . . . . . . . . . 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))))
67	53, 55, 65, 66	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))) ↔ (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘(𝐺 DProd 𝑆))))
68	52, 67	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
69	68	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
70	26, 69	syldan 591	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (♯‘𝐵)) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
71		pceq0 16842	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → ((𝑞 pCnt (♯‘𝐵)) = 0 ↔ ¬ 𝑞 ∥ (♯‘𝐵)))
72	56, 63, 71	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt (♯‘𝐵)) = 0 ↔ ¬ 𝑞 ∥ (♯‘𝐵)))
73	72	biimpar 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (♯‘𝐵)) → (𝑞 pCnt (♯‘𝐵)) = 0)
74		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
75	74	subg0cl 19066	. . . . . . . . . . . . . 14 ⊢ ((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝐺 DProd 𝑆))
76		ne0i 4304	. . . . . . . . . . . . . 14 ⊢ ((0_g‘𝐺) ∈ (𝐺 DProd 𝑆) → (𝐺 DProd 𝑆) ≠ ∅)
77	17, 75, 76	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 DProd 𝑆) ≠ ∅)
78		hashnncl 14331	. . . . . . . . . . . . . 14 ⊢ ((𝐺 DProd 𝑆) ∈ Fin → ((♯‘(𝐺 DProd 𝑆)) ∈ ℕ ↔ (𝐺 DProd 𝑆) ≠ ∅))
79	6, 78	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((♯‘(𝐺 DProd 𝑆)) ∈ ℕ ↔ (𝐺 DProd 𝑆) ≠ ∅))
80	77, 79	mpbird 257	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ)
81	80	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ)
82	56, 81	pccld 16821	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))) ∈ ℕ₀)
83	82	nn0ge0d 12506	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → 0 ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
84	83	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (♯‘𝐵)) → 0 ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
85	73, 84	eqbrtrd 5129	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (♯‘𝐵)) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
86	70, 85	pm2.61dan 812	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
87	86	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ ℙ (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
88	10	nn0zd 12555	. . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
89		pc2dvds 16850	. . . . . 6 ⊢ (((♯‘𝐵) ∈ ℤ ∧ (♯‘(𝐺 DProd 𝑆)) ∈ ℤ) → ((♯‘𝐵) ∥ (♯‘(𝐺 DProd 𝑆)) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆)))))
90	88, 54, 89	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((♯‘𝐵) ∥ (♯‘(𝐺 DProd 𝑆)) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆)))))
91	87, 90	mpbird 257	. . . 4 ⊢ (𝜑 → (♯‘𝐵) ∥ (♯‘(𝐺 DProd 𝑆)))
92		dvdseq 16284	. . . 4 ⊢ ((((♯‘(𝐺 DProd 𝑆)) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) ∧ ((♯‘(𝐺 DProd 𝑆)) ∥ (♯‘𝐵) ∧ (♯‘𝐵) ∥ (♯‘(𝐺 DProd 𝑆)))) → (♯‘(𝐺 DProd 𝑆)) = (♯‘𝐵))
93	8, 10, 19, 91, 92	syl22anc 838	. . 3 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) = (♯‘𝐵))
94		hashen 14312	. . . 4 ⊢ (((𝐺 DProd 𝑆) ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘(𝐺 DProd 𝑆)) = (♯‘𝐵) ↔ (𝐺 DProd 𝑆) ≈ 𝐵))
95	6, 1, 94	syl2anc 584	. . 3 ⊢ (𝜑 → ((♯‘(𝐺 DProd 𝑆)) = (♯‘𝐵) ↔ (𝐺 DProd 𝑆) ≈ 𝐵))
96	93, 95	mpbid 232	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ≈ 𝐵)
97		fisseneq 9204	. 2 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd 𝑆) ⊆ 𝐵 ∧ (𝐺 DProd 𝑆) ≈ 𝐵) → (𝐺 DProd 𝑆) = 𝐵)
98	1, 4, 96, 97	syl3anc 1373	1 ⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 {crab 3405 ⊆ wss 3914 ∅c0 4296 class class class wbr 5107 ↦ cmpt 5188 dom cdm 5638 ‘cfv 6511 (class class class)co 7387 ≈ cen 8915 Fincfn 8918 0cc0 11068 ≤ cle 11209 ℕcn 12186 ℕ₀cn0 12442 ℤcz 12529 ↑cexp 14026 ♯chash 14295 ∥ cdvds 16222 ℙcprime 16641 pCnt cpc 16807 Basecbs 17179 ↾_s cress 17200 0_gc0g 17402 Grpcgrp 18865 SubGrpcsubg 19052 odcod 19454 Abelcabl 19711 DProd cdprd 19925

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-inf2 9594 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 ax-pre-sup 11146

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-iin 4958 df-disj 5075 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-of 7653 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-2o 8435 df-oadd 8438 df-omul 8439 df-er 8671 df-ec 8673 df-qs 8677 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-xnn0 12516 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-dvds 16223 df-gcd 16465 df-prm 16642 df-pc 16808 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-0g 17404 df-gsum 17405 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-eqg 19057 df-ghm 19145 df-gim 19191 df-ga 19222 df-cntz 19249 df-oppg 19278 df-od 19458 df-lsm 19566 df-pj1 19567 df-cmn 19712 df-abl 19713 df-dprd 19927

This theorem is referenced by: ablfaclem2 20018

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