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Theorem ablfac1c 20096

Description: The factors of ablfac1b 20095 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 20041	. . 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 595	. . . . 5 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ Fin)
7		hashcl 14366	. . . . 5 ⊢ ((𝐺 DProd 𝑆) ∈ Fin → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ₀)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ₀)
9		hashcl 14366	. . . . 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 20095	. . . . . 6 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
16		dprdsubg 20049	. . . . . 6 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
17	15, 16	syl 17	. . . . 5 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
18	2	lagsubg 19219	. . . . 5 ⊢ (((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) ∧ 𝐵 ∈ Fin) → (♯‘(𝐺 DProd 𝑆)) ∥ (♯‘𝐵))
19	17, 1, 18	syl2anc 593	. . . 4 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∥ (♯‘𝐵))
20		breq1 5102	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑞 → (𝑤 ∥ (♯‘𝐵) ↔ 𝑞 ∥ (♯‘𝐵)))
21		ablfac1c.d	. . . . . . . . . . 11 ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
22	20, 21	elrab2 3653	. . . . . . . . . 10 ⊢ (𝑞 ∈ 𝐷 ↔ (𝑞 ∈ ℙ ∧ 𝑞 ∥ (♯‘𝐵)))
23		ablfac1.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
24	23	sseld 3935	. . . . . . . . . 10 ⊢ (𝜑 → (𝑞 ∈ 𝐷 → 𝑞 ∈ 𝐴))
25	22, 24	biimtrrid 245	. . . . . . . . 9 ⊢ (𝜑 → ((𝑞 ∈ ℙ ∧ 𝑞 ∥ (♯‘𝐵)) → 𝑞 ∈ 𝐴))
26	25	impl 459	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (♯‘𝐵)) → 𝑞 ∈ 𝐴)
27	2, 11, 12, 13, 1, 14	ablfac1a 20094	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
28	2	fvexi 6877	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐵 ∈ V
29	28	rabex 5294	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
30	29, 12	dmmpti 6661	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑆 = 𝐴
31	30	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝑆 = 𝐴)
32	15, 31	dprdf2 20032	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
33	32	ffvelcdmda 7061	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
34	15	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd 𝑆)
35	30	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → dom 𝑆 = 𝐴)
36		simpr 488	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴)
37	34, 35, 36	dprdub 20050	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ⊆ (𝐺 DProd 𝑆))
38	17	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
39		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ↾_s (𝐺 DProd 𝑆)) = (𝐺 ↾_s (𝐺 DProd 𝑆))
40	39	subsubg 19174	. . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘(𝐺 ↾_s (𝐺 DProd 𝑆))))
43	39	subgbas 19155	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) ∈ Fin)
46	44, 45	eqeltrrd 2862	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ∈ Fin)
47		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) = (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))
48	47	lagsubg 19219	. . . . . . . . . . . . 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 593	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) ∥ (♯‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
50	44	fveq2d 6867	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd 𝑆)) = (♯‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
51	49, 50	breqtrrd 5127	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) ∥ (♯‘(𝐺 DProd 𝑆)))
52	27, 51	eqbrtrrd 5123	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘(𝐺 DProd 𝑆)))
53	14	sselda 3936	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℙ)
54	8	nn0zd 12590	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∈ ℤ)
55	54	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd 𝑆)) ∈ ℤ)
56		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → 𝑞 ∈ ℙ)
57		ablgrp 19808	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
58	2	grpbn0 18991	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
59	13, 57, 58	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≠ ∅)
60		hashnncl 14376	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
61	1, 60	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
62	59, 61	mpbird 259	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
63	62	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (♯‘𝐵) ∈ ℕ)
64	56, 63	pccld 16869	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ₀)
65	53, 64	syldan 600	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ₀)
66		pcdvdsb 16888	. . . . . . . . . . 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 1389	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))) ↔ (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘(𝐺 DProd 𝑆))))
68	52, 67	mpbird 259	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
69	68	adantlr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
70	26, 69	syldan 600	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (♯‘𝐵)) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
71		pceq0 16890	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → ((𝑞 pCnt (♯‘𝐵)) = 0 ↔ ¬ 𝑞 ∥ (♯‘𝐵)))
72	56, 63, 71	syl2anc 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt (♯‘𝐵)) = 0 ↔ ¬ 𝑞 ∥ (♯‘𝐵)))
73	72	biimpar 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (♯‘𝐵)) → (𝑞 pCnt (♯‘𝐵)) = 0)
74		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
75	74	subg0cl 19159	. . . . . . . . . . . . . 14 ⊢ ((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝐺 DProd 𝑆))
76		ne0i 4293	. . . . . . . . . . . . . 14 ⊢ ((0_g‘𝐺) ∈ (𝐺 DProd 𝑆) → (𝐺 DProd 𝑆) ≠ ∅)
77	17, 75, 76	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 DProd 𝑆) ≠ ∅)
78		hashnncl 14376	. . . . . . . . . . . . . 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 259	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ)
81	80	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ)
82	56, 81	pccld 16869	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))) ∈ ℕ₀)
83	82	nn0ge0d 12542	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → 0 ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
84	83	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (♯‘𝐵)) → 0 ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
85	73, 84	eqbrtrd 5121	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (♯‘𝐵)) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
86	70, 85	pm2.61dan 822	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
87	86	ralrimiva 3153	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ ℙ (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
88	10	nn0zd 12590	. . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
89		pc2dvds 16898	. . . . . 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 593	. . . . 5 ⊢ (𝜑 → ((♯‘𝐵) ∥ (♯‘(𝐺 DProd 𝑆)) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆)))))
91	87, 90	mpbird 259	. . . 4 ⊢ (𝜑 → (♯‘𝐵) ∥ (♯‘(𝐺 DProd 𝑆)))
92		dvdseq 16331	. . . 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 849	. . 3 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) = (♯‘𝐵))
94		hashen 14357	. . . 4 ⊢ (((𝐺 DProd 𝑆) ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘(𝐺 DProd 𝑆)) = (♯‘𝐵) ↔ (𝐺 DProd 𝑆) ≈ 𝐵))
95	6, 1, 94	syl2anc 593	. . 3 ⊢ (𝜑 → ((♯‘(𝐺 DProd 𝑆)) = (♯‘𝐵) ↔ (𝐺 DProd 𝑆) ≈ 𝐵))
96	93, 95	mpbid 234	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ≈ 𝐵)
97		fisseneq 9203	. 2 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd 𝑆) ⊆ 𝐵 ∧ (𝐺 DProd 𝑆) ≈ 𝐵) → (𝐺 DProd 𝑆) = 𝐵)
98	1, 4, 96, 97	syl3anc 1389	1 ⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 {crab 3413 ⊆ wss 3904 ∅c0 4285 class class class wbr 5099 ↦ cmpt 5180 dom cdm 5645 ‘cfv 6517 (class class class)co 7392 ≈ cen 8920 Fincfn 8923 0cc0 11070 ≤ cle 11214 ℕcn 12207 ℕ₀cn0 12478 ℤcz 12565 ↑cexp 14071 ♯chash 14340 ∥ cdvds 16269 ℙcprime 16688 pCnt cpc 16855 Basecbs 17228 ↾_s cress 17249 0_gc0g 17451 Grpcgrp 18958 SubGrpcsubg 19145 odcod 19547 Abelcabl 19804 DProd cdprd 20018

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-inf2 9593 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-disj 5067 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-tpos 8201 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-oadd 8436 df-omul 8437 df-er 8673 df-ec 8675 df-qs 8679 df-map 8805 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-sup 9385 df-inf 9386 df-oi 9455 df-dju 9856 df-card 9894 df-acn 9897 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-n0 12479 df-xnn0 12552 df-z 12566 df-uz 12837 df-q 12947 df-rp 12991 df-fz 13510 df-fzo 13657 df-fl 13799 df-mod 13877 df-seq 14012 df-exp 14072 df-fac 14284 df-bc 14313 df-hash 14341 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-clim 15498 df-sum 15697 df-dvds 16270 df-gcd 16512 df-prm 16689 df-pc 16856 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-0g 17453 df-gsum 17454 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-mhm 18800 df-submnd 18801 df-grp 18961 df-minusg 18962 df-sbg 18963 df-mulg 19093 df-subg 19148 df-eqg 19150 df-ghm 19237 df-gim 19282 df-ga 19313 df-cntz 19340 df-oppg 19369 df-od 19551 df-lsm 19659 df-pj1 19660 df-cmn 19805 df-abl 19806 df-dprd 20020

This theorem is referenced by: ablfaclem2 20111

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