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Theorem ablfac1c 19412

Description: The factors of ablfac1b 19411 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 19357	. . 3 ⊢ (𝐺 DProd 𝑆) ⊆ 𝐵
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝐵)
5		ssfi 8829	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd 𝑆) ⊆ 𝐵) → (𝐺 DProd 𝑆) ∈ Fin)
6	1, 3, 5	sylancl 589	. . . . 5 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ Fin)
7		hashcl 13888	. . . . 5 ⊢ ((𝐺 DProd 𝑆) ∈ Fin → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ₀)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ₀)
9		hashcl 13888	. . . . 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 19411	. . . . . 6 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
16		dprdsubg 19365	. . . . . 6 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
17	15, 16	syl 17	. . . . 5 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
18	2	lagsubg 18560	. . . . 5 ⊢ (((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) ∧ 𝐵 ∈ Fin) → (♯‘(𝐺 DProd 𝑆)) ∥ (♯‘𝐵))
19	17, 1, 18	syl2anc 587	. . . 4 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∥ (♯‘𝐵))
20		breq1 5042	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑞 → (𝑤 ∥ (♯‘𝐵) ↔ 𝑞 ∥ (♯‘𝐵)))
21		ablfac1c.d	. . . . . . . . . . 11 ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
22	20, 21	elrab2 3594	. . . . . . . . . 10 ⊢ (𝑞 ∈ 𝐷 ↔ (𝑞 ∈ ℙ ∧ 𝑞 ∥ (♯‘𝐵)))
23		ablfac1.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
24	23	sseld 3886	. . . . . . . . . 10 ⊢ (𝜑 → (𝑞 ∈ 𝐷 → 𝑞 ∈ 𝐴))
25	22, 24	syl5bir 246	. . . . . . . . 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 19410	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
28	2	fvexi 6709	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐵 ∈ V
29	28	rabex 5210	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
30	29, 12	dmmpti 6500	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑆 = 𝐴
31	30	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝑆 = 𝐴)
32	15, 31	dprdf2 19348	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
33	32	ffvelrnda 6882	. . . . . . . . . . . . . 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 19366	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ⊆ (𝐺 DProd 𝑆))
38	17	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
39		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ↾_s (𝐺 DProd 𝑆)) = (𝐺 ↾_s (𝐺 DProd 𝑆))
40	39	subsubg 18520	. . . . . . . . . . . . . . 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 18501	. . . . . . . . . . . . . . 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 2832	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ∈ Fin)
47		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) = (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))
48	47	lagsubg 18560	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) ∥ (♯‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
50	44	fveq2d 6699	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd 𝑆)) = (♯‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
51	49, 50	breqtrrd 5067	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) ∥ (♯‘(𝐺 DProd 𝑆)))
52	27, 51	eqbrtrrd 5063	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘(𝐺 DProd 𝑆)))
53	14	sselda 3887	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℙ)
54	8	nn0zd 12245	. . . . . . . . . . . 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 19129	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
58	2	grpbn0 18350	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
59	13, 57, 58	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≠ ∅)
60		hashnncl 13898	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
61	1, 60	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
62	59, 61	mpbird 260	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
63	62	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (♯‘𝐵) ∈ ℕ)
64	56, 63	pccld 16366	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ₀)
65	53, 64	syldan 594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ₀)
66		pcdvdsb 16385	. . . . . . . . . . 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 260	. . . . . . . . 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 594	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (♯‘𝐵)) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
71		pceq0 16387	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → ((𝑞 pCnt (♯‘𝐵)) = 0 ↔ ¬ 𝑞 ∥ (♯‘𝐵)))
72	56, 63, 71	syl2anc 587	. . . . . . . . 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 2736	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
75	74	subg0cl 18505	. . . . . . . . . . . . . 14 ⊢ ((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝐺 DProd 𝑆))
76		ne0i 4235	. . . . . . . . . . . . . 14 ⊢ ((0_g‘𝐺) ∈ (𝐺 DProd 𝑆) → (𝐺 DProd 𝑆) ≠ ∅)
77	17, 75, 76	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 DProd 𝑆) ≠ ∅)
78		hashnncl 13898	. . . . . . . . . . . . . 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 260	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ)
81	80	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ)
82	56, 81	pccld 16366	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))) ∈ ℕ₀)
83	82	nn0ge0d 12118	. . . . . . . . 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 5061	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (♯‘𝐵)) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
86	70, 85	pm2.61dan 813	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
87	86	ralrimiva 3095	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ ℙ (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
88	10	nn0zd 12245	. . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
89		pc2dvds 16395	. . . . . 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 587	. . . . 5 ⊢ (𝜑 → ((♯‘𝐵) ∥ (♯‘(𝐺 DProd 𝑆)) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆)))))
91	87, 90	mpbird 260	. . . 4 ⊢ (𝜑 → (♯‘𝐵) ∥ (♯‘(𝐺 DProd 𝑆)))
92		dvdseq 15838	. . . 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 839	. . 3 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) = (♯‘𝐵))
94		hashen 13878	. . . 4 ⊢ (((𝐺 DProd 𝑆) ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘(𝐺 DProd 𝑆)) = (♯‘𝐵) ↔ (𝐺 DProd 𝑆) ≈ 𝐵))
95	6, 1, 94	syl2anc 587	. . 3 ⊢ (𝜑 → ((♯‘(𝐺 DProd 𝑆)) = (♯‘𝐵) ↔ (𝐺 DProd 𝑆) ≈ 𝐵))
96	93, 95	mpbid 235	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ≈ 𝐵)
97		fisseneq 8865	. 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 {crab 3055 ⊆ wss 3853 ∅c0 4223 class class class wbr 5039 ↦ cmpt 5120 dom cdm 5536 ‘cfv 6358 (class class class)co 7191 ≈ cen 8601 Fincfn 8604 0cc0 10694 ≤ cle 10833 ℕcn 11795 ℕ₀cn0 12055 ℤcz 12141 ↑cexp 13600 ♯chash 13861 ∥ cdvds 15778 ℙcprime 16191 pCnt cpc 16352 Basecbs 16666 ↾_s cress 16667 0_gc0g 16898 Grpcgrp 18319 SubGrpcsubg 18491 odcod 18870 Abelcabl 19125 DProd cdprd 19334

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-disj 5005 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-oadd 8184 df-omul 8185 df-er 8369 df-ec 8371 df-qs 8375 df-map 8488 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-sup 9036 df-inf 9037 df-oi 9104 df-dju 9482 df-card 9520 df-acn 9523 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-xnn0 12128 df-z 12142 df-uz 12404 df-q 12510 df-rp 12552 df-fz 13061 df-fzo 13204 df-fl 13332 df-mod 13408 df-seq 13540 df-exp 13601 df-fac 13805 df-bc 13834 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-clim 15014 df-sum 15215 df-dvds 15779 df-gcd 16017 df-prm 16192 df-pc 16353 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-0g 16900 df-gsum 16901 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-eqg 18496 df-ghm 18574 df-gim 18617 df-ga 18638 df-cntz 18665 df-oppg 18692 df-od 18874 df-lsm 18979 df-pj1 18980 df-cmn 19126 df-abl 19127 df-dprd 19336

This theorem is referenced by: ablfaclem2 19427

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