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Theorem ablfac1c 20174

Description: The factors of ablfac1b 20173 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 20119	. . 3 ⊢ (𝐺 DProd 𝑆) ⊆ 𝐵
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝐵)
5		ssfi 9167	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd 𝑆) ⊆ 𝐵) → (𝐺 DProd 𝑆) ∈ Fin)
6	1, 3, 5	sylancl 598	. . . . 5 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ Fin)
7		hashcl 14412	. . . . 5 ⊢ ((𝐺 DProd 𝑆) ∈ Fin → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ₀)
8	6, 7	syl 18	. . . 4 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ₀)
9		hashcl 14412	. . . . 5 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
10	1, 9	syl 18	. . . 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 20173	. . . . . 6 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
16		dprdsubg 20127	. . . . . 6 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
17	15, 16	syl 18	. . . . 5 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
18	2	lagsubg 19297	. . . . 5 ⊢ (((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) ∧ 𝐵 ∈ Fin) → (♯‘(𝐺 DProd 𝑆)) ∥ (♯‘𝐵))
19	17, 1, 18	syl2anc 596	. . . 4 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∥ (♯‘𝐵))
20		breq1 5117	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑞 → (𝑤 ∥ (♯‘𝐵) ↔ 𝑞 ∥ (♯‘𝐵)))
21		ablfac1c.d	. . . . . . . . . . 11 ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
22	20, 21	elrab2 3657	. . . . . . . . . 10 ⊢ (𝑞 ∈ 𝐷 ↔ (𝑞 ∈ ℙ ∧ 𝑞 ∥ (♯‘𝐵)))
23		ablfac1.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
24	23	sseld 3939	. . . . . . . . . 10 ⊢ (𝜑 → (𝑞 ∈ 𝐷 → 𝑞 ∈ 𝐴))
25	22, 24	biimtrrid 246	. . . . . . . . 9 ⊢ (𝜑 → ((𝑞 ∈ ℙ ∧ 𝑞 ∥ (♯‘𝐵)) → 𝑞 ∈ 𝐴))
26	25	impl 461	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (♯‘𝐵)) → 𝑞 ∈ 𝐴)
27	2, 11, 12, 13, 1, 14	ablfac1a 20172	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
28	2	fvexi 6902	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐵 ∈ V
29	28	rabex 5314	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
30	29, 12	dmmpti 6686	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑆 = 𝐴
31	30	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝑆 = 𝐴)
32	15, 31	dprdf2 20110	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
33	32	ffvelcdmda 7086	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
34	15	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd 𝑆)
35	30	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → dom 𝑆 = 𝐴)
36		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴)
37	34, 35, 36	dprdub 20128	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ⊆ (𝐺 DProd 𝑆))
38	17	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
39		eqid 2766	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ↾_s (𝐺 DProd 𝑆)) = (𝐺 ↾_s (𝐺 DProd 𝑆))
40	39	subsubg 19247	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘(𝐺 ↾_s (𝐺 DProd 𝑆))))
43	39	subgbas 19227	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) → (𝐺 DProd 𝑆) = (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))))
44	38, 43	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) = (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))))
45	6	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) ∈ Fin)
46	44, 45	eqeltrrd 2867	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ∈ Fin)
47		eqid 2766	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) = (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))
48	47	lagsubg 19297	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) ∥ (♯‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
50	44	fveq2d 6892	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd 𝑆)) = (♯‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
51	49, 50	breqtrrd 5144	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) ∥ (♯‘(𝐺 DProd 𝑆)))
52	27, 51	eqbrtrrd 5140	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘(𝐺 DProd 𝑆)))
53	14	sselda 3940	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℙ)
54	8	nn0zd 12634	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∈ ℤ)
55	54	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd 𝑆)) ∈ ℤ)
56		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → 𝑞 ∈ ℙ)
57		ablgrp 19886	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
58	2	grpbn0 19064	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
59	13, 57, 58	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≠ ∅)
60		hashnncl 14422	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
61	1, 60	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
62	59, 61	mpbird 260	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
63	62	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (♯‘𝐵) ∈ ℕ)
64	56, 63	pccld 16935	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ₀)
65	53, 64	syldan 603	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ₀)
66		pcdvdsb 16954	. . . . . . . . . . 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 1398	. . . . . . . . . 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 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
70	26, 69	syldan 603	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (♯‘𝐵)) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
71		pceq0 16956	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → ((𝑞 pCnt (♯‘𝐵)) = 0 ↔ ¬ 𝑞 ∥ (♯‘𝐵)))
72	56, 63, 71	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt (♯‘𝐵)) = 0 ↔ ¬ 𝑞 ∥ (♯‘𝐵)))
73	72	biimpar 483	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (♯‘𝐵)) → (𝑞 pCnt (♯‘𝐵)) = 0)
74		eqid 2766	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
75	74	subg0cl 19231	. . . . . . . . . . . . . 14 ⊢ ((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝐺 DProd 𝑆))
76		ne0i 4297	. . . . . . . . . . . . . 14 ⊢ ((0_g‘𝐺) ∈ (𝐺 DProd 𝑆) → (𝐺 DProd 𝑆) ≠ ∅)
77	17, 75, 76	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 DProd 𝑆) ≠ ∅)
78		hashnncl 14422	. . . . . . . . . . . . . 14 ⊢ ((𝐺 DProd 𝑆) ∈ Fin → ((♯‘(𝐺 DProd 𝑆)) ∈ ℕ ↔ (𝐺 DProd 𝑆) ≠ ∅))
79	6, 78	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((♯‘(𝐺 DProd 𝑆)) ∈ ℕ ↔ (𝐺 DProd 𝑆) ≠ ∅))
80	77, 79	mpbird 260	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ)
81	80	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (♯‘(𝐺 DProd 𝑆)) ∈ ℕ)
82	56, 81	pccld 16935	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))) ∈ ℕ₀)
83	82	nn0ge0d 12586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → 0 ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
84	83	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (♯‘𝐵)) → 0 ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
85	73, 84	eqbrtrd 5138	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (♯‘𝐵)) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
86	70, 85	pm2.61dan 825	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
87	86	ralrimiva 3160	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ ℙ (𝑞 pCnt (♯‘𝐵)) ≤ (𝑞 pCnt (♯‘(𝐺 DProd 𝑆))))
88	10	nn0zd 12634	. . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
89		pc2dvds 16964	. . . . . 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 596	. . . . 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 16397	. . . 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 852	. . 3 ⊢ (𝜑 → (♯‘(𝐺 DProd 𝑆)) = (♯‘𝐵))
94		hashen 14403	. . . 4 ⊢ (((𝐺 DProd 𝑆) ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘(𝐺 DProd 𝑆)) = (♯‘𝐵) ↔ (𝐺 DProd 𝑆) ≈ 𝐵))
95	6, 1, 94	syl2anc 596	. . 3 ⊢ (𝜑 → ((♯‘(𝐺 DProd 𝑆)) = (♯‘𝐵) ↔ (𝐺 DProd 𝑆) ≈ 𝐵))
96	93, 95	mpbid 235	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ≈ 𝐵)
97		fisseneq 9233	. 2 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd 𝑆) ⊆ 𝐵 ∧ (𝐺 DProd 𝑆) ≈ 𝐵) → (𝐺 DProd 𝑆) = 𝐵)
98	1, 4, 96, 97	syl3anc 1398	1 ⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∀wral 3082 {crab 3419 ⊆ wss 3908 ∅c0 4289 class class class wbr 5114 ↦ cmpt 5197 dom cdm 5666 ‘cfv 6543 (class class class)co 7423 ≈ cen 8949 Fincfn 8952 0cc0 11118 ≤ cle 11262 ℕcn 12251 ℕ₀cn0 12522 ℤcz 12609 ↑cexp 14117 ♯chash 14386 ∥ cdvds 16335 ℙcprime 16754 pCnt cpc 16921 Basecbs 17294 ↾_s cress 17315 0_gc0g 17517 Grpcgrp 19031 SubGrpcsubg 19217 odcod 19625 Abelcabl 19882 DProd cdprd 20096

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-disj 5082 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-tpos 8231 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8703 df-ec 8705 df-qs 8709 df-map 8835 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9906 df-card 9944 df-acn 9947 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-xnn0 12596 df-z 12610 df-uz 12881 df-q 12991 df-rp 13035 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-fac 14330 df-bc 14359 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-sum 15764 df-dvds 16336 df-gcd 16578 df-prm 16755 df-pc 16922 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-0g 17519 df-gsum 17520 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-mhm 18872 df-submnd 18873 df-grp 19034 df-minusg 19035 df-sbg 19036 df-mulg 19165 df-subg 19220 df-eqg 19222 df-ghm 19315 df-gim 19360 df-ga 19391 df-cntz 19418 df-oppg 19447 df-od 19629 df-lsm 19737 df-pj1 19738 df-cmn 19883 df-abl 19884 df-dprd 20098

This theorem is used by: ablfaclem2 20189

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