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Theorem ablfac1b 18292

Description: Any abelian group is the direct product of factors of prime power order (with the exact order further matching the prime factorization of the group order). (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)

**Assertion**
Ref	Expression
ablfac1b	⊢ (𝜑 → 𝐺dom DProd 𝑆)

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

Proof of Theorem ablfac1b Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2610	. 2 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
3		eqid 2610	. 2 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		ablfac1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Abel)
5		ablgrp 18021	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		ablfac1.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
8		nnex 10903	. . . . 5 ⊢ ℕ ∈ V
9		prmnn 15226	. . . . . 6 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
10	9	ssriv 3572	. . . . 5 ⊢ ℙ ⊆ ℕ
11	8, 10	ssexi 4731	. . . 4 ⊢ ℙ ∈ V
12	11	ssex 4730	. . 3 ⊢ (𝐴 ⊆ ℙ → 𝐴 ∈ V)
13	7, 12	syl 17	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
14	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝐺 ∈ Abel)
15	7	sselda 3568	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ ℙ)
16	15, 9	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ ℕ)
17		ablfac1.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐺)
18	17	grpbn0 17274	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
19	6, 18	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≠ ∅)
20		ablfac1.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Fin)
21		hashnncl 13018	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → ((#‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
22	20, 21	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((#‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
23	19, 22	mpbird 246	. . . . . . . 8 ⊢ (𝜑 → (#‘𝐵) ∈ ℕ)
24	23	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (#‘𝐵) ∈ ℕ)
25	15, 24	pccld 15393	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝 pCnt (#‘𝐵)) ∈ ℕ₀)
26	16, 25	nnexpcld 12892	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝↑(𝑝 pCnt (#‘𝐵))) ∈ ℕ)
27	26	nnzd 11357	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝↑(𝑝 pCnt (#‘𝐵))) ∈ ℤ)
28		ablfac1.o	. . . . 5 ⊢ 𝑂 = (od‘𝐺)
29	28, 17	oddvdssubg 18081	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ (𝑝↑(𝑝 pCnt (#‘𝐵))) ∈ ℤ) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} ∈ (SubGrp‘𝐺))
30	14, 27, 29	syl2anc 691	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} ∈ (SubGrp‘𝐺))
31		ablfac1.s	. . 3 ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})
32	30, 31	fmptd 6292	. 2 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
33	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝐺 ∈ Abel)
34	32	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑆:𝐴⟶(SubGrp‘𝐺))
35		simpr1 1060	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑎 ∈ 𝐴)
36	34, 35	ffvelrnd 6268	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑎) ∈ (SubGrp‘𝐺))
37		simpr2 1061	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑏 ∈ 𝐴)
38	34, 37	ffvelrnd 6268	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑏) ∈ (SubGrp‘𝐺))
39	1, 33, 36, 38	ablcntzd 18083	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑎) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑏)))
40		id 22	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → 𝑝 = 𝑎)
41		oveq1 6556	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → (𝑝 pCnt (#‘𝐵)) = (𝑎 pCnt (#‘𝐵)))
42	40, 41	oveq12d 6567	. . . . . . . . 9 ⊢ (𝑝 = 𝑎 → (𝑝↑(𝑝 pCnt (#‘𝐵))) = (𝑎↑(𝑎 pCnt (#‘𝐵))))
43	42	breq2d 4595	. . . . . . . 8 ⊢ (𝑝 = 𝑎 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵))) ↔ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))))
44	43	rabbidv 3164	. . . . . . 7 ⊢ (𝑝 = 𝑎 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))})
45		fvex 6113	. . . . . . . . 9 ⊢ (Base‘𝐺) ∈ V
46	17, 45	eqeltri 2684	. . . . . . . 8 ⊢ 𝐵 ∈ V
47	46	rabex 4740	. . . . . . 7 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} ∈ V
48	44, 31, 47	fvmpt3i 6196	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → (𝑆‘𝑎) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))})
49	48	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))})
50		eqimss 3620	. . . . 5 ⊢ ((𝑆‘𝑎) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))} → (𝑆‘𝑎) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))})
51	49, 50	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))})
52	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐺 ∈ Abel)
53		eqid 2610	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
54	53	subgacs 17452	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
55		acsmre 16136	. . . . . 6 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
56	52, 5, 54, 55	4syl 19	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
57		df-ima 5051	. . . . . . 7 ⊢ (𝑆 “ (𝐴 ∖ {𝑎})) = ran (𝑆 ↾ (𝐴 ∖ {𝑎}))
58	7	sselda 3568	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℙ)
59	58	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ∈ ℙ)
60	23	ad3antrrr 762	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (#‘𝐵) ∈ ℕ)
61		pcdvds 15406	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℙ ∧ (#‘𝐵) ∈ ℕ) → (𝑎↑(𝑎 pCnt (#‘𝐵))) ∥ (#‘𝐵))
62	59, 60, 61	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (#‘𝐵))) ∥ (#‘𝐵))
63	7	ad3antrrr 762	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝐴 ⊆ ℙ)
64		eldifi 3694	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ (𝐴 ∖ {𝑎}) → 𝑝 ∈ 𝐴)
65	64	ad2antlr 759	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ 𝐴)
66	63, 65	sseldd 3569	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ ℙ)
67		pcdvds 15406	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ ℙ ∧ (#‘𝐵) ∈ ℕ) → (𝑝↑(𝑝 pCnt (#‘𝐵))) ∥ (#‘𝐵))
68	66, 60, 67	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (#‘𝐵))) ∥ (#‘𝐵))
69		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎↑(𝑎 pCnt (#‘𝐵))) = (𝑎↑(𝑎 pCnt (#‘𝐵)))
70		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) = ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))
71	17, 28, 31, 4, 20, 7, 69, 70	ablfac1lem 18290	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (((𝑎↑(𝑎 pCnt (#‘𝐵))) ∈ ℕ ∧ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) ∈ ℕ) ∧ ((𝑎↑(𝑎 pCnt (#‘𝐵))) gcd ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))) = 1 ∧ (#‘𝐵) = ((𝑎↑(𝑎 pCnt (#‘𝐵))) · ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))))))
72	71	simp1d 1066	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑎↑(𝑎 pCnt (#‘𝐵))) ∈ ℕ ∧ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) ∈ ℕ))
73	72	simpld 474	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎↑(𝑎 pCnt (#‘𝐵))) ∈ ℕ)
74	73	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (#‘𝐵))) ∈ ℕ)
75	74	nnzd 11357	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (#‘𝐵))) ∈ ℤ)
76	66, 9	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ ℕ)
77	66, 60	pccld 15393	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝 pCnt (#‘𝐵)) ∈ ℕ₀)
78	76, 77	nnexpcld 12892	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (#‘𝐵))) ∈ ℕ)
79	78	nnzd 11357	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (#‘𝐵))) ∈ ℤ)
80	60	nnzd 11357	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (#‘𝐵) ∈ ℤ)
81		eldifsni 4261	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (𝐴 ∖ {𝑎}) → 𝑝 ≠ 𝑎)
82	81	ad2antlr 759	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ≠ 𝑎)
83	82	necomd 2837	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ≠ 𝑝)
84		prmrp 15262	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ ℙ ∧ 𝑝 ∈ ℙ) → ((𝑎 gcd 𝑝) = 1 ↔ 𝑎 ≠ 𝑝))
85	59, 66, 84	syl2anc 691	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎 gcd 𝑝) = 1 ↔ 𝑎 ≠ 𝑝))
86	83, 85	mpbird 246	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎 gcd 𝑝) = 1)
87		prmz 15227	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℙ → 𝑎 ∈ ℤ)
88	59, 87	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ∈ ℤ)
89		prmz 15227	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
90	66, 89	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ ℤ)
91	59, 60	pccld 15393	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎 pCnt (#‘𝐵)) ∈ ℕ₀)
92		rpexp12i 15272	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ ℤ ∧ 𝑝 ∈ ℤ ∧ ((𝑎 pCnt (#‘𝐵)) ∈ ℕ₀ ∧ (𝑝 pCnt (#‘𝐵)) ∈ ℕ₀)) → ((𝑎 gcd 𝑝) = 1 → ((𝑎↑(𝑎 pCnt (#‘𝐵))) gcd (𝑝↑(𝑝 pCnt (#‘𝐵)))) = 1))
93	88, 90, 91, 77, 92	syl112anc 1322	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎 gcd 𝑝) = 1 → ((𝑎↑(𝑎 pCnt (#‘𝐵))) gcd (𝑝↑(𝑝 pCnt (#‘𝐵)))) = 1))
94	86, 93	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎↑(𝑎 pCnt (#‘𝐵))) gcd (𝑝↑(𝑝 pCnt (#‘𝐵)))) = 1)
95		coprmdvds2 15206	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑎↑(𝑎 pCnt (#‘𝐵))) ∈ ℤ ∧ (𝑝↑(𝑝 pCnt (#‘𝐵))) ∈ ℤ ∧ (#‘𝐵) ∈ ℤ) ∧ ((𝑎↑(𝑎 pCnt (#‘𝐵))) gcd (𝑝↑(𝑝 pCnt (#‘𝐵)))) = 1) → (((𝑎↑(𝑎 pCnt (#‘𝐵))) ∥ (#‘𝐵) ∧ (𝑝↑(𝑝 pCnt (#‘𝐵))) ∥ (#‘𝐵)) → ((𝑎↑(𝑎 pCnt (#‘𝐵))) · (𝑝↑(𝑝 pCnt (#‘𝐵)))) ∥ (#‘𝐵)))
96	75, 79, 80, 94, 95	syl31anc 1321	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (((𝑎↑(𝑎 pCnt (#‘𝐵))) ∥ (#‘𝐵) ∧ (𝑝↑(𝑝 pCnt (#‘𝐵))) ∥ (#‘𝐵)) → ((𝑎↑(𝑎 pCnt (#‘𝐵))) · (𝑝↑(𝑝 pCnt (#‘𝐵)))) ∥ (#‘𝐵)))
97	62, 68, 96	mp2and 711	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎↑(𝑎 pCnt (#‘𝐵))) · (𝑝↑(𝑝 pCnt (#‘𝐵)))) ∥ (#‘𝐵))
98	71	simp3d 1068	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (#‘𝐵) = ((𝑎↑(𝑎 pCnt (#‘𝐵))) · ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))))
99	98	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (#‘𝐵) = ((𝑎↑(𝑎 pCnt (#‘𝐵))) · ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))))
100	97, 99	breqtrd 4609	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎↑(𝑎 pCnt (#‘𝐵))) · (𝑝↑(𝑝 pCnt (#‘𝐵)))) ∥ ((𝑎↑(𝑎 pCnt (#‘𝐵))) · ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))))
101	72	simprd 478	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) ∈ ℕ)
102	101	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) ∈ ℕ)
103	102	nnzd 11357	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) ∈ ℤ)
104	74	nnne0d 10942	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (#‘𝐵))) ≠ 0)
105		dvdscmulr 14848	. . . . . . . . . . . . . 14 ⊢ (((𝑝↑(𝑝 pCnt (#‘𝐵))) ∈ ℤ ∧ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) ∈ ℤ ∧ ((𝑎↑(𝑎 pCnt (#‘𝐵))) ∈ ℤ ∧ (𝑎↑(𝑎 pCnt (#‘𝐵))) ≠ 0)) → (((𝑎↑(𝑎 pCnt (#‘𝐵))) · (𝑝↑(𝑝 pCnt (#‘𝐵)))) ∥ ((𝑎↑(𝑎 pCnt (#‘𝐵))) · ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))) ↔ (𝑝↑(𝑝 pCnt (#‘𝐵))) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))))
106	79, 103, 75, 104, 105	syl112anc 1322	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (((𝑎↑(𝑎 pCnt (#‘𝐵))) · (𝑝↑(𝑝 pCnt (#‘𝐵)))) ∥ ((𝑎↑(𝑎 pCnt (#‘𝐵))) · ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))) ↔ (𝑝↑(𝑝 pCnt (#‘𝐵))) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))))
107	100, 106	mpbid 221	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (#‘𝐵))) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))))
108	17, 28	odcl 17778	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐵 → (𝑂‘𝑥) ∈ ℕ₀)
109	108	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑂‘𝑥) ∈ ℕ₀)
110	109	nn0zd 11356	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑂‘𝑥) ∈ ℤ)
111		dvdstr 14856	. . . . . . . . . . . . 13 ⊢ (((𝑂‘𝑥) ∈ ℤ ∧ (𝑝↑(𝑝 pCnt (#‘𝐵))) ∈ ℤ ∧ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) ∈ ℤ) → (((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵))) ∧ (𝑝↑(𝑝 pCnt (#‘𝐵))) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))) → (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))))
112	110, 79, 103, 111	syl3anc 1318	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵))) ∧ (𝑝↑(𝑝 pCnt (#‘𝐵))) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))) → (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))))
113	107, 112	mpan2d 706	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵))) → (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))))
114	113	ss2rabdv 3646	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
115	47	elpw 4114	. . . . . . . . . 10 ⊢ ({𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} ∈ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))} ↔ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
116	114, 115	sylibr 223	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} ∈ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
117	31	reseq1i 5313	. . . . . . . . . 10 ⊢ (𝑆 ↾ (𝐴 ∖ {𝑎})) = ((𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))}) ↾ (𝐴 ∖ {𝑎}))
118		difss 3699	. . . . . . . . . . 11 ⊢ (𝐴 ∖ {𝑎}) ⊆ 𝐴
119		resmpt 5369	. . . . . . . . . . 11 ⊢ ((𝐴 ∖ {𝑎}) ⊆ 𝐴 → ((𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))}) ↾ (𝐴 ∖ {𝑎})) = (𝑝 ∈ (𝐴 ∖ {𝑎}) ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))}))
120	118, 119	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))}) ↾ (𝐴 ∖ {𝑎})) = (𝑝 ∈ (𝐴 ∖ {𝑎}) ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})
121	117, 120	eqtri 2632	. . . . . . . . 9 ⊢ (𝑆 ↾ (𝐴 ∖ {𝑎})) = (𝑝 ∈ (𝐴 ∖ {𝑎}) ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})
122	116, 121	fmptd 6292	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆 ↾ (𝐴 ∖ {𝑎})):(𝐴 ∖ {𝑎})⟶𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
123		frn 5966	. . . . . . . 8 ⊢ ((𝑆 ↾ (𝐴 ∖ {𝑎})):(𝐴 ∖ {𝑎})⟶𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))} → ran (𝑆 ↾ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
124	122, 123	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ran (𝑆 ↾ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
125	57, 124	syl5eqss 3612	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
126		sspwuni 4547	. . . . . 6 ⊢ ((𝑆 “ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))} ↔ ∪ (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
127	125, 126	sylib 207	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∪ (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
128	101	nnzd 11357	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) ∈ ℤ)
129	28, 17	oddvdssubg 18081	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) ∈ ℤ) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))} ∈ (SubGrp‘𝐺))
130	52, 128, 129	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))} ∈ (SubGrp‘𝐺))
131	3	mrcsscl 16103	. . . . 5 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))} ∧ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))} ∈ (SubGrp‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐴 ∖ {𝑎}))) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
132	56, 127, 130, 131	syl3anc 1318	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐴 ∖ {𝑎}))) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
133		ss2in 3802	. . . 4 ⊢ (((𝑆‘𝑎) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))} ∧ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐴 ∖ {𝑎}))) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))}) → ((𝑆‘𝑎) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐴 ∖ {𝑎})))) ⊆ ({𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))} ∩ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))}))
134	51, 132, 133	syl2anc 691	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑆‘𝑎) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐴 ∖ {𝑎})))) ⊆ ({𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))} ∩ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))}))
135		eqid 2610	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))}
136		eqid 2610	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))}
137	71	simp2d 1067	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑎↑(𝑎 pCnt (#‘𝐵))) gcd ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))) = 1)
138		eqid 2610	. . . . 5 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
139	17, 28, 135, 136, 52, 73, 101, 137, 98, 2, 138	ablfacrp 18288	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (({𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))} ∩ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))}) = {(0_g‘𝐺)} ∧ ({𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))} (LSSum‘𝐺){𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))}) = 𝐵))
140	139	simpld 474	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ({𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))} ∩ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))}) = {(0_g‘𝐺)})
141	134, 140	sseqtrd 3604	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑆‘𝑎) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐴 ∖ {𝑎})))) ⊆ {(0_g‘𝐺)})
142	1, 2, 3, 6, 13, 32, 39, 141	dmdprdd 18221	1 ⊢ (𝜑 → 𝐺dom DProd 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 0cc0 9815 1c1 9816 · cmul 9820 / cdiv 10563 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ↑cexp 12722 #chash 12979 ∥ cdvds 14821 gcd cgcd 15054 ℙcprime 15223 pCnt cpc 15379 Basecbs 15695 0_gc0g 15923 Moorecmre 16065 mrClscmrc 16066 ACScacs 16068 Grpcgrp 17245 SubGrpcsubg 17411 Cntzccntz 17571 odcod 17767 LSSumclsm 17872 Abelcabl 18017 DProd cdprd 18215

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-dvds 14822 df-gcd 15055 df-prm 15224 df-pc 15380 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-eqg 17416 df-cntz 17573 df-od 17771 df-lsm 17874 df-cmn 18018 df-abl 18019 df-dprd 18217

This theorem is referenced by: ablfac1c 18293 ablfac1eu 18295 ablfaclem2 18308 ablfaclem3 18309

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