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Theorem ablfac1b 18469

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 2622	. 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2622	. 2 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
3		eqid 2622	. 2 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		ablfac1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Abel)
5		ablgrp 18198	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		ablfac1.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
8		nnex 11026	. . . . 5 ⊢ ℕ ∈ V
9		prmnn 15388	. . . . . 6 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
10	9	ssriv 3607	. . . . 5 ⊢ ℙ ⊆ ℕ
11	8, 10	ssexi 4803	. . . 4 ⊢ ℙ ∈ V
12	11	ssex 4802	. . 3 ⊢ (𝐴 ⊆ ℙ → 𝐴 ∈ V)
13	7, 12	syl 17	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
14	4	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝐺 ∈ Abel)
15	7	sselda 3603	. . . . . . 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 17451	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
19	6, 18	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≠ ∅)
20		ablfac1.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Fin)
21		hashnncl 13157	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → ((#‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
22	20, 21	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((#‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
23	19, 22	mpbird 247	. . . . . . . 8 ⊢ (𝜑 → (#‘𝐵) ∈ ℕ)
24	23	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (#‘𝐵) ∈ ℕ)
25	15, 24	pccld 15555	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝 pCnt (#‘𝐵)) ∈ ℕ₀)
26	16, 25	nnexpcld 13030	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝↑(𝑝 pCnt (#‘𝐵))) ∈ ℕ)
27	26	nnzd 11481	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝↑(𝑝 pCnt (#‘𝐵))) ∈ ℤ)
28		ablfac1.o	. . . . 5 ⊢ 𝑂 = (od‘𝐺)
29	28, 17	oddvdssubg 18258	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ (𝑝↑(𝑝 pCnt (#‘𝐵))) ∈ ℤ) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} ∈ (SubGrp‘𝐺))
30	14, 27, 29	syl2anc 693	. . 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 6385	. 2 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
33	4	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝐺 ∈ Abel)
34	32	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑆:𝐴⟶(SubGrp‘𝐺))
35		simpr1 1067	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑎 ∈ 𝐴)
36	34, 35	ffvelrnd 6360	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑎) ∈ (SubGrp‘𝐺))
37		simpr2 1068	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑏 ∈ 𝐴)
38	34, 37	ffvelrnd 6360	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑏) ∈ (SubGrp‘𝐺))
39	1, 33, 36, 38	ablcntzd 18260	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑎) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑏)))
40		id 22	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → 𝑝 = 𝑎)
41		oveq1 6657	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → (𝑝 pCnt (#‘𝐵)) = (𝑎 pCnt (#‘𝐵)))
42	40, 41	oveq12d 6668	. . . . . . . . 9 ⊢ (𝑝 = 𝑎 → (𝑝↑(𝑝 pCnt (#‘𝐵))) = (𝑎↑(𝑎 pCnt (#‘𝐵))))
43	42	breq2d 4665	. . . . . . . 8 ⊢ (𝑝 = 𝑎 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵))) ↔ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))))
44	43	rabbidv 3189	. . . . . . 7 ⊢ (𝑝 = 𝑎 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))})
45		fvex 6201	. . . . . . . . 9 ⊢ (Base‘𝐺) ∈ V
46	17, 45	eqeltri 2697	. . . . . . . 8 ⊢ 𝐵 ∈ V
47	46	rabex 4813	. . . . . . 7 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} ∈ V
48	44, 31, 47	fvmpt3i 6287	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → (𝑆‘𝑎) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))})
49	48	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))})
50		eqimss 3657	. . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐺 ∈ Abel)
53		eqid 2622	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
54	53	subgacs 17629	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
55		acsmre 16313	. . . . . 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 5127	. . . . . . 7 ⊢ (𝑆 “ (𝐴 ∖ {𝑎})) = ran (𝑆 ↾ (𝐴 ∖ {𝑎}))
58	7	sselda 3603	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℙ)
59	58	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ∈ ℙ)
60	23	ad3antrrr 766	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (#‘𝐵) ∈ ℕ)
61		pcdvds 15568	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℙ ∧ (#‘𝐵) ∈ ℕ) → (𝑎↑(𝑎 pCnt (#‘𝐵))) ∥ (#‘𝐵))
62	59, 60, 61	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (#‘𝐵))) ∥ (#‘𝐵))
63	7	ad3antrrr 766	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝐴 ⊆ ℙ)
64		eldifi 3732	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ (𝐴 ∖ {𝑎}) → 𝑝 ∈ 𝐴)
65	64	ad2antlr 763	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ 𝐴)
66	63, 65	sseldd 3604	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ ℙ)
67		pcdvds 15568	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ ℙ ∧ (#‘𝐵) ∈ ℕ) → (𝑝↑(𝑝 pCnt (#‘𝐵))) ∥ (#‘𝐵))
68	66, 60, 67	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (#‘𝐵))) ∥ (#‘𝐵))
69		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎↑(𝑎 pCnt (#‘𝐵))) = (𝑎↑(𝑎 pCnt (#‘𝐵)))
70		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) = ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))
71	17, 28, 31, 4, 20, 7, 69, 70	ablfac1lem 18467	. . . . . . . . . . . . . . . . . . . 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 1073	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑎↑(𝑎 pCnt (#‘𝐵))) ∈ ℕ ∧ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) ∈ ℕ))
73	72	simpld 475	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎↑(𝑎 pCnt (#‘𝐵))) ∈ ℕ)
74	73	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (#‘𝐵))) ∈ ℕ)
75	74	nnzd 11481	. . . . . . . . . . . . . . . 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 15555	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝 pCnt (#‘𝐵)) ∈ ℕ₀)
78	76, 77	nnexpcld 13030	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (#‘𝐵))) ∈ ℕ)
79	78	nnzd 11481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (#‘𝐵))) ∈ ℤ)
80	60	nnzd 11481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (#‘𝐵) ∈ ℤ)
81		eldifsni 4320	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (𝐴 ∖ {𝑎}) → 𝑝 ≠ 𝑎)
82	81	ad2antlr 763	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ≠ 𝑎)
83	82	necomd 2849	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ≠ 𝑝)
84		prmrp 15424	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ ℙ ∧ 𝑝 ∈ ℙ) → ((𝑎 gcd 𝑝) = 1 ↔ 𝑎 ≠ 𝑝))
85	59, 66, 84	syl2anc 693	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎 gcd 𝑝) = 1 ↔ 𝑎 ≠ 𝑝))
86	83, 85	mpbird 247	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎 gcd 𝑝) = 1)
87		prmz 15389	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℙ → 𝑎 ∈ ℤ)
88	59, 87	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ∈ ℤ)
89		prmz 15389	. . . . . . . . . . . . . . . . . . 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 15555	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎 pCnt (#‘𝐵)) ∈ ℕ₀)
92		rpexp12i 15434	. . . . . . . . . . . . . . . . . 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 1330	. . . . . . . . . . . . . . . . 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 15368	. . . . . . . . . . . . . . . 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 1329	. . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎↑(𝑎 pCnt (#‘𝐵))) · (𝑝↑(𝑝 pCnt (#‘𝐵)))) ∥ (#‘𝐵))
98	71	simp3d 1075	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (#‘𝐵) = ((𝑎↑(𝑎 pCnt (#‘𝐵))) · ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))))
99	98	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (#‘𝐵) = ((𝑎↑(𝑎 pCnt (#‘𝐵))) · ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))))
100	97, 99	breqtrd 4679	. . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) ∈ ℕ)
102	101	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) ∈ ℕ)
103	102	nnzd 11481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) ∈ ℤ)
104	74	nnne0d 11065	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (#‘𝐵))) ≠ 0)
105		dvdscmulr 15010	. . . . . . . . . . . . . 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 1330	. . . . . . . . . . . . 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 222	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (#‘𝐵))) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))))
108	17, 28	odcl 17955	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐵 → (𝑂‘𝑥) ∈ ℕ₀)
109	108	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑂‘𝑥) ∈ ℕ₀)
110	109	nn0zd 11480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑂‘𝑥) ∈ ℤ)
111		dvdstr 15018	. . . . . . . . . . . . 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 1326	. . . . . . . . . . . 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 710	. . . . . . . . . . 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 3683	. . . . . . . . . 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 4164	. . . . . . . . . 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 224	. . . . . . . . 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 5392	. . . . . . . . . 10 ⊢ (𝑆 ↾ (𝐴 ∖ {𝑎})) = ((𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))}) ↾ (𝐴 ∖ {𝑎}))
118		difss 3737	. . . . . . . . . . 11 ⊢ (𝐴 ∖ {𝑎}) ⊆ 𝐴
119		resmpt 5449	. . . . . . . . . . 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 2644	. . . . . . . . 9 ⊢ (𝑆 ↾ (𝐴 ∖ {𝑎})) = (𝑝 ∈ (𝐴 ∖ {𝑎}) ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})
122	116, 121	fmptd 6385	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆 ↾ (𝐴 ∖ {𝑎})):(𝐴 ∖ {𝑎})⟶𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
123		frn 6053	. . . . . . . 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 3649	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
126		sspwuni 4611	. . . . . 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 208	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∪ (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
128	101	nnzd 11481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵)))) ∈ ℤ)
129	28, 17	oddvdssubg 18258	. . . . . 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 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))} ∈ (SubGrp‘𝐺))
131	3	mrcsscl 16280	. . . . 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 1326	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐴 ∖ {𝑎}))) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))})
133		ss2in 3840	. . . 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 693	. . 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 2622	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (#‘𝐵)))}
136		eqid 2622	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))}
137	71	simp2d 1074	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑎↑(𝑎 pCnt (#‘𝐵))) gcd ((#‘𝐵) / (𝑎↑(𝑎 pCnt (#‘𝐵))))) = 1)
138		eqid 2622	. . . . 5 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
139	17, 28, 135, 136, 52, 73, 101, 137, 98, 2, 138	ablfacrp 18465	. . . 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 475	. . 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 3641	. 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 18398	1 ⊢ (𝜑 → 𝐺dom DProd 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {crab 2916 Vcvv 3200 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 ∪ cuni 4436 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 0cc0 9936 1c1 9937 · cmul 9941 / cdiv 10684 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ↑cexp 12860 #chash 13117 ∥ cdvds 14983 gcd cgcd 15216 ℙcprime 15385 pCnt cpc 15541 Basecbs 15857 0_gc0g 16100 Moorecmre 16242 mrClscmrc 16243 ACScacs 16245 Grpcgrp 17422 SubGrpcsubg 17588 Cntzccntz 17748 odcod 17944 LSSumclsm 18049 Abelcabl 18194 DProd cdprd 18392

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-eqg 17593 df-cntz 17750 df-od 17948 df-lsm 18051 df-cmn 18195 df-abl 18196 df-dprd 18394

This theorem is referenced by: ablfac1c 18470 ablfac1eu 18472 ablfaclem2 18485 ablfaclem3 18486

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