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Theorem ablfac1b 18913

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 2797	. 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2797	. 2 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
3		eqid 2797	. 2 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		ablfac1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Abel)
5		ablgrp 18642	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		ablfac1.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
8		prmex 15854	. . . 4 ⊢ ℙ ∈ V
9	8	ssex 5123	. . 3 ⊢ (𝐴 ⊆ ℙ → 𝐴 ∈ V)
10	7, 9	syl 17	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
11	4	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝐺 ∈ Abel)
12	7	sselda 3895	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ ℙ)
13		prmnn 15851	. . . . . . 7 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
14	12, 13	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ ℕ)
15		ablfac1.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐺)
16	15	grpbn0 17894	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
17	6, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≠ ∅)
18		ablfac1.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Fin)
19		hashnncl 13581	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
20	18, 19	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
21	17, 20	mpbird 258	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
22	21	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (♯‘𝐵) ∈ ℕ)
23	12, 22	pccld 16020	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝 pCnt (♯‘𝐵)) ∈ ℕ₀)
24	14, 23	nnexpcld 13460	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℕ)
25	24	nnzd 11940	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℤ)
26		ablfac1.o	. . . . 5 ⊢ 𝑂 = (od‘𝐺)
27	26, 15	oddvdssubg 18702	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℤ) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ (SubGrp‘𝐺))
28	11, 25, 27	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ (SubGrp‘𝐺))
29		ablfac1.s	. . 3 ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
30	28, 29	fmptd 6748	. 2 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
31	4	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝐺 ∈ Abel)
32	30	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑆:𝐴⟶(SubGrp‘𝐺))
33		simpr1 1187	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑎 ∈ 𝐴)
34	32, 33	ffvelrnd 6724	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑎) ∈ (SubGrp‘𝐺))
35		simpr2 1188	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑏 ∈ 𝐴)
36	32, 35	ffvelrnd 6724	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑏) ∈ (SubGrp‘𝐺))
37	1, 31, 34, 36	ablcntzd 18704	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑎) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑏)))
38		id 22	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → 𝑝 = 𝑎)
39		oveq1 7030	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → (𝑝 pCnt (♯‘𝐵)) = (𝑎 pCnt (♯‘𝐵)))
40	38, 39	oveq12d 7041	. . . . . . . . 9 ⊢ (𝑝 = 𝑎 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑎↑(𝑎 pCnt (♯‘𝐵))))
41	40	breq2d 4980	. . . . . . . 8 ⊢ (𝑝 = 𝑎 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))))
42	41	rabbidv 3428	. . . . . . 7 ⊢ (𝑝 = 𝑎 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))})
43	15	fvexi 6559	. . . . . . . 8 ⊢ 𝐵 ∈ V
44	43	rabex 5133	. . . . . . 7 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
45	42, 29, 44	fvmpt3i 6647	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → (𝑆‘𝑎) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))})
46	45	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))})
47		eqimss 3950	. . . . 5 ⊢ ((𝑆‘𝑎) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))} → (𝑆‘𝑎) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))})
48	46, 47	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))})
49	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐺 ∈ Abel)
50		eqid 2797	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
51	50	subgacs 18072	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
52		acsmre 16756	. . . . . 6 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
53	49, 5, 51, 52	4syl 19	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
54		df-ima 5463	. . . . . . 7 ⊢ (𝑆 “ (𝐴 ∖ {𝑎})) = ran (𝑆 ↾ (𝐴 ∖ {𝑎}))
55	7	sselda 3895	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℙ)
56	55	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ∈ ℙ)
57	21	ad3antrrr 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (♯‘𝐵) ∈ ℕ)
58		pcdvds 16033	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
59	56, 57, 58	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
60	7	ad3antrrr 726	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝐴 ⊆ ℙ)
61		eldifi 4030	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ (𝐴 ∖ {𝑎}) → 𝑝 ∈ 𝐴)
62	61	ad2antlr 723	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ 𝐴)
63	60, 62	sseldd 3896	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ ℙ)
64		pcdvds 16033	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
65	63, 57, 64	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
66		eqid 2797	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎↑(𝑎 pCnt (♯‘𝐵))) = (𝑎↑(𝑎 pCnt (♯‘𝐵)))
67		eqid 2797	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) = ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))
68	15, 26, 29, 4, 18, 7, 66, 67	ablfac1lem 18911	. . . . . . . . . . . . . . . . . . . 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)))))))
69	68	simp1d 1135	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑎↑(𝑎 pCnt (♯‘𝐵))) ∈ ℕ ∧ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℕ))
70	69	simpld 495	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∈ ℕ)
71	70	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∈ ℕ)
72	71	nnzd 11940	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∈ ℤ)
73	63, 13	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ ℕ)
74	63, 57	pccld 16020	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝 pCnt (♯‘𝐵)) ∈ ℕ₀)
75	73, 74	nnexpcld 13460	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℕ)
76	75	nnzd 11940	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℤ)
77	57	nnzd 11940	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (♯‘𝐵) ∈ ℤ)
78		eldifsni 4635	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (𝐴 ∖ {𝑎}) → 𝑝 ≠ 𝑎)
79	78	ad2antlr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ≠ 𝑎)
80	79	necomd 3041	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ≠ 𝑝)
81		prmrp 15889	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ ℙ ∧ 𝑝 ∈ ℙ) → ((𝑎 gcd 𝑝) = 1 ↔ 𝑎 ≠ 𝑝))
82	56, 63, 81	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎 gcd 𝑝) = 1 ↔ 𝑎 ≠ 𝑝))
83	80, 82	mpbird 258	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎 gcd 𝑝) = 1)
84		prmz 15852	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℙ → 𝑎 ∈ ℤ)
85	56, 84	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ∈ ℤ)
86		prmz 15852	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
87	63, 86	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ ℤ)
88	56, 57	pccld 16020	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎 pCnt (♯‘𝐵)) ∈ ℕ₀)
89		rpexp12i 15899	. . . . . . . . . . . . . . . . . 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))
90	85, 87, 88, 74, 89	syl112anc 1367	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎 gcd 𝑝) = 1 → ((𝑎↑(𝑎 pCnt (♯‘𝐵))) gcd (𝑝↑(𝑝 pCnt (♯‘𝐵)))) = 1))
91	83, 90	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎↑(𝑎 pCnt (♯‘𝐵))) gcd (𝑝↑(𝑝 pCnt (♯‘𝐵)))) = 1)
92		coprmdvds2 15831	. . . . . . . . . . . . . . . 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)))
93	72, 76, 77, 91, 92	syl31anc 1366	. . . . . . . . . . . . . . 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)))
94	59, 65, 93	mp2and 695	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎↑(𝑎 pCnt (♯‘𝐵))) · (𝑝↑(𝑝 pCnt (♯‘𝐵)))) ∥ (♯‘𝐵))
95	68	simp3d 1137	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (♯‘𝐵) = ((𝑎↑(𝑎 pCnt (♯‘𝐵))) · ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))))
96	95	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (♯‘𝐵) = ((𝑎↑(𝑎 pCnt (♯‘𝐵))) · ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))))
97	94, 96	breqtrd 4994	. . . . . . . . . . . . 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))))))
98	69	simprd 496	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℕ)
99	98	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℕ)
100	99	nnzd 11940	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℤ)
101	71	nnne0d 11541	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ≠ 0)
102		dvdscmulr 15475	. . . . . . . . . . . . . 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))))))
103	76, 100, 72, 101, 102	syl112anc 1367	. . . . . . . . . . . . 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))))))
104	97, 103	mpbid 233	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))))
105	15, 26	odcl 18399	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐵 → (𝑂‘𝑥) ∈ ℕ₀)
106	105	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑂‘𝑥) ∈ ℕ₀)
107	106	nn0zd 11939	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑂‘𝑥) ∈ ℤ)
108		dvdstr 15483	. . . . . . . . . . . . 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))))))
109	107, 76, 100, 108	syl3anc 1364	. . . . . . . . . . . 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))))))
110	104, 109	mpan2d 690	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) → (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))))
111	110	ss2rabdv 3979	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
112	44	elpw 4465	. . . . . . . . . 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))))})
113	111, 112	sylibr 235	. . . . . . . . 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))))})
114	29	reseq1i 5737	. . . . . . . . . 10 ⊢ (𝑆 ↾ (𝐴 ∖ {𝑎})) = ((𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) ↾ (𝐴 ∖ {𝑎}))
115		difss 4035	. . . . . . . . . . 11 ⊢ (𝐴 ∖ {𝑎}) ⊆ 𝐴
116		resmpt 5793	. . . . . . . . . . 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)))}))
117	115, 116	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)))})
118	114, 117	eqtri 2821	. . . . . . . . 9 ⊢ (𝑆 ↾ (𝐴 ∖ {𝑎})) = (𝑝 ∈ (𝐴 ∖ {𝑎}) ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
119	113, 118	fmptd 6748	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆 ↾ (𝐴 ∖ {𝑎})):(𝐴 ∖ {𝑎})⟶𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
120	119	frnd 6396	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ran (𝑆 ↾ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
121	54, 120	eqsstrid 3942	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
122		sspwuni 4927	. . . . . 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))))})
123	121, 122	sylib 219	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∪ (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
124	98	nnzd 11940	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℤ)
125	26, 15	oddvdssubg 18702	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℤ) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))} ∈ (SubGrp‘𝐺))
126	49, 124, 125	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))} ∈ (SubGrp‘𝐺))
127	3	mrcsscl 16724	. . . . 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))))})
128	53, 123, 126, 127	syl3anc 1364	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐴 ∖ {𝑎}))) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
129		ss2in 4139	. . . 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))))}))
130	48, 128, 129	syl2anc 584	. . 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))))}))
131		eqid 2797	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))}
132		eqid 2797	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))}
133	68	simp2d 1136	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑎↑(𝑎 pCnt (♯‘𝐵))) gcd ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))) = 1)
134		eqid 2797	. . . . 5 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
135	15, 26, 131, 132, 49, 70, 98, 133, 95, 2, 134	ablfacrp 18909	. . . 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))
136	135	simpld 495	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ({𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))} ∩ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))}) = {(0_g‘𝐺)})
137	130, 136	sseqtrd 3934	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑆‘𝑎) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐴 ∖ {𝑎})))) ⊆ {(0_g‘𝐺)})
138	1, 2, 3, 6, 10, 30, 37, 137	dmdprdd 18842	1 ⊢ (𝜑 → 𝐺dom DProd 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 {crab 3111 Vcvv 3440 ∖ cdif 3862 ∩ cin 3864 ⊆ wss 3865 ∅c0 4217 𝒫 cpw 4459 {csn 4478 ∪ cuni 4751 class class class wbr 4968 ↦ cmpt 5047 dom cdm 5450 ran crn 5451 ↾ cres 5452 “ cima 5453 ⟶wf 6228 ‘cfv 6232 (class class class)co 7023 Fincfn 8364 0cc0 10390 1c1 10391 · cmul 10395 / cdiv 11151 ℕcn 11492 ℕ₀cn0 11751 ℤcz 11835 ↑cexp 13283 ♯chash 13544 ∥ cdvds 15444 gcd cgcd 15680 ℙcprime 15848 pCnt cpc 16006 Basecbs 16316 0_gc0g 16546 Moorecmre 16686 mrClscmrc 16687 ACScacs 16689 Grpcgrp 17865 SubGrpcsubg 18031 Cntzccntz 18190 odcod 18387 LSSumclsm 18493 Abelcabl 18638 DProd cdprd 18836

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-disj 4937 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-2o 7961 df-oadd 7964 df-omul 7965 df-er 8146 df-ec 8148 df-qs 8152 df-map 8265 df-ixp 8318 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-sup 8759 df-inf 8760 df-oi 8827 df-card 9221 df-acn 9224 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-n0 11752 df-z 11836 df-uz 12098 df-q 12202 df-rp 12244 df-fz 12747 df-fzo 12888 df-fl 13016 df-mod 13092 df-seq 13224 df-exp 13284 df-hash 13545 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-clim 14683 df-sum 14881 df-dvds 15445 df-gcd 15681 df-prm 15849 df-pc 16007 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-0g 16548 df-mre 16690 df-mrc 16691 df-acs 16693 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-grp 17868 df-minusg 17869 df-sbg 17870 df-mulg 17986 df-subg 18034 df-eqg 18036 df-cntz 18192 df-od 18391 df-lsm 18495 df-cmn 18639 df-abl 18640 df-dprd 18838

This theorem is referenced by: ablfac1c 18914 ablfac1eu 18916 ablfaclem2 18929 ablfaclem3 18930

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