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Theorem ablfac1b 20328

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 2763	. 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2763	. 2 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
3		eqid 2763	. 2 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		ablfac1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Abel)
5		ablgrp 20041	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
6	4, 5	syl 18	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		ablfac1.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
8		prmex 16893	. . . 4 ⊢ ℙ ∈ V
9	8	ssex 5287	. . 3 ⊢ (𝐴 ⊆ ℙ → 𝐴 ∈ V)
10	7, 9	syl 18	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
11	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝐺 ∈ Abel)
12	7	sselda 3913	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ ℙ)
13		prmnn 16890	. . . . . . 7 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
14	12, 13	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ ℕ)
15		ablfac1.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐺)
16	15	grpbn0 19218	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
17	6, 16	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≠ ∅)
18		ablfac1.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Fin)
19		hashnncl 14549	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
20	18, 19	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
21	17, 20	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
22	21	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (♯‘𝐵) ∈ ℕ)
23	12, 22	pccld 17069	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝 pCnt (♯‘𝐵)) ∈ ℕ₀)
24	14, 23	nnexpcld 14428	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℕ)
25	24	nnzd 12757	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℤ)
26		ablfac1.o	. . . . 5 ⊢ 𝑂 = (od‘𝐺)
27	26, 15	oddvdssubg 20111	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℤ) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ (SubGrp‘𝐺))
28	11, 25, 27	syl2anc 596	. . 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 7144	. 2 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
31	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝐺 ∈ Abel)
32	30	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑆:𝐴⟶(SubGrp‘𝐺))
33		simpr1 1213	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑎 ∈ 𝐴)
34	32, 33	ffvelcdmd 7115	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑎) ∈ (SubGrp‘𝐺))
35		simpr2 1214	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑏 ∈ 𝐴)
36	32, 35	ffvelcdmd 7115	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑏) ∈ (SubGrp‘𝐺))
37	1, 31, 34, 36	ablcntzd 20113	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑎) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑏)))
38		id 23	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → 𝑝 = 𝑎)
39		oveq1 7456	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → (𝑝 pCnt (♯‘𝐵)) = (𝑎 pCnt (♯‘𝐵)))
40	38, 39	oveq12d 7467	. . . . . . . . 9 ⊢ (𝑝 = 𝑎 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑎↑(𝑎 pCnt (♯‘𝐵))))
41	40	breq2d 5118	. . . . . . . 8 ⊢ (𝑝 = 𝑎 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))))
42	41	rabbidv 3423	. . . . . . 7 ⊢ (𝑝 = 𝑎 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))})
43	15	fvexi 6928	. . . . . . . 8 ⊢ 𝐵 ∈ V
44	43	rabex 5305	. . . . . . 7 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
45	42, 29, 44	fvmpt3i 7028	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → (𝑆‘𝑎) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))})
46	45	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))})
47		eqimss 3971	. . . . 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 18	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))})
49	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐺 ∈ Abel)
50		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
51	50	subgacs 19412	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
52		acsmre 17867	. . . . . 6 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
53	49, 5, 51, 52	4syl 20	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
54		df-ima 5669	. . . . . . 7 ⊢ (𝑆 “ (𝐴 ∖ {𝑎})) = ran (𝑆 ↾ (𝐴 ∖ {𝑎}))
55	7	sselda 3913	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℙ)
56	55	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ∈ ℙ)
57	21	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (♯‘𝐵) ∈ ℕ)
58		pcdvds 17083	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
59	56, 57, 58	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
60	7	ad3antrrr 743	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝐴 ⊆ ℙ)
61		eldifi 4070	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ (𝐴 ∖ {𝑎}) → 𝑝 ∈ 𝐴)
62	61	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ 𝐴)
63	60, 62	sseldd 3914	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ ℙ)
64		pcdvds 17083	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
65	63, 57, 64	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
66		eqid 2763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎↑(𝑎 pCnt (♯‘𝐵))) = (𝑎↑(𝑎 pCnt (♯‘𝐵)))
67		eqid 2763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) = ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))
68	15, 26, 29, 4, 18, 7, 66, 67	ablfac1lem 20326	. . . . . . . . . . . . . . . . . . . 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 1160	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑎↑(𝑎 pCnt (♯‘𝐵))) ∈ ℕ ∧ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℕ))
70	69	simpld 500	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∈ ℕ)
71	70	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∈ ℕ)
72	71	nnzd 12757	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∈ ℤ)
73	63, 13	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ ℕ)
74	63, 57	pccld 17069	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝 pCnt (♯‘𝐵)) ∈ ℕ₀)
75	73, 74	nnexpcld 14428	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℕ)
76	75	nnzd 12757	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℤ)
77	57	nnzd 12757	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (♯‘𝐵) ∈ ℤ)
78		eldifsni 4753	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (𝐴 ∖ {𝑎}) → 𝑝 ≠ 𝑎)
79	78	ad2antlr 740	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ≠ 𝑎)
80	79	necomd 3014	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ≠ 𝑝)
81		prmrp 16929	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ ℙ ∧ 𝑝 ∈ ℙ) → ((𝑎 gcd 𝑝) = 1 ↔ 𝑎 ≠ 𝑝))
82	56, 63, 81	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎 gcd 𝑝) = 1 ↔ 𝑎 ≠ 𝑝))
83	80, 82	mpbird 260	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎 gcd 𝑝) = 1)
84		prmz 16891	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℙ → 𝑎 ∈ ℤ)
85	56, 84	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ∈ ℤ)
86		prmz 16891	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
87	63, 86	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ ℤ)
88	56, 57	pccld 17069	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎 pCnt (♯‘𝐵)) ∈ ℕ₀)
89		rpexp12i 16941	. . . . . . . . . . . . . . . . . 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 1402	. . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎↑(𝑎 pCnt (♯‘𝐵))) gcd (𝑝↑(𝑝 pCnt (♯‘𝐵)))) = 1)
92		coprmdvds2 16870	. . . . . . . . . . . . . . . 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 1401	. . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎↑(𝑎 pCnt (♯‘𝐵))) · (𝑝↑(𝑝 pCnt (♯‘𝐵)))) ∥ (♯‘𝐵))
95	68	simp3d 1162	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (♯‘𝐵) = ((𝑎↑(𝑎 pCnt (♯‘𝐵))) · ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))))
96	95	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (♯‘𝐵) = ((𝑎↑(𝑎 pCnt (♯‘𝐵))) · ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))))
97	94, 96	breqtrd 5135	. . . . . . . . . . . . 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 501	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℕ)
99	98	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℕ)
100	99	nnzd 12757	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℤ)
101	71	nnne0d 12426	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ≠ 0)
102		dvdscmulr 16495	. . . . . . . . . . . . . 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 1402	. . . . . . . . . . . . 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 235	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))))
105	15, 26	odcl 19792	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐵 → (𝑂‘𝑥) ∈ ℕ₀)
106	105	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑂‘𝑥) ∈ ℕ₀)
107	106	nn0zd 12756	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑂‘𝑥) ∈ ℤ)
108		dvdstr 16505	. . . . . . . . . . . . 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 1399	. . . . . . . . . . . 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 707	. . . . . . . . . . 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 4003	. . . . . . . . . 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 4564	. . . . . . . . . 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 237	. . . . . . . . 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 5980	. . . . . . . . . 10 ⊢ (𝑆 ↾ (𝐴 ∖ {𝑎})) = ((𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) ↾ (𝐴 ∖ {𝑎}))
115		difss 4075	. . . . . . . . . . 11 ⊢ (𝐴 ∖ {𝑎}) ⊆ 𝐴
116		resmpt 6045	. . . . . . . . . . 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 2786	. . . . . . . . 9 ⊢ (𝑆 ↾ (𝐴 ∖ {𝑎})) = (𝑝 ∈ (𝐴 ∖ {𝑎}) ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
119	113, 118	fmptd 7144	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆 ↾ (𝐴 ∖ {𝑎})):(𝐴 ∖ {𝑎})⟶𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
120	119	frnd 6746	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ran (𝑆 ↾ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
121	54, 120	eqsstrid 3951	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
122		sspwuni 5063	. . . . . 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 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∪ (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
124	98	nnzd 12757	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℤ)
125	26, 15	oddvdssubg 20111	. . . . . 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 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))} ∈ (SubGrp‘𝐺))
127	3	mrcsscl 17835	. . . . 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 1399	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐴 ∖ {𝑎}))) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
129		ss2in 4144	. . . 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 596	. . 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 2763	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))}
132		eqid 2763	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))}
133	68	simp2d 1161	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑎↑(𝑎 pCnt (♯‘𝐵))) gcd ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))) = 1)
134		eqid 2763	. . . . 5 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
135	15, 26, 131, 132, 49, 70, 98, 133, 95, 2, 134	ablfacrp 20324	. . . 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 500	. . 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 3949	. 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 20257	1 ⊢ (𝜑 → 𝐺dom DProd 𝑆)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 {crab 3416 Vcvv 3454 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 ∅c0 4281 𝒫 cpw 4560 {csn 4588 ∪ cuni 4870 class class class wbr 5106 ↦ cmpt 5190 dom cdm 5656 ran crn 5657 ↾ cres 5658 “ cima 5659 ⟶wf 6560 ‘cfv 6564 (class class class)co 7449 Fincfn 9009 0cc0 11237 1c1 11238 · cmul 11242 / cdiv 12011 ℕcn 12373 ℕ₀cn0 12644 ℤcz 12731 ↑cexp 14242 ♯chash 14513 ∥ cdvds 16463 gcd cgcd 16705 ℙcprime 16887 pCnt cpc 17055 Basecbs 17428 0_gc0g 17651 Moorecmre 17793 mrClscmrc 17794 ACScacs 17796 Grpcgrp 19185 SubGrpcsubg 19371 Cntzccntz 19571 odcod 19780 LSSumclsm 19890 Abelcabl 20037 DProd cdprd 20251

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7781 ax-inf2 9677 ax-cnex 11293 ax-resscn 11294 ax-1cn 11295 ax-icn 11296 ax-addcl 11297 ax-addrcl 11298 ax-mulcl 11299 ax-mulrcl 11300 ax-mulcom 11301 ax-addass 11302 ax-mulass 11303 ax-distr 11304 ax-i2m1 11305 ax-1ne0 11306 ax-1rid 11307 ax-rnegex 11308 ax-rrecex 11309 ax-cnre 11310 ax-pre-lttri 11311 ax-pre-lttrn 11312 ax-pre-ltadd 11313 ax-pre-mulgt0 11314 ax-pre-sup 11315

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-disj 5074 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6328 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6518 df-fun 6566 df-fn 6567 df-f 6568 df-f1 6569 df-fo 6570 df-f1o 6571 df-fv 6572 df-isom 6573 df-riota 7406 df-ov 7452 df-oprab 7453 df-mpo 7454 df-om 7910 df-1st 8033 df-2nd 8034 df-frecs 8335 df-wrecs 8366 df-recs 8415 df-rdg 8454 df-1o 8512 df-2o 8513 df-oadd 8516 df-omul 8517 df-er 8753 df-ec 8755 df-qs 8759 df-map 8885 df-ixp 8962 df-en 9010 df-dom 9011 df-sdom 9012 df-fin 9013 df-sup 9469 df-inf 9470 df-oi 9539 df-card 10055 df-acn 10058 df-pnf 11382 df-mnf 11383 df-xr 11384 df-ltxr 11385 df-le 11386 df-sub 11580 df-neg 11581 df-div 12012 df-nn 12374 df-2 12443 df-3 12444 df-n0 12645 df-z 12732 df-uz 13004 df-q 13114 df-rp 13159 df-fz 13678 df-fzo 13827 df-fl 13970 df-mod 14048 df-seq 14183 df-exp 14243 df-hash 14514 df-cj 15307 df-re 15308 df-im 15309 df-sqrt 15443 df-abs 15444 df-clim 15696 df-sum 15895 df-dvds 16464 df-gcd 16706 df-prm 16888 df-pc 17056 df-sets 17383 df-slot 17401 df-ndx 17413 df-base 17429 df-ress 17450 df-plusg 17482 df-0g 17653 df-mre 17797 df-mrc 17798 df-acs 17800 df-mgm 18857 df-sgrp 18949 df-mnd 18965 df-submnd 19020 df-grp 19188 df-minusg 19189 df-sbg 19190 df-mulg 19319 df-subg 19374 df-eqg 19376 df-cntz 19573 df-od 19784 df-lsm 19892 df-cmn 20038 df-abl 20039 df-dprd 20253

This theorem is used by: ablfac1c 20329 ablfac1eu 20331 ablfaclem2 20344 ablfaclem3 20345

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