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Theorem ablfac1b 20199

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 2760	. 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2760	. 2 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
3		eqid 2760	. 2 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		ablfac1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Abel)
5		ablgrp 19912	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
6	4, 5	syl 18	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		ablfac1.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
8		prmex 16767	. . . 4 ⊢ ℙ ∈ V
9	8	ssex 5285	. . 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 3931	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ ℙ)
13		prmnn 16764	. . . . . . 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 19090	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
17	6, 16	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≠ ∅)
18		ablfac1.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Fin)
19		hashnncl 14430	. . . . . . . . . 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 16942	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝 pCnt (♯‘𝐵)) ∈ ℕ₀)
24	14, 23	nnexpcld 14309	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℕ)
25	24	nnzd 12641	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℤ)
26		ablfac1.o	. . . . 5 ⊢ 𝑂 = (od‘𝐺)
27	26, 15	oddvdssubg 19982	. . . 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 7107	. 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 7078	. . 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 7078	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑏) ∈ (SubGrp‘𝐺))
37	1, 31, 34, 36	ablcntzd 19984	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑎) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑏)))
38		id 23	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → 𝑝 = 𝑎)
39		oveq1 7420	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → (𝑝 pCnt (♯‘𝐵)) = (𝑎 pCnt (♯‘𝐵)))
40	38, 39	oveq12d 7431	. . . . . . . . 9 ⊢ (𝑝 = 𝑎 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑎↑(𝑎 pCnt (♯‘𝐵))))
41	40	breq2d 5115	. . . . . . . 8 ⊢ (𝑝 = 𝑎 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))))
42	41	rabbidv 3419	. . . . . . 7 ⊢ (𝑝 = 𝑎 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))})
43	15	fvexi 6892	. . . . . . . 8 ⊢ 𝐵 ∈ V
44	43	rabex 5303	. . . . . . 7 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
45	42, 29, 44	fvmpt3i 6992	. . . . . 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 3989	. . . . 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 2760	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
51	50	subgacs 19284	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
52		acsmre 17740	. . . . . 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 5668	. . . . . . 7 ⊢ (𝑆 “ (𝐴 ∖ {𝑎})) = ran (𝑆 ↾ (𝐴 ∖ {𝑎}))
55	7	sselda 3931	. . . . . . . . . . . . . . . . 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 16956	. . . . . . . . . . . . . . . 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 4078	. . . . . . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ ℙ)
64		pcdvds 16956	. . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎↑(𝑎 pCnt (♯‘𝐵))) = (𝑎↑(𝑎 pCnt (♯‘𝐵)))
67		eqid 2760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) = ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))
68	15, 26, 29, 4, 18, 7, 66, 67	ablfac1lem 20197	. . . . . . . . . . . . . . . . . . . 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 12641	. . . . . . . . . . . . . . . 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 16942	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝 pCnt (♯‘𝐵)) ∈ ℕ₀)
75	73, 74	nnexpcld 14309	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℕ)
76	75	nnzd 12641	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℤ)
77	57	nnzd 12641	. . . . . . . . . . . . . . . 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 3010	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ≠ 𝑝)
81		prmrp 16803	. . . . . . . . . . . . . . . . . . 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 16765	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℙ → 𝑎 ∈ ℤ)
85	56, 84	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ∈ ℤ)
86		prmz 16765	. . . . . . . . . . . . . . . . . . 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 16942	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎 pCnt (♯‘𝐵)) ∈ ℕ₀)
89		rpexp12i 16815	. . . . . . . . . . . . . . . . . 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 1401	. . . . . . . . . . . . . . . . 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 16744	. . . . . . . . . . . . . . . 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 1400	. . . . . . . . . . . . . . 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 5131	. . . . . . . . . . . . 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 12641	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℤ)
101	71	nnne0d 12310	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ≠ 0)
102		dvdscmulr 16374	. . . . . . . . . . . . . 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 1401	. . . . . . . . . . . . 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 19663	. . . . . . . . . . . . . . 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 12640	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑂‘𝑥) ∈ ℤ)
108		dvdstr 16384	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 4023	. . . . . . . . . 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 4561	. . . . . . . . . 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 5968	. . . . . . . . . 10 ⊢ (𝑆 ↾ (𝐴 ∖ {𝑎})) = ((𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) ↾ (𝐴 ∖ {𝑎}))
115		difss 4083	. . . . . . . . . . 11 ⊢ (𝐴 ∖ {𝑎}) ⊆ 𝐴
116		resmpt 6033	. . . . . . . . . . 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 2783	. . . . . . . . 9 ⊢ (𝑆 ↾ (𝐴 ∖ {𝑎})) = (𝑝 ∈ (𝐴 ∖ {𝑎}) ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
119	113, 118	fmptd 7107	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆 ↾ (𝐴 ∖ {𝑎})):(𝐴 ∖ {𝑎})⟶𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
120	119	frnd 6711	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ran (𝑆 ↾ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
121	54, 120	eqsstrid 3969	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
122		sspwuni 5060	. . . . . 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 12641	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℤ)
125	26, 15	oddvdssubg 19982	. . . . . 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 17708	. . . . 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 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐴 ∖ {𝑎}))) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
129		ss2in 4190	. . . 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 2760	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))}
132		eqid 2760	. . . . 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 2760	. . . . 5 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
135	15, 26, 131, 132, 49, 70, 98, 133, 95, 2, 134	ablfacrp 20195	. . . 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 3967	. 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 20128	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 2145 ≠ wne 2955 {crab 3412 Vcvv 3450 ∖ cdif 3896 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 𝒫 cpw 4557 {csn 4584 ∪ cuni 4867 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5655 ran crn 5656 ↾ cres 5657 “ cima 5658 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 Fincfn 8952 0cc0 11124 1c1 11125 · cmul 11129 / cdiv 11895 ℕcn 12257 ℕ₀cn0 12528 ℤcz 12615 ↑cexp 14125 ♯chash 14394 ∥ cdvds 16342 gcd cgcd 16584 ℙcprime 16761 pCnt cpc 16928 Basecbs 17301 0_gc0g 17524 Moorecmre 17666 mrClscmrc 17667 ACScacs 17669 Grpcgrp 19057 SubGrpcsubg 19243 Cntzccntz 19442 odcod 19651 LSSumclsm 19761 Abelcabl 19908 DProd cdprd 20122

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-omul 8460 df-er 8696 df-ec 8698 df-qs 8702 df-map 8828 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-acn 9947 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-n0 12529 df-z 12616 df-uz 12888 df-q 12998 df-rp 13043 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-clim 15575 df-sum 15774 df-dvds 16343 df-gcd 16585 df-prm 16762 df-pc 16929 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-0g 17526 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-grp 19060 df-minusg 19061 df-sbg 19062 df-mulg 19191 df-subg 19246 df-eqg 19248 df-cntz 19444 df-od 19655 df-lsm 19763 df-cmn 19909 df-abl 19910 df-dprd 20124

This theorem is used by: ablfac1c 20200 ablfac1eu 20202 ablfaclem2 20215 ablfaclem3 20216

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