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Theorem ablfac1b 19992

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 2726	. 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2726	. 2 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
3		eqid 2726	. 2 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		ablfac1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Abel)
5		ablgrp 19705	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		ablfac1.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
8		prmex 16621	. . . 4 ⊢ ℙ ∈ V
9	8	ssex 5314	. . 3 ⊢ (𝐴 ⊆ ℙ → 𝐴 ∈ V)
10	7, 9	syl 17	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
11	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝐺 ∈ Abel)
12	7	sselda 3977	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ ℙ)
13		prmnn 16618	. . . . . . 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 18896	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
17	6, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≠ ∅)
18		ablfac1.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Fin)
19		hashnncl 14331	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
20	18, 19	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
21	17, 20	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
22	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (♯‘𝐵) ∈ ℕ)
23	12, 22	pccld 16792	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝 pCnt (♯‘𝐵)) ∈ ℕ₀)
24	14, 23	nnexpcld 14213	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℕ)
25	24	nnzd 12589	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℤ)
26		ablfac1.o	. . . . 5 ⊢ 𝑂 = (od‘𝐺)
27	26, 15	oddvdssubg 19775	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℤ) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ (SubGrp‘𝐺))
28	11, 25, 27	syl2anc 583	. . 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 7109	. 2 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
31	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝐺 ∈ Abel)
32	30	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑆:𝐴⟶(SubGrp‘𝐺))
33		simpr1 1191	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑎 ∈ 𝐴)
34	32, 33	ffvelcdmd 7081	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑎) ∈ (SubGrp‘𝐺))
35		simpr2 1192	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑏 ∈ 𝐴)
36	32, 35	ffvelcdmd 7081	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑏) ∈ (SubGrp‘𝐺))
37	1, 31, 34, 36	ablcntzd 19777	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑎) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑏)))
38		id 22	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → 𝑝 = 𝑎)
39		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → (𝑝 pCnt (♯‘𝐵)) = (𝑎 pCnt (♯‘𝐵)))
40	38, 39	oveq12d 7423	. . . . . . . . 9 ⊢ (𝑝 = 𝑎 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑎↑(𝑎 pCnt (♯‘𝐵))))
41	40	breq2d 5153	. . . . . . . 8 ⊢ (𝑝 = 𝑎 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))))
42	41	rabbidv 3434	. . . . . . 7 ⊢ (𝑝 = 𝑎 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))})
43	15	fvexi 6899	. . . . . . . 8 ⊢ 𝐵 ∈ V
44	43	rabex 5325	. . . . . . 7 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
45	42, 29, 44	fvmpt3i 6997	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → (𝑆‘𝑎) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))})
46	45	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))})
47		eqimss 4035	. . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐺 ∈ Abel)
50		eqid 2726	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
51	50	subgacs 19088	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
52		acsmre 17605	. . . . . 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 5682	. . . . . . 7 ⊢ (𝑆 “ (𝐴 ∖ {𝑎})) = ran (𝑆 ↾ (𝐴 ∖ {𝑎}))
55	7	sselda 3977	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℙ)
56	55	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ∈ ℙ)
57	21	ad3antrrr 727	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (♯‘𝐵) ∈ ℕ)
58		pcdvds 16806	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
59	56, 57, 58	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
60	7	ad3antrrr 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝐴 ⊆ ℙ)
61		eldifi 4121	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ (𝐴 ∖ {𝑎}) → 𝑝 ∈ 𝐴)
62	61	ad2antlr 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ 𝐴)
63	60, 62	sseldd 3978	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ ℙ)
64		pcdvds 16806	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
65	63, 57, 64	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
66		eqid 2726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎↑(𝑎 pCnt (♯‘𝐵))) = (𝑎↑(𝑎 pCnt (♯‘𝐵)))
67		eqid 2726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) = ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))
68	15, 26, 29, 4, 18, 7, 66, 67	ablfac1lem 19990	. . . . . . . . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑎↑(𝑎 pCnt (♯‘𝐵))) ∈ ℕ ∧ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℕ))
70	69	simpld 494	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∈ ℕ)
71	70	ad2antrr 723	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∈ ℕ)
72	71	nnzd 12589	. . . . . . . . . . . . . . . 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 16792	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝 pCnt (♯‘𝐵)) ∈ ℕ₀)
75	73, 74	nnexpcld 14213	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℕ)
76	75	nnzd 12589	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℤ)
77	57	nnzd 12589	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (♯‘𝐵) ∈ ℤ)
78		eldifsni 4788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (𝐴 ∖ {𝑎}) → 𝑝 ≠ 𝑎)
79	78	ad2antlr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ≠ 𝑎)
80	79	necomd 2990	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ≠ 𝑝)
81		prmrp 16656	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ ℙ ∧ 𝑝 ∈ ℙ) → ((𝑎 gcd 𝑝) = 1 ↔ 𝑎 ≠ 𝑝))
82	56, 63, 81	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎 gcd 𝑝) = 1 ↔ 𝑎 ≠ 𝑝))
83	80, 82	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎 gcd 𝑝) = 1)
84		prmz 16619	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℙ → 𝑎 ∈ ℤ)
85	56, 84	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ∈ ℤ)
86		prmz 16619	. . . . . . . . . . . . . . . . . . 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 16792	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎 pCnt (♯‘𝐵)) ∈ ℕ₀)
89		rpexp12i 16669	. . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . 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 16598	. . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . 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 696	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎↑(𝑎 pCnt (♯‘𝐵))) · (𝑝↑(𝑝 pCnt (♯‘𝐵)))) ∥ (♯‘𝐵))
95	68	simp3d 1141	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (♯‘𝐵) = ((𝑎↑(𝑎 pCnt (♯‘𝐵))) · ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))))
96	95	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (♯‘𝐵) = ((𝑎↑(𝑎 pCnt (♯‘𝐵))) · ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))))
97	94, 96	breqtrd 5167	. . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℕ)
99	98	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℕ)
100	99	nnzd 12589	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℤ)
101	71	nnne0d 12266	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ≠ 0)
102		dvdscmulr 16235	. . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))))
105	15, 26	odcl 19456	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐵 → (𝑂‘𝑥) ∈ ℕ₀)
106	105	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑂‘𝑥) ∈ ℕ₀)
107	106	nn0zd 12588	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑂‘𝑥) ∈ ℤ)
108		dvdstr 16244	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 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 691	. . . . . . . . . . 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 4068	. . . . . . . . . 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 4601	. . . . . . . . . 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 233	. . . . . . . . 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 5971	. . . . . . . . . 10 ⊢ (𝑆 ↾ (𝐴 ∖ {𝑎})) = ((𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) ↾ (𝐴 ∖ {𝑎}))
115		difss 4126	. . . . . . . . . . 11 ⊢ (𝐴 ∖ {𝑎}) ⊆ 𝐴
116		resmpt 6031	. . . . . . . . . . 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 2754	. . . . . . . . 9 ⊢ (𝑆 ↾ (𝐴 ∖ {𝑎})) = (𝑝 ∈ (𝐴 ∖ {𝑎}) ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
119	113, 118	fmptd 7109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆 ↾ (𝐴 ∖ {𝑎})):(𝐴 ∖ {𝑎})⟶𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
120	119	frnd 6719	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ran (𝑆 ↾ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
121	54, 120	eqsstrid 4025	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
122		sspwuni 5096	. . . . . 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 217	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∪ (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
124	98	nnzd 12589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℤ)
125	26, 15	oddvdssubg 19775	. . . . . 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 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))} ∈ (SubGrp‘𝐺))
127	3	mrcsscl 17573	. . . . 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 1368	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐴 ∖ {𝑎}))) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
129		ss2in 4231	. . . 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 583	. . 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 2726	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))}
132		eqid 2726	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))}
133	68	simp2d 1140	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑎↑(𝑎 pCnt (♯‘𝐵))) gcd ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))) = 1)
134		eqid 2726	. . . . 5 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
135	15, 26, 131, 132, 49, 70, 98, 133, 95, 2, 134	ablfacrp 19988	. . . 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 494	. . 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 4017	. 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 19921	1 ⊢ (𝜑 → 𝐺dom DProd 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 {crab 3426 Vcvv 3468 ∖ cdif 3940 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 𝒫 cpw 4597 {csn 4623 ∪ cuni 4902 class class class wbr 5141 ↦ cmpt 5224 dom cdm 5669 ran crn 5670 ↾ cres 5671 “ cima 5672 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 Fincfn 8941 0cc0 11112 1c1 11113 · cmul 11117 / cdiv 11875 ℕcn 12216 ℕ₀cn0 12476 ℤcz 12562 ↑cexp 14032 ♯chash 14295 ∥ cdvds 16204 gcd cgcd 16442 ℙcprime 16615 pCnt cpc 16778 Basecbs 17153 0_gc0g 17394 Moorecmre 17535 mrClscmrc 17536 ACScacs 17538 Grpcgrp 18863 SubGrpcsubg 19047 Cntzccntz 19231 odcod 19444 LSSumclsm 19554 Abelcabl 19701 DProd cdprd 19915

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-omul 8472 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-dvds 16205 df-gcd 16443 df-prm 16616 df-pc 16779 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-eqg 19052 df-cntz 19233 df-od 19448 df-lsm 19556 df-cmn 19702 df-abl 19703 df-dprd 19917

This theorem is referenced by: ablfac1c 19993 ablfac1eu 19995 ablfaclem2 20008 ablfaclem3 20009

W3C validator