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Theorem ablfac1b 19985

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 2731	. 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2731	. 2 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
3		eqid 2731	. 2 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		ablfac1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Abel)
5		ablgrp 19698	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		ablfac1.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
8		prmex 16588	. . . 4 ⊢ ℙ ∈ V
9	8	ssex 5259	. . 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 3934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ ℙ)
13		prmnn 16585	. . . . . . 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 18879	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
17	6, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≠ ∅)
18		ablfac1.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Fin)
19		hashnncl 14273	. . . . . . . . . 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 16762	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝 pCnt (♯‘𝐵)) ∈ ℕ₀)
24	14, 23	nnexpcld 14152	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℕ)
25	24	nnzd 12495	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℤ)
26		ablfac1.o	. . . . 5 ⊢ 𝑂 = (od‘𝐺)
27	26, 15	oddvdssubg 19768	. . . 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 7047	. 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 1195	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑎 ∈ 𝐴)
34	32, 33	ffvelcdmd 7018	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑎) ∈ (SubGrp‘𝐺))
35		simpr2 1196	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → 𝑏 ∈ 𝐴)
36	32, 35	ffvelcdmd 7018	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑏) ∈ (SubGrp‘𝐺))
37	1, 31, 34, 36	ablcntzd 19770	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑎 ≠ 𝑏)) → (𝑆‘𝑎) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑏)))
38		id 22	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → 𝑝 = 𝑎)
39		oveq1 7353	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → (𝑝 pCnt (♯‘𝐵)) = (𝑎 pCnt (♯‘𝐵)))
40	38, 39	oveq12d 7364	. . . . . . . . 9 ⊢ (𝑝 = 𝑎 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑎↑(𝑎 pCnt (♯‘𝐵))))
41	40	breq2d 5103	. . . . . . . 8 ⊢ (𝑝 = 𝑎 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))))
42	41	rabbidv 3402	. . . . . . 7 ⊢ (𝑝 = 𝑎 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))})
43	15	fvexi 6836	. . . . . . . 8 ⊢ 𝐵 ∈ V
44	43	rabex 5277	. . . . . . 7 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
45	42, 29, 44	fvmpt3i 6934	. . . . . 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 3993	. . . . 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 2731	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
51	50	subgacs 19074	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
52		acsmre 17558	. . . . . 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 5629	. . . . . . 7 ⊢ (𝑆 “ (𝐴 ∖ {𝑎})) = ran (𝑆 ↾ (𝐴 ∖ {𝑎}))
55	7	sselda 3934	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℙ)
56	55	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ∈ ℙ)
57	21	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (♯‘𝐵) ∈ ℕ)
58		pcdvds 16776	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝐴 ⊆ ℙ)
61		eldifi 4081	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ (𝐴 ∖ {𝑎}) → 𝑝 ∈ 𝐴)
62	61	ad2antlr 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ 𝐴)
63	60, 62	sseldd 3935	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ∈ ℙ)
64		pcdvds 16776	. . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎↑(𝑎 pCnt (♯‘𝐵))) = (𝑎↑(𝑎 pCnt (♯‘𝐵)))
67		eqid 2731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) = ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))
68	15, 26, 29, 4, 18, 7, 66, 67	ablfac1lem 19983	. . . . . . . . . . . . . . . . . . . 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 1142	. . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ∈ ℕ)
72	71	nnzd 12495	. . . . . . . . . . . . . . . 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 16762	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝 pCnt (♯‘𝐵)) ∈ ℕ₀)
75	73, 74	nnexpcld 14152	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℕ)
76	75	nnzd 12495	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∈ ℤ)
77	57	nnzd 12495	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (♯‘𝐵) ∈ ℤ)
78		eldifsni 4742	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (𝐴 ∖ {𝑎}) → 𝑝 ≠ 𝑎)
79	78	ad2antlr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑝 ≠ 𝑎)
80	79	necomd 2983	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ≠ 𝑝)
81		prmrp 16623	. . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎 gcd 𝑝) = 1)
84		prmz 16586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℙ → 𝑎 ∈ ℤ)
85	56, 84	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → 𝑎 ∈ ℤ)
86		prmz 16586	. . . . . . . . . . . . . . . . . . 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 16762	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎 pCnt (♯‘𝐵)) ∈ ℕ₀)
89		rpexp12i 16635	. . . . . . . . . . . . . . . . . 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 1376	. . . . . . . . . . . . . . . . 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 16565	. . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . 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 699	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((𝑎↑(𝑎 pCnt (♯‘𝐵))) · (𝑝↑(𝑝 pCnt (♯‘𝐵)))) ∥ (♯‘𝐵))
95	68	simp3d 1144	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (♯‘𝐵) = ((𝑎↑(𝑎 pCnt (♯‘𝐵))) · ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))))
96	95	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (♯‘𝐵) = ((𝑎↑(𝑎 pCnt (♯‘𝐵))) · ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))))
97	94, 96	breqtrd 5117	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℕ)
100	99	nnzd 12495	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℤ)
101	71	nnne0d 12175	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑎↑(𝑎 pCnt (♯‘𝐵))) ≠ 0)
102		dvdscmulr 16195	. . . . . . . . . . . . . 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 1376	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑝↑(𝑝 pCnt (♯‘𝐵))) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))))
105	15, 26	odcl 19449	. . . . . . . . . . . . . . 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 12494	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 ∈ (𝐴 ∖ {𝑎})) ∧ 𝑥 ∈ 𝐵) → (𝑂‘𝑥) ∈ ℤ)
108		dvdstr 16205	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 694	. . . . . . . . . . 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 4026	. . . . . . . . . 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 4554	. . . . . . . . . 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 234	. . . . . . . . 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 5924	. . . . . . . . . 10 ⊢ (𝑆 ↾ (𝐴 ∖ {𝑎})) = ((𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) ↾ (𝐴 ∖ {𝑎}))
115		difss 4086	. . . . . . . . . . 11 ⊢ (𝐴 ∖ {𝑎}) ⊆ 𝐴
116		resmpt 5986	. . . . . . . . . . 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 7047	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆 ↾ (𝐴 ∖ {𝑎})):(𝐴 ∖ {𝑎})⟶𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
120	119	frnd 6659	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ran (𝑆 ↾ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
121	54, 120	eqsstrid 3973	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ 𝒫 {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
122		sspwuni 5048	. . . . . 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 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∪ (𝑆 “ (𝐴 ∖ {𝑎})) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
124	98	nnzd 12495	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵)))) ∈ ℤ)
125	26, 15	oddvdssubg 19768	. . . . . 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 17526	. . . . 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 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐴 ∖ {𝑎}))) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))})
129		ss2in 4195	. . . 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 2731	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑎↑(𝑎 pCnt (♯‘𝐵)))}
132		eqid 2731	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))}
133	68	simp2d 1143	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑎↑(𝑎 pCnt (♯‘𝐵))) gcd ((♯‘𝐵) / (𝑎↑(𝑎 pCnt (♯‘𝐵))))) = 1)
134		eqid 2731	. . . . 5 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
135	15, 26, 131, 132, 49, 70, 98, 133, 95, 2, 134	ablfacrp 19981	. . . 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 3971	. 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 19914	1 ⊢ (𝜑 → 𝐺dom DProd 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 {crab 3395 Vcvv 3436 ∖ cdif 3899 ∩ cin 3901 ⊆ wss 3902 ∅c0 4283 𝒫 cpw 4550 {csn 4576 ∪ cuni 4859 class class class wbr 5091 ↦ cmpt 5172 dom cdm 5616 ran crn 5617 ↾ cres 5618 “ cima 5619 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 Fincfn 8869 0cc0 11006 1c1 11007 · cmul 11011 / cdiv 11774 ℕcn 12125 ℕ₀cn0 12381 ℤcz 12468 ↑cexp 13968 ♯chash 14237 ∥ cdvds 16163 gcd cgcd 16405 ℙcprime 16582 pCnt cpc 16748 Basecbs 17120 0_gc0g 17343 Moorecmre 17484 mrClscmrc 17485 ACScacs 17487 Grpcgrp 18846 SubGrpcsubg 19033 Cntzccntz 19228 odcod 19437 LSSumclsm 19547 Abelcabl 19694 DProd cdprd 19908

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-disj 5059 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-omul 8390 df-er 8622 df-ec 8624 df-qs 8628 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-acn 9835 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-z 12469 df-uz 12733 df-q 12847 df-rp 12891 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-dvds 16164 df-gcd 16406 df-prm 16583 df-pc 16749 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-0g 17345 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-eqg 19038 df-cntz 19230 df-od 19441 df-lsm 19549 df-cmn 19695 df-abl 19696 df-dprd 19910

This theorem is referenced by: ablfac1c 19986 ablfac1eu 19988 ablfaclem2 20001 ablfaclem3 20002

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