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Theorem dchrisum0flblem2 27480

Description: Lemma for dchrisum0flb 27481. Induction over relatively prime factors, with the prime power case handled in dchrisum0flblem1 . (Contributed by Mario Carneiro, 5-May-2016.) Replace reference to OLD theorem. (Revised by Wolf Lammen, 8-Sep-2020.)

**Hypotheses**
Ref	Expression
rpvmasum.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l	⊢ 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a	⊢ (𝜑 → 𝑁 ∈ ℕ)
rpvmasum2.g	⊢ 𝐺 = (DChr‘𝑁)
rpvmasum2.d	⊢ 𝐷 = (Base‘𝐺)
rpvmasum2.1	⊢ 1 = (0_g‘𝐺)
dchrisum0f.f	⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)))
dchrisum0f.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrisum0flb.r	⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
dchrisum0flb.1	⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘2))
dchrisum0flb.2	⊢ (𝜑 → 𝑃 ∈ ℙ)
dchrisum0flb.3	⊢ (𝜑 → 𝑃 ∥ 𝐴)
dchrisum0flb.4	⊢ (𝜑 → ∀𝑦 ∈ (1..^𝐴)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))

**Assertion**
Ref	Expression
dchrisum0flblem2	⊢ (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ (𝐹‘𝐴))

Distinct variable groups: 𝑦, 1 𝑦,𝐹 𝑞,𝑏,𝑣,𝑦,𝐴 𝑁,𝑞,𝑦 𝑃,𝑏,𝑞,𝑣,𝑦 𝑦,𝑍 𝑦,𝐷 𝐿,𝑏,𝑣,𝑦 𝑋,𝑏,𝑣,𝑦 Allowed substitution hints: 𝜑(𝑦,𝑣,𝑞,𝑏) 𝐷(𝑣,𝑞,𝑏) 1 (𝑣,𝑞,𝑏) 𝐹(𝑣,𝑞,𝑏) 𝐺(𝑦,𝑣,𝑞,𝑏) 𝐿(𝑞) 𝑁(𝑣,𝑏) 𝑋(𝑞) 𝑍(𝑣,𝑞,𝑏)

**Proof of Theorem dchrisum0flblem2**
Step	Hyp	Ref	Expression
1		breq1 5102	. . 3 ⊢ (1 = if((√‘𝐴) ∈ ℕ, 1, 0) → (1 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))) ↔ if((√‘𝐴) ∈ ℕ, 1, 0) ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))))
2		breq1 5102	. . 3 ⊢ (0 = if((√‘𝐴) ∈ ℕ, 1, 0) → (0 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))) ↔ if((√‘𝐴) ∈ ℕ, 1, 0) ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))))
3		1t1e1 12306	. . . 4 ⊢ (1 · 1) = 1
4		dchrisum0flb.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℙ)
5	4	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℙ)
6		nnq 12879	. . . . . . . . . . . . . . 15 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ∈ ℚ)
7	6	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ∈ ℚ)
8		nnne0 12183	. . . . . . . . . . . . . . 15 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ≠ 0)
9	8	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ≠ 0)
10		2z 12527	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ
11	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℤ)
12		pcexp 16791	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ ((√‘𝐴) ∈ ℚ ∧ (√‘𝐴) ≠ 0) ∧ 2 ∈ ℤ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (2 · (𝑃 pCnt (√‘𝐴))))
13	5, 7, 9, 11, 12	syl121anc 1378	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (2 · (𝑃 pCnt (√‘𝐴))))
14		dchrisum0flb.1	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘2))
15		eluz2nn 12805	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℕ)
16	14, 15	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℕ)
17	16	nncnd 12165	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℂ)
18	17	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝐴 ∈ ℂ)
19	18	sqsqrtd 15369	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((√‘𝐴)↑2) = 𝐴)
20	19	oveq2d 7376	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (𝑃 pCnt 𝐴))
21		2cnd 12227	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℂ)
22		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ∈ ℕ)
23	5, 22	pccld 16782	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt (√‘𝐴)) ∈ ℕ₀)
24	23	nn0cnd 12468	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt (√‘𝐴)) ∈ ℂ)
25	21, 24	mulcomd 11157	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (2 · (𝑃 pCnt (√‘𝐴))) = ((𝑃 pCnt (√‘𝐴)) · 2))
26	13, 20, 25	3eqtr3rd 2781	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((𝑃 pCnt (√‘𝐴)) · 2) = (𝑃 pCnt 𝐴))
27	26	oveq2d 7376	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑((𝑃 pCnt (√‘𝐴)) · 2)) = (𝑃↑(𝑃 pCnt 𝐴)))
28		prmnn 16605	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
29	5, 28	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℕ)
30	29	nncnd 12165	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℂ)
31		2nn0 12422	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
32	31	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℕ₀)
33	30, 32, 23	expmuld 14076	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑((𝑃 pCnt (√‘𝐴)) · 2)) = ((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2))
34	27, 33	eqtr3d 2774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) = ((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2))
35	34	fveq2d 6839	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝑃↑(𝑃 pCnt 𝐴))) = (√‘((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2)))
36	29, 23	nnexpcld 14172	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℕ)
37	36	nnrpd 12951	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℝ⁺)
38	37	rprege0d 12960	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℝ ∧ 0 ≤ (𝑃↑(𝑃 pCnt (√‘𝐴)))))
39		sqrtsq 15196	. . . . . . . . . 10 ⊢ (((𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℝ ∧ 0 ≤ (𝑃↑(𝑃 pCnt (√‘𝐴)))) → (√‘((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2)) = (𝑃↑(𝑃 pCnt (√‘𝐴))))
40	38, 39	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2)) = (𝑃↑(𝑃 pCnt (√‘𝐴))))
41	35, 40	eqtrd 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝑃↑(𝑃 pCnt 𝐴))) = (𝑃↑(𝑃 pCnt (√‘𝐴))))
42	41, 36	eqeltrd 2837	. . . . . . 7 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ)
43	42	iftrued 4488	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) = 1)
44		rpvmasum.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
45		rpvmasum.l	. . . . . . . 8 ⊢ 𝐿 = (ℤRHom‘𝑍)
46		rpvmasum.a	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
47		rpvmasum2.g	. . . . . . . 8 ⊢ 𝐺 = (DChr‘𝑁)
48		rpvmasum2.d	. . . . . . . 8 ⊢ 𝐷 = (Base‘𝐺)
49		rpvmasum2.1	. . . . . . . 8 ⊢ 1 = (0_g‘𝐺)
50		dchrisum0f.f	. . . . . . . 8 ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)))
51		dchrisum0f.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
52		dchrisum0flb.r	. . . . . . . 8 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
53	4, 16	pccld 16782	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈ ℕ₀)
54	44, 45, 46, 47, 48, 49, 50, 51, 52, 4, 53	dchrisum0flblem1 27479	. . . . . . 7 ⊢ (𝜑 → if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
55	54	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
56	43, 55	eqbrtrrd 5123	. . . . 5 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
57		pcdvds 16796	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
58	4, 16, 57	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
59	4, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ)
60	59, 53	nnexpcld 14172	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
61		nndivdvds 16192	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ) → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴 ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ))
62	16, 60, 61	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴 ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ))
63	58, 62	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ)
64	63	nnzd 12518	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ)
65	64	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ)
66	16	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝐴 ∈ ℕ)
67	66	nnrpd 12951	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝐴 ∈ ℝ⁺)
68	67	rprege0d 12960	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
69	60	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
70	69	nnrpd 12951	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℝ⁺)
71		sqrtdiv 15192	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℝ⁺) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))))
72	68, 70, 71	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))))
73		nnz 12513	. . . . . . . . . . 11 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ∈ ℤ)
74		znq 12869	. . . . . . . . . . 11 ⊢ (((√‘𝐴) ∈ ℤ ∧ (√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ) → ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ)
75	73, 42, 74	syl2an2 687	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ)
76	72, 75	eqeltrd 2837	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ)
77		zsqrtelqelz 16689	. . . . . . . . 9 ⊢ (((𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ ∧ (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ)
78	65, 76, 77	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ)
79	63	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ)
80	79	nnrpd 12951	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ⁺)
81	80	sqrtgt0d 15340	. . . . . . . 8 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 0 < (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
82		elnnz 12502	. . . . . . . 8 ⊢ ((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ ↔ ((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ ∧ 0 < (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
83	78, 81, 82	sylanbrc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ)
84	83	iftrued 4488	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) = 1)
85		fveq2 6835	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → (√‘𝑦) = (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
86	85	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → ((√‘𝑦) ∈ ℕ ↔ (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ))
87	86	ifbid 4504	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → if((√‘𝑦) ∈ ℕ, 1, 0) = if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0))
88		fveq2 6835	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → (𝐹‘𝑦) = (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
89	87, 88	breq12d 5112	. . . . . . . 8 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → (if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) ↔ if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
90		dchrisum0flb.4	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ (1..^𝐴)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))
91		nnuz 12794	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
92	63, 91	eleqtrdi 2847	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ (ℤ_≥‘1))
93	16	nnzd 12518	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℤ)
94	59	nnred 12164	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℝ)
95		dchrisum0flb.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∥ 𝐴)
96		pcelnn 16802	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑃 pCnt 𝐴) ∈ ℕ ↔ 𝑃 ∥ 𝐴))
97	4, 16, 96	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑃 pCnt 𝐴) ∈ ℕ ↔ 𝑃 ∥ 𝐴))
98	95, 97	mpbird 257	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈ ℕ)
99		prmuz2 16627	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
100		eluz2gt1 12837	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
101	4, 99, 100	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 < 𝑃)
102		expgt1 14027	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℝ ∧ (𝑃 pCnt 𝐴) ∈ ℕ ∧ 1 < 𝑃) → 1 < (𝑃↑(𝑃 pCnt 𝐴)))
103	94, 98, 101, 102	syl3anc 1374	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < (𝑃↑(𝑃 pCnt 𝐴)))
104		1red 11137	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
105		0lt1 11663	. . . . . . . . . . . . 13 ⊢ 0 < 1
106	105	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
107	60	nnred 12164	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℝ)
108	60	nngt0d 12198	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < (𝑃↑(𝑃 pCnt 𝐴)))
109	16	nnred 12164	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
110	16	nngt0d 12198	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝐴)
111		ltdiv2 12032	. . . . . . . . . . . 12 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ ((𝑃↑(𝑃 pCnt 𝐴)) ∈ ℝ ∧ 0 < (𝑃↑(𝑃 pCnt 𝐴))) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 < (𝑃↑(𝑃 pCnt 𝐴)) ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < (𝐴 / 1)))
112	104, 106, 107, 108, 109, 110, 111	syl222anc 1389	. . . . . . . . . . 11 ⊢ (𝜑 → (1 < (𝑃↑(𝑃 pCnt 𝐴)) ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < (𝐴 / 1)))
113	103, 112	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < (𝐴 / 1))
114	17	div1d 11913	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 / 1) = 𝐴)
115	113, 114	breqtrd 5125	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < 𝐴)
116		elfzo2 13582	. . . . . . . . 9 ⊢ ((𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ (1..^𝐴) ↔ ((𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ (ℤ_≥‘1) ∧ 𝐴 ∈ ℤ ∧ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < 𝐴))
117	92, 93, 115, 116	syl3anbrc 1345	. . . . . . . 8 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ (1..^𝐴))
118	89, 90, 117	rspcdva 3578	. . . . . . 7 ⊢ (𝜑 → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
119	118	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
120	84, 119	eqbrtrrd 5123	. . . . 5 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
121		1re 11136	. . . . . . . 8 ⊢ 1 ∈ ℝ
122		0le1 11664	. . . . . . . 8 ⊢ 0 ≤ 1
123	121, 122	pm3.2i 470	. . . . . . 7 ⊢ (1 ∈ ℝ ∧ 0 ≤ 1)
124	123	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (1 ∈ ℝ ∧ 0 ≤ 1))
125	44, 45, 46, 47, 48, 49, 50, 51, 52	dchrisum0ff 27478	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
126	125, 60	ffvelcdmd 7032	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ)
127	126	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ)
128	125, 63	ffvelcdmd 7032	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℝ)
129	128	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℝ)
130		lemul12a 12003	. . . . . 6 ⊢ ((((1 ∈ ℝ ∧ 0 ≤ 1) ∧ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ) ∧ ((1 ∈ ℝ ∧ 0 ≤ 1) ∧ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℝ)) → ((1 ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) ∧ 1 ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))) → (1 · 1) ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))))
131	124, 127, 124, 129, 130	syl22anc 839	. . . . 5 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((1 ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) ∧ 1 ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))) → (1 · 1) ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))))
132	56, 120, 131	mp2and 700	. . . 4 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (1 · 1) ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
133	3, 132	eqbrtrrid 5135	. . 3 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
134		0red 11139	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
135		0re 11138	. . . . . . . 8 ⊢ 0 ∈ ℝ
136	121, 135	ifcli 4528	. . . . . . 7 ⊢ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) ∈ ℝ
137	136	a1i 11	. . . . . 6 ⊢ (𝜑 → if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) ∈ ℝ)
138		breq2 5103	. . . . . . . 8 ⊢ (1 = if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) → (0 ≤ 1 ↔ 0 ≤ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0)))
139		breq2 5103	. . . . . . . 8 ⊢ (0 = if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) → (0 ≤ 0 ↔ 0 ≤ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0)))
140		0le0 12250	. . . . . . . 8 ⊢ 0 ≤ 0
141	138, 139, 122, 140	keephyp 4552	. . . . . . 7 ⊢ 0 ≤ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0)
142	141	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ≤ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0))
143	134, 137, 126, 142, 54	letrd 11294	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
144	121, 135	ifcli 4528	. . . . . . 7 ⊢ if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ∈ ℝ
145	144	a1i 11	. . . . . 6 ⊢ (𝜑 → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ∈ ℝ)
146		breq2 5103	. . . . . . . 8 ⊢ (1 = if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) → (0 ≤ 1 ↔ 0 ≤ if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0)))
147		breq2 5103	. . . . . . . 8 ⊢ (0 = if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) → (0 ≤ 0 ↔ 0 ≤ if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0)))
148	146, 147, 122, 140	keephyp 4552	. . . . . . 7 ⊢ 0 ≤ if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0)
149	148	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ≤ if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0))
150	134, 145, 128, 149, 118	letrd 11294	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
151	126, 128, 143, 150	mulge0d 11718	. . . 4 ⊢ (𝜑 → 0 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
152	151	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ (√‘𝐴) ∈ ℕ) → 0 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
153	1, 2, 133, 152	ifbothda 4519	. 2 ⊢ (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
154	60	nncnd 12165	. . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℂ)
155	60	nnne0d 12199	. . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ≠ 0)
156	17, 154, 155	divcan2d 11923	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 𝐴)
157	156	fveq2d 6839	. . 3 ⊢ (𝜑 → (𝐹‘((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))) = (𝐹‘𝐴))
158		pcndvds2 16800	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))
159	4, 16, 158	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))
160		coprm 16642	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ) → (¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ↔ (𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1))
161	4, 64, 160	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ↔ (𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1))
162	159, 161	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1)
163		prmz 16606	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
164	4, 163	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℤ)
165		rpexp1i 16654	. . . . . 6 ⊢ ((𝑃 ∈ ℤ ∧ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ ∧ (𝑃 pCnt 𝐴) ∈ ℕ₀) → ((𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1 → ((𝑃↑(𝑃 pCnt 𝐴)) gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1))
166	164, 64, 53, 165	syl3anc 1374	. . . . 5 ⊢ (𝜑 → ((𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1 → ((𝑃↑(𝑃 pCnt 𝐴)) gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1))
167	162, 166	mpd 15	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐴)) gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1)
168	44, 45, 46, 47, 48, 49, 50, 51, 60, 63, 167	dchrisum0fmul 27477	. . 3 ⊢ (𝜑 → (𝐹‘((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))) = ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
169	157, 168	eqtr3d 2774	. 2 ⊢ (𝜑 → (𝐹‘𝐴) = ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
170	153, 169	breqtrrd 5127	1 ⊢ (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ (𝐹‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 {crab 3400 ifcif 4480 class class class wbr 5099 ↦ cmpt 5180 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 ℂcc 11028 ℝcr 11029 0cc0 11030 1c1 11031 · cmul 11035 < clt 11170 ≤ cle 11171 / cdiv 11798 ℕcn 12149 2c2 12204 ℕ₀cn0 12405 ℤcz 12492 ℤ_≥cuz 12755 ℚcq 12865 ℝ⁺crp 12909 ..^cfzo 13574 ↑cexp 13988 √csqrt 15160 Σcsu 15613 ∥ cdvds 16183 gcd cgcd 16425 ℙcprime 16602 pCnt cpc 16768 Basecbs 17140 0_gc0g 17363 ℤRHomczrh 21458 ℤ/nℤczn 21461 DChrcdchr 27203

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109 ax-mulf 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-disj 5067 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-omul 8404 df-er 8637 df-ec 8639 df-qs 8643 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9855 df-acn 9858 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13269 df-ioc 13270 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-fl 13716 df-mod 13794 df-seq 13929 df-exp 13989 df-fac 14201 df-bc 14230 df-hash 14258 df-shft 14994 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-limsup 15398 df-clim 15415 df-rlim 15416 df-sum 15614 df-ef 15994 df-sin 15996 df-cos 15997 df-pi 15999 df-dvds 16184 df-gcd 16426 df-prm 16603 df-numer 16666 df-denom 16667 df-pc 16769 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-xrs 17427 df-qtop 17432 df-imas 17433 df-qus 17434 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18712 df-submnd 18713 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-nsg 19058 df-eqg 19059 df-ghm 19146 df-cntz 19250 df-od 19461 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20277 df-dvdsr 20297 df-unit 20298 df-invr 20328 df-dvr 20341 df-rhm 20412 df-subrng 20483 df-subrg 20507 df-drng 20668 df-lmod 20817 df-lss 20887 df-lsp 20927 df-sra 21129 df-rgmod 21130 df-lidl 21167 df-rsp 21168 df-2idl 21209 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-zring 21406 df-zrh 21462 df-zn 21465 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24268 df-ms 24269 df-tms 24270 df-cncf 24831 df-limc 25827 df-dv 25828 df-log 26525 df-cxp 26526 df-dchr 27204

This theorem is referenced by: dchrisum0flb 27481

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