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Theorem dchrisum0flblem2 27836

Description: Lemma for dchrisum0flb 27837. 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 5105	. . 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 5105	. . 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 12504	. . . 4 ⊢ (1 · 1) = 1
4		dchrisum0flb.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℙ)
5	4	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℙ)
6		nnq 13089	. . . . . . . . . . . . . . 15 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ∈ ℚ)
7	6	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ∈ ℚ)
8		nnne0 12372	. . . . . . . . . . . . . . 15 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ≠ 0)
9	8	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ≠ 0)
10		2z 12728	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ
11	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℤ)
12		pcexp 17037	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ ((√‘𝐴) ∈ ℚ ∧ (√‘𝐴) ≠ 0) ∧ 2 ∈ ℤ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (2 · (𝑃 pCnt (√‘𝐴))))
13	5, 7, 9, 11, 12	syl121anc 1402	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (2 · (𝑃 pCnt (√‘𝐴))))
14		dchrisum0flb.1	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘2))
15		eluz2nn 13015	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℕ)
16	14, 15	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℕ)
17	16	nncnd 12351	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℂ)
18	17	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝐴 ∈ ℂ)
19	18	sqsqrtd 15609	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((√‘𝐴)↑2) = 𝐴)
20	19	oveq2d 7436	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (𝑃 pCnt 𝐴))
21		2cnd 12421	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℂ)
22		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ∈ ℕ)
23	5, 22	pccld 17028	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt (√‘𝐴)) ∈ ℕ₀)
24	23	nn0cnd 12669	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt (√‘𝐴)) ∈ ℂ)
25	21, 24	mulcomd 11330	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (2 · (𝑃 pCnt (√‘𝐴))) = ((𝑃 pCnt (√‘𝐴)) · 2))
26	13, 20, 25	3eqtr3rd 2804	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((𝑃 pCnt (√‘𝐴)) · 2) = (𝑃 pCnt 𝐴))
27	26	oveq2d 7436	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑((𝑃 pCnt (√‘𝐴)) · 2)) = (𝑃↑(𝑃 pCnt 𝐴)))
28		prmnn 16849	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
29	5, 28	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℕ)
30	29	nncnd 12351	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℂ)
31		2nn0 12623	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
32	31	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℕ₀)
33	30, 32, 23	expmuld 14292	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑((𝑃 pCnt (√‘𝐴)) · 2)) = ((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2))
34	27, 33	eqtr3d 2797	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) = ((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2))
35	34	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝑃↑(𝑃 pCnt 𝐴))) = (√‘((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2)))
36	29, 23	nnexpcld 14389	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℕ)
37	36	nnrpd 13162	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℝ⁺)
38	37	rprege0d 13171	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℝ ∧ 0 ≤ (𝑃↑(𝑃 pCnt (√‘𝐴)))))
39		sqrtsq 15436	. . . . . . . . . 10 ⊢ (((𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℝ ∧ 0 ≤ (𝑃↑(𝑃 pCnt (√‘𝐴)))) → (√‘((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2)) = (𝑃↑(𝑃 pCnt (√‘𝐴))))
40	38, 39	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2)) = (𝑃↑(𝑃 pCnt (√‘𝐴))))
41	35, 40	eqtrd 2795	. . . . . . . 8 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝑃↑(𝑃 pCnt 𝐴))) = (𝑃↑(𝑃 pCnt (√‘𝐴))))
42	41, 36	eqeltrd 2860	. . . . . . 7 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ)
43	42	iftrued 4489	. . . . . 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 17028	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈ ℕ₀)
54	44, 45, 46, 47, 48, 49, 50, 51, 52, 4, 53	dchrisum0flblem1 27835	. . . . . . 7 ⊢ (𝜑 → if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
55	54	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
56	43, 55	eqbrtrrd 5128	. . . . 5 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
57		pcdvds 17042	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
58	4, 16, 57	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
59	4, 28	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ)
60	59, 53	nnexpcld 14389	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
61		nndivdvds 16431	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ) → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴 ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ))
62	16, 60, 61	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴 ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ))
63	58, 62	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ)
64	63	nnzd 12719	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ)
65	64	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ)
66	16	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝐴 ∈ ℕ)
67	66	nnrpd 13162	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝐴 ∈ ℝ⁺)
68	67	rprege0d 13171	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
69	60	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
70	69	nnrpd 13162	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℝ⁺)
71		sqrtdiv 15432	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))))
73		nnz 12714	. . . . . . . . . . 11 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ∈ ℤ)
74		znq 13079	. . . . . . . . . . 11 ⊢ (((√‘𝐴) ∈ ℤ ∧ (√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ) → ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ)
75	73, 42, 74	syl2an2 699	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ)
76	72, 75	eqeltrd 2860	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ)
77		zsqrtelqelz 16934	. . . . . . . . 9 ⊢ (((𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ ∧ (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ)
78	65, 76, 77	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ)
79	63	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ)
80	79	nnrpd 13162	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ⁺)
81	80	sqrtgt0d 15580	. . . . . . . 8 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 0 < (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
82		elnnz 12703	. . . . . . . 8 ⊢ ((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ ↔ ((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ ∧ 0 < (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
83	78, 81, 82	sylanbrc 595	. . . . . . 7 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ)
84	83	iftrued 4489	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) = 1)
85		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → (√‘𝑦) = (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
86	85	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → ((√‘𝑦) ∈ ℕ ↔ (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ))
87	86	ifbid 4505	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → if((√‘𝑦) ∈ ℕ, 1, 0) = if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0))
88		fveq2 6885	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → (𝐹‘𝑦) = (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
89	87, 88	breq12d 5115	. . . . . . . 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 13004	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
92	63, 91	eleqtrdi 2870	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ (ℤ_≥‘1))
93	16	nnzd 12719	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℤ)
94	59	nnred 12350	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℝ)
95		dchrisum0flb.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∥ 𝐴)
96		pcelnn 17048	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑃 pCnt 𝐴) ∈ ℕ ↔ 𝑃 ∥ 𝐴))
97	4, 16, 96	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑃 pCnt 𝐴) ∈ ℕ ↔ 𝑃 ∥ 𝐴))
98	95, 97	mpbird 260	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈ ℕ)
99		prmuz2 16871	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
100		eluz2gt1 13047	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
101	4, 99, 100	3syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 < 𝑃)
102		expgt1 14243	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℝ ∧ (𝑃 pCnt 𝐴) ∈ ℕ ∧ 1 < 𝑃) → 1 < (𝑃↑(𝑃 pCnt 𝐴)))
103	94, 98, 101, 102	syl3anc 1398	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < (𝑃↑(𝑃 pCnt 𝐴)))
104		1red 11309	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
105		0lt1 11838	. . . . . . . . . . . . 13 ⊢ 0 < 1
106	105	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
107	60	nnred 12350	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℝ)
108	60	nngt0d 12387	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < (𝑃↑(𝑃 pCnt 𝐴)))
109	16	nnred 12350	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
110	16	nngt0d 12387	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝐴)
111		ltdiv2 12203	. . . . . . . . . . . 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 1413	. . . . . . . . . . 11 ⊢ (𝜑 → (1 < (𝑃↑(𝑃 pCnt 𝐴)) ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < (𝐴 / 1)))
113	103, 112	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < (𝐴 / 1))
114	17	div1d 12085	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 / 1) = 𝐴)
115	113, 114	breqtrd 5130	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < 𝐴)
116		elfzo2 13796	. . . . . . . . 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 1362	. . . . . . . 8 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ (1..^𝐴))
118	89, 90, 117	rspcdva 3577	. . . . . . 7 ⊢ (𝜑 → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
119	118	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
120	84, 119	eqbrtrrd 5128	. . . . 5 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
121		1re 11308	. . . . . . . 8 ⊢ 1 ∈ ℝ
122		0le1 11839	. . . . . . . 8 ⊢ 0 ≤ 1
123	121, 122	pm3.2i 476	. . . . . . 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 27834	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
126	125, 60	ffvelcdmd 7085	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ)
127	126	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ)
128	125, 63	ffvelcdmd 7085	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℝ)
129	128	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℝ)
130		lemul12a 12175	. . . . . 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 852	. . . . 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 712	. . . 4 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (1 · 1) ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
133	3, 132	eqbrtrrid 5140	. . 3 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
134		0red 11311	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
135		0re 11310	. . . . . . . 8 ⊢ 0 ∈ ℝ
136	121, 135	ifcli 4529	. . . . . . 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 5106	. . . . . . . 8 ⊢ (1 = if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) → (0 ≤ 1 ↔ 0 ≤ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0)))
139		breq2 5106	. . . . . . . 8 ⊢ (0 = if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) → (0 ≤ 0 ↔ 0 ≤ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0)))
140		0le0 12444	. . . . . . . 8 ⊢ 0 ≤ 0
141	138, 139, 122, 140	keephyp 4553	. . . . . . 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 11467	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
144	121, 135	ifcli 4529	. . . . . . 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 5106	. . . . . . . 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 5106	. . . . . . . 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 4553	. . . . . . 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 11467	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
151	126, 128, 143, 150	mulge0d 11893	. . . 4 ⊢ (𝜑 → 0 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
152	151	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ (√‘𝐴) ∈ ℕ) → 0 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
153	1, 2, 133, 152	ifbothda 4520	. 2 ⊢ (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
154	60	nncnd 12351	. . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℂ)
155	60	nnne0d 12388	. . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ≠ 0)
156	17, 154, 155	divcan2d 12095	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 𝐴)
157	156	fveq2d 6889	. . 3 ⊢ (𝜑 → (𝐹‘((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))) = (𝐹‘𝐴))
158		pcndvds2 17046	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))
159	4, 16, 158	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))
160		coprm 16887	. . . . . . 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 596	. . . . . 6 ⊢ (𝜑 → (¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ↔ (𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1))
162	159, 161	mpbid 235	. . . . 5 ⊢ (𝜑 → (𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1)
163		prmz 16850	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
164	4, 163	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℤ)
165		rpexp1i 16899	. . . . . 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 1398	. . . . 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 16	. . . 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 27833	. . 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 2797	. 2 ⊢ (𝜑 → (𝐹‘𝐴) = ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
170	153, 169	breqtrrd 5132	1 ⊢ (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ (𝐹‘𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 {crab 3412 ifcif 4481 class class class wbr 5102 ↦ cmpt 5185 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 · cmul 11205 < clt 11343 ≤ cle 11344 / cdiv 11973 ℕcn 12335 2c2 12397 ℕ₀cn0 12606 ℤcz 12693 ℤ_≥cuz 12965 ℚcq 13075 ℝ⁺crp 13120 ..^cfzo 13788 ↑cexp 14204 √csqrt 15400 Σcsu 15853 ∥ cdvds 16422 gcd cgcd 16664 ℙcprime 16846 pCnt cpc 17014 Basecbs 17387 0_gc0g 17610 ℤRHomczrh 21805 ℤ/nℤczn 21808 DChrcdchr 27559

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279 ax-mulf 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 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 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-tpos 8243 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-oadd 8480 df-omul 8481 df-er 8717 df-ec 8719 df-qs 8723 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-acn 10023 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-shft 15220 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-limsup 15638 df-clim 15655 df-rlim 15656 df-sum 15854 df-ef 16233 df-sin 16235 df-cos 16236 df-pi 16238 df-dvds 16423 df-gcd 16665 df-prm 16847 df-numer 16911 df-denom 16912 df-pc 17015 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-qus 17681 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-submnd 18979 df-grp 19147 df-minusg 19148 df-sbg 19149 df-mulg 19278 df-subg 19333 df-nsg 19334 df-eqg 19335 df-ghm 19428 df-cntz 19531 df-od 19742 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-ring 20461 df-cring 20462 df-oppr 20567 df-dvdsr 20587 df-unit 20588 df-invr 20618 df-dvr 20631 df-rhm 20702 df-subrng 20798 df-subrg 20822 df-drng 20982 df-lmod 21137 df-lss 21207 df-lsp 21247 df-sra 21448 df-rgmod 21449 df-lidl 21486 df-rsp 21487 df-2idl 21543 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-zring 21753 df-zrh 21809 df-zn 21812 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-limc 26186 df-dv 26187 df-log 26884 df-cxp 26885 df-dchr 27560

This theorem is used by: dchrisum0flb 27837

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