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Theorem dchrisum0flblem2 25198

Description: Lemma for dchrisum0flb 25199. 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 4656	. . 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 4656	. . 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 11175	. . . 4 ⊢ (1 · 1) = 1
4		dchrisum0flb.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℙ)
5	4	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℙ)
6		nnq 11801	. . . . . . . . . . . . . . 15 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ∈ ℚ)
7	6	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ∈ ℚ)
8		nnne0 11053	. . . . . . . . . . . . . . 15 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ≠ 0)
9	8	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ≠ 0)
10		2z 11409	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ
11	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℤ)
12		pcexp 15564	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ ((√‘𝐴) ∈ ℚ ∧ (√‘𝐴) ≠ 0) ∧ 2 ∈ ℤ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (2 · (𝑃 pCnt (√‘𝐴))))
13	5, 7, 9, 11, 12	syl121anc 1331	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (2 · (𝑃 pCnt (√‘𝐴))))
14		dchrisum0flb.1	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘2))
15		eluz2nn 11726	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℕ)
16	14, 15	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℕ)
17	16	nncnd 11036	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℂ)
18	17	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝐴 ∈ ℂ)
19	18	sqsqrtd 14178	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((√‘𝐴)↑2) = 𝐴)
20	19	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (𝑃 pCnt 𝐴))
21		2cnd 11093	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℂ)
22		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ∈ ℕ)
23	5, 22	pccld 15555	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt (√‘𝐴)) ∈ ℕ₀)
24	23	nn0cnd 11353	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt (√‘𝐴)) ∈ ℂ)
25	21, 24	mulcomd 10061	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (2 · (𝑃 pCnt (√‘𝐴))) = ((𝑃 pCnt (√‘𝐴)) · 2))
26	13, 20, 25	3eqtr3rd 2665	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((𝑃 pCnt (√‘𝐴)) · 2) = (𝑃 pCnt 𝐴))
27	26	oveq2d 6666	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑((𝑃 pCnt (√‘𝐴)) · 2)) = (𝑃↑(𝑃 pCnt 𝐴)))
28		prmnn 15388	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
29	5, 28	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℕ)
30	29	nncnd 11036	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℂ)
31		2nn0 11309	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
32	31	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℕ₀)
33	30, 32, 23	expmuld 13011	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑((𝑃 pCnt (√‘𝐴)) · 2)) = ((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2))
34	27, 33	eqtr3d 2658	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) = ((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2))
35	34	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝑃↑(𝑃 pCnt 𝐴))) = (√‘((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2)))
36	29, 23	nnexpcld 13030	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℕ)
37	36	nnrpd 11870	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℝ⁺)
38	37	rprege0d 11879	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℝ ∧ 0 ≤ (𝑃↑(𝑃 pCnt (√‘𝐴)))))
39		sqrtsq 14010	. . . . . . . . . 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 2656	. . . . . . . 8 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝑃↑(𝑃 pCnt 𝐴))) = (𝑃↑(𝑃 pCnt (√‘𝐴))))
42	41, 36	eqeltrd 2701	. . . . . . 7 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ)
43	42	iftrued 4094	. . . . . 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 15555	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈ ℕ₀)
54	44, 45, 46, 47, 48, 49, 50, 51, 52, 4, 53	dchrisum0flblem1 25197	. . . . . . 7 ⊢ (𝜑 → if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
55	54	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
56	43, 55	eqbrtrrd 4677	. . . . 5 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
57		pcdvds 15568	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
58	4, 16, 57	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
59	4, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ)
60	59, 53	nnexpcld 13030	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
61		nndivdvds 14989	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ) → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴 ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ))
62	16, 60, 61	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴 ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ))
63	58, 62	mpbid 222	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ)
64	63	nnzd 11481	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ)
65	64	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ)
66	16	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝐴 ∈ ℕ)
67	66	nnrpd 11870	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝐴 ∈ ℝ⁺)
68	67	rprege0d 11879	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
69	60	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
70	69	nnrpd 11870	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℝ⁺)
71		sqrtdiv 14006	. . . . . . . . . . 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 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))))
73		nnz 11399	. . . . . . . . . . . 12 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ∈ ℤ)
74	73	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ∈ ℤ)
75		znq 11792	. . . . . . . . . . 11 ⊢ (((√‘𝐴) ∈ ℤ ∧ (√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ) → ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ)
76	74, 42, 75	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ)
77	72, 76	eqeltrd 2701	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ)
78		zsqrtelqelz 15466	. . . . . . . . 9 ⊢ (((𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ ∧ (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ)
79	65, 77, 78	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ)
80	63	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ)
81	80	nnrpd 11870	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ⁺)
82	81	sqrtgt0d 14151	. . . . . . . 8 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 0 < (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
83		elnnz 11387	. . . . . . . 8 ⊢ ((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ ↔ ((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ ∧ 0 < (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
84	79, 82, 83	sylanbrc 698	. . . . . . 7 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ)
85	84	iftrued 4094	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) = 1)
86		nnuz 11723	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
87	63, 86	syl6eleq 2711	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ (ℤ_≥‘1))
88	16	nnzd 11481	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℤ)
89	59	nnred 11035	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℝ)
90		dchrisum0flb.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∥ 𝐴)
91		pcelnn 15574	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑃 pCnt 𝐴) ∈ ℕ ↔ 𝑃 ∥ 𝐴))
92	4, 16, 91	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑃 pCnt 𝐴) ∈ ℕ ↔ 𝑃 ∥ 𝐴))
93	90, 92	mpbird 247	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈ ℕ)
94		prmuz2 15408	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
95		eluz2b2 11761	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℤ_≥‘2) ↔ (𝑃 ∈ ℕ ∧ 1 < 𝑃))
96	95	simprbi 480	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
97	4, 94, 96	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 < 𝑃)
98		expgt1 12898	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℝ ∧ (𝑃 pCnt 𝐴) ∈ ℕ ∧ 1 < 𝑃) → 1 < (𝑃↑(𝑃 pCnt 𝐴)))
99	89, 93, 97, 98	syl3anc 1326	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < (𝑃↑(𝑃 pCnt 𝐴)))
100		1red 10055	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
101		0lt1 10550	. . . . . . . . . . . . 13 ⊢ 0 < 1
102	101	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
103	60	nnred 11035	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℝ)
104	60	nngt0d 11064	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < (𝑃↑(𝑃 pCnt 𝐴)))
105	16	nnred 11035	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
106	16	nngt0d 11064	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝐴)
107		ltdiv2 10909	. . . . . . . . . . . 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)))
108	100, 102, 103, 104, 105, 106, 107	syl222anc 1342	. . . . . . . . . . 11 ⊢ (𝜑 → (1 < (𝑃↑(𝑃 pCnt 𝐴)) ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < (𝐴 / 1)))
109	99, 108	mpbid 222	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < (𝐴 / 1))
110	17	div1d 10793	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 / 1) = 𝐴)
111	109, 110	breqtrd 4679	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < 𝐴)
112		elfzo2 12473	. . . . . . . . 9 ⊢ ((𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ (1..^𝐴) ↔ ((𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ (ℤ_≥‘1) ∧ 𝐴 ∈ ℤ ∧ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < 𝐴))
113	87, 88, 111, 112	syl3anbrc 1246	. . . . . . . 8 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ (1..^𝐴))
114		dchrisum0flb.4	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ (1..^𝐴)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦))
115		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → (√‘𝑦) = (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
116	115	eleq1d 2686	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → ((√‘𝑦) ∈ ℕ ↔ (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ))
117	116	ifbid 4108	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → if((√‘𝑦) ∈ ℕ, 1, 0) = if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0))
118		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → (𝐹‘𝑦) = (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
119	117, 118	breq12d 4666	. . . . . . . . 9 ⊢ (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))))))
120	119	rspcv 3305	. . . . . . . 8 ⊢ ((𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ (1..^𝐴) → (∀𝑦 ∈ (1..^𝐴)if((√‘𝑦) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑦) → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
121	113, 114, 120	sylc 65	. . . . . . 7 ⊢ (𝜑 → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
122	121	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
123	85, 122	eqbrtrrd 4677	. . . . 5 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
124		1re 10039	. . . . . . . 8 ⊢ 1 ∈ ℝ
125		0le1 10551	. . . . . . . 8 ⊢ 0 ≤ 1
126	124, 125	pm3.2i 471	. . . . . . 7 ⊢ (1 ∈ ℝ ∧ 0 ≤ 1)
127	126	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (1 ∈ ℝ ∧ 0 ≤ 1))
128	44, 45, 46, 47, 48, 49, 50, 51, 52	dchrisum0ff 25196	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
129	128, 60	ffvelrnd 6360	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ)
130	129	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ)
131	128, 63	ffvelrnd 6360	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℝ)
132	131	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℝ)
133		lemul12a 10881	. . . . . 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)))))))
134	127, 130, 127, 132, 133	syl22anc 1327	. . . . 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)))))))
135	56, 123, 134	mp2and 715	. . . 4 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (1 · 1) ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
136	3, 135	syl5eqbrr 4689	. . 3 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
137		0red 10041	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
138		0re 10040	. . . . . . . 8 ⊢ 0 ∈ ℝ
139	124, 138	keepel 4155	. . . . . . 7 ⊢ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) ∈ ℝ
140	139	a1i 11	. . . . . 6 ⊢ (𝜑 → if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) ∈ ℝ)
141		breq2 4657	. . . . . . . 8 ⊢ (1 = if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) → (0 ≤ 1 ↔ 0 ≤ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0)))
142		breq2 4657	. . . . . . . 8 ⊢ (0 = if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) → (0 ≤ 0 ↔ 0 ≤ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0)))
143		0le0 11110	. . . . . . . 8 ⊢ 0 ≤ 0
144	141, 142, 125, 143	keephyp 4152	. . . . . . 7 ⊢ 0 ≤ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0)
145	144	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ≤ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0))
146	137, 140, 129, 145, 54	letrd 10194	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
147	124, 138	keepel 4155	. . . . . . 7 ⊢ if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ∈ ℝ
148	147	a1i 11	. . . . . 6 ⊢ (𝜑 → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ∈ ℝ)
149		breq2 4657	. . . . . . . 8 ⊢ (1 = if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) → (0 ≤ 1 ↔ 0 ≤ if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0)))
150		breq2 4657	. . . . . . . 8 ⊢ (0 = if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) → (0 ≤ 0 ↔ 0 ≤ if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0)))
151	149, 150, 125, 143	keephyp 4152	. . . . . . 7 ⊢ 0 ≤ if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0)
152	151	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ≤ if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0))
153	137, 148, 131, 152, 121	letrd 10194	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
154	129, 131, 146, 153	mulge0d 10604	. . . 4 ⊢ (𝜑 → 0 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
155	154	adantr 481	. . 3 ⊢ ((𝜑 ∧ ¬ (√‘𝐴) ∈ ℕ) → 0 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
156	1, 2, 136, 155	ifbothda 4123	. 2 ⊢ (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
157	60	nncnd 11036	. . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℂ)
158	60	nnne0d 11065	. . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ≠ 0)
159	17, 157, 158	divcan2d 10803	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 𝐴)
160	159	fveq2d 6195	. . 3 ⊢ (𝜑 → (𝐹‘((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))) = (𝐹‘𝐴))
161		pcndvds2 15572	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))
162	4, 16, 161	syl2anc 693	. . . . . 6 ⊢ (𝜑 → ¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))
163		coprm 15423	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ) → (¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ↔ (𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1))
164	4, 64, 163	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ↔ (𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1))
165	162, 164	mpbid 222	. . . . 5 ⊢ (𝜑 → (𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1)
166		prmz 15389	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
167	4, 166	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℤ)
168		rpexp1i 15433	. . . . . 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))
169	167, 64, 53, 168	syl3anc 1326	. . . . 5 ⊢ (𝜑 → ((𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1 → ((𝑃↑(𝑃 pCnt 𝐴)) gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1))
170	165, 169	mpd 15	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐴)) gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1)
171	44, 45, 46, 47, 48, 49, 50, 51, 60, 63, 170	dchrisum0fmul 25195	. . 3 ⊢ (𝜑 → (𝐹‘((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))) = ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
172	160, 171	eqtr3d 2658	. 2 ⊢ (𝜑 → (𝐹‘𝐴) = ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
173	156, 172	breqtrrd 4681	1 ⊢ (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ (𝐹‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 ifcif 4086 class class class wbr 4653 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 < clt 10074 ≤ cle 10075 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ℚcq 11788 ℝ⁺crp 11832 ..^cfzo 12465 ↑cexp 12860 √csqrt 13973 Σcsu 14416 ∥ cdvds 14983 gcd cgcd 15216 ℙcprime 15385 pCnt cpc 15541 Basecbs 15857 0_gc0g 16100 ℤRHomczrh 19848 ℤ/nℤczn 19851 DChrcdchr 24957

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-dvds 14984 df-gcd 15217 df-prm 15386 df-numer 15443 df-denom 15444 df-pc 15542 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-qus 16169 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-nsg 17592 df-eqg 17593 df-ghm 17658 df-cntz 17750 df-od 17948 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-rnghom 18715 df-drng 18749 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-sra 19172 df-rgmod 19173 df-lidl 19174 df-rsp 19175 df-2idl 19232 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-zring 19819 df-zrh 19852 df-zn 19855 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304 df-dchr 24958

This theorem is referenced by: dchrisum0flb 25199

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