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Theorem dchrisum0flblem2 26079

Description: Lemma for dchrisum0flb 26080. 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 5061	. . 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 5061	. . 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 11793	. . . 4 ⊢ (1 · 1) = 1
4		dchrisum0flb.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℙ)
5	4	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℙ)
6		nnq 12355	. . . . . . . . . . . . . . 15 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ∈ ℚ)
7	6	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ∈ ℚ)
8		nnne0 11665	. . . . . . . . . . . . . . 15 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ≠ 0)
9	8	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ≠ 0)
10		2z 12008	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ
11	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℤ)
12		pcexp 16190	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ ((√‘𝐴) ∈ ℚ ∧ (√‘𝐴) ≠ 0) ∧ 2 ∈ ℤ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (2 · (𝑃 pCnt (√‘𝐴))))
13	5, 7, 9, 11, 12	syl121anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (2 · (𝑃 pCnt (√‘𝐴))))
14		dchrisum0flb.1	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘2))
15		eluz2nn 12278	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℕ)
16	14, 15	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℕ)
17	16	nncnd 11648	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℂ)
18	17	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝐴 ∈ ℂ)
19	18	sqsqrtd 14793	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((√‘𝐴)↑2) = 𝐴)
20	19	oveq2d 7166	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (𝑃 pCnt 𝐴))
21		2cnd 11709	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℂ)
22		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ∈ ℕ)
23	5, 22	pccld 16181	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt (√‘𝐴)) ∈ ℕ₀)
24	23	nn0cnd 11951	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt (√‘𝐴)) ∈ ℂ)
25	21, 24	mulcomd 10656	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (2 · (𝑃 pCnt (√‘𝐴))) = ((𝑃 pCnt (√‘𝐴)) · 2))
26	13, 20, 25	3eqtr3rd 2865	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((𝑃 pCnt (√‘𝐴)) · 2) = (𝑃 pCnt 𝐴))
27	26	oveq2d 7166	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑((𝑃 pCnt (√‘𝐴)) · 2)) = (𝑃↑(𝑃 pCnt 𝐴)))
28		prmnn 16012	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
29	5, 28	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℕ)
30	29	nncnd 11648	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℂ)
31		2nn0 11908	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
32	31	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℕ₀)
33	30, 32, 23	expmuld 13507	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑((𝑃 pCnt (√‘𝐴)) · 2)) = ((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2))
34	27, 33	eqtr3d 2858	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) = ((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2))
35	34	fveq2d 6668	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝑃↑(𝑃 pCnt 𝐴))) = (√‘((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2)))
36	29, 23	nnexpcld 13600	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℕ)
37	36	nnrpd 12423	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℝ⁺)
38	37	rprege0d 12432	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℝ ∧ 0 ≤ (𝑃↑(𝑃 pCnt (√‘𝐴)))))
39		sqrtsq 14623	. . . . . . . . . 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 2856	. . . . . . . 8 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝑃↑(𝑃 pCnt 𝐴))) = (𝑃↑(𝑃 pCnt (√‘𝐴))))
42	41, 36	eqeltrd 2913	. . . . . . 7 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ)
43	42	iftrued 4474	. . . . . 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 16181	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈ ℕ₀)
54	44, 45, 46, 47, 48, 49, 50, 51, 52, 4, 53	dchrisum0flblem1 26078	. . . . . . 7 ⊢ (𝜑 → if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
55	54	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
56	43, 55	eqbrtrrd 5082	. . . . 5 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
57		pcdvds 16194	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
58	4, 16, 57	syl2anc 586	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
59	4, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ)
60	59, 53	nnexpcld 13600	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
61		nndivdvds 15610	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ) → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴 ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ))
62	16, 60, 61	syl2anc 586	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴 ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ))
63	58, 62	mpbid 234	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ)
64	63	nnzd 12080	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ)
65	64	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ)
66	16	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝐴 ∈ ℕ)
67	66	nnrpd 12423	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝐴 ∈ ℝ⁺)
68	67	rprege0d 12432	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
69	60	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
70	69	nnrpd 12423	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℝ⁺)
71		sqrtdiv 14619	. . . . . . . . . . 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 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))))
73		nnz 11998	. . . . . . . . . . 11 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ∈ ℤ)
74		znq 12346	. . . . . . . . . . 11 ⊢ (((√‘𝐴) ∈ ℤ ∧ (√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ) → ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ)
75	73, 42, 74	syl2an2 684	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ)
76	72, 75	eqeltrd 2913	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ)
77		zsqrtelqelz 16092	. . . . . . . . 9 ⊢ (((𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ ∧ (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ)
78	65, 76, 77	syl2anc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ)
79	63	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ)
80	79	nnrpd 12423	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ⁺)
81	80	sqrtgt0d 14766	. . . . . . . 8 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 0 < (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
82		elnnz 11985	. . . . . . . 8 ⊢ ((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ ↔ ((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ ∧ 0 < (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
83	78, 81, 82	sylanbrc 585	. . . . . . 7 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ)
84	83	iftrued 4474	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) = 1)
85		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → (√‘𝑦) = (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
86	85	eleq1d 2897	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → ((√‘𝑦) ∈ ℕ ↔ (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ))
87	86	ifbid 4488	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → if((√‘𝑦) ∈ ℕ, 1, 0) = if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0))
88		fveq2 6664	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → (𝐹‘𝑦) = (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
89	87, 88	breq12d 5071	. . . . . . . 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 12275	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
92	63, 91	eleqtrdi 2923	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ (ℤ_≥‘1))
93	16	nnzd 12080	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℤ)
94	59	nnred 11647	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℝ)
95		dchrisum0flb.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∥ 𝐴)
96		pcelnn 16200	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑃 pCnt 𝐴) ∈ ℕ ↔ 𝑃 ∥ 𝐴))
97	4, 16, 96	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑃 pCnt 𝐴) ∈ ℕ ↔ 𝑃 ∥ 𝐴))
98	95, 97	mpbird 259	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈ ℕ)
99		prmuz2 16034	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
100		eluz2gt1 12314	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
101	4, 99, 100	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 < 𝑃)
102		expgt1 13461	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℝ ∧ (𝑃 pCnt 𝐴) ∈ ℕ ∧ 1 < 𝑃) → 1 < (𝑃↑(𝑃 pCnt 𝐴)))
103	94, 98, 101, 102	syl3anc 1367	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < (𝑃↑(𝑃 pCnt 𝐴)))
104		1red 10636	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
105		0lt1 11156	. . . . . . . . . . . . 13 ⊢ 0 < 1
106	105	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
107	60	nnred 11647	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℝ)
108	60	nngt0d 11680	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < (𝑃↑(𝑃 pCnt 𝐴)))
109	16	nnred 11647	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
110	16	nngt0d 11680	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝐴)
111		ltdiv2 11520	. . . . . . . . . . . 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 1382	. . . . . . . . . . 11 ⊢ (𝜑 → (1 < (𝑃↑(𝑃 pCnt 𝐴)) ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < (𝐴 / 1)))
113	103, 112	mpbid 234	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < (𝐴 / 1))
114	17	div1d 11402	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 / 1) = 𝐴)
115	113, 114	breqtrd 5084	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < 𝐴)
116		elfzo2 13035	. . . . . . . . 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 1339	. . . . . . . 8 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ (1..^𝐴))
118	89, 90, 117	rspcdva 3624	. . . . . . 7 ⊢ (𝜑 → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
119	118	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
120	84, 119	eqbrtrrd 5082	. . . . 5 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
121		1re 10635	. . . . . . . 8 ⊢ 1 ∈ ℝ
122		0le1 11157	. . . . . . . 8 ⊢ 0 ≤ 1
123	121, 122	pm3.2i 473	. . . . . . 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 26077	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
126	125, 60	ffvelrnd 6846	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ)
127	126	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ)
128	125, 63	ffvelrnd 6846	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℝ)
129	128	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℝ)
130		lemul12a 11492	. . . . . 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 836	. . . . 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 697	. . . 4 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (1 · 1) ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
133	3, 132	eqbrtrrid 5094	. . 3 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
134		0red 10638	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
135		0re 10637	. . . . . . . 8 ⊢ 0 ∈ ℝ
136	121, 135	ifcli 4512	. . . . . . 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 5062	. . . . . . . 8 ⊢ (1 = if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) → (0 ≤ 1 ↔ 0 ≤ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0)))
139		breq2 5062	. . . . . . . 8 ⊢ (0 = if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) → (0 ≤ 0 ↔ 0 ≤ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0)))
140		0le0 11732	. . . . . . . 8 ⊢ 0 ≤ 0
141	138, 139, 122, 140	keephyp 4535	. . . . . . 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 10791	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
144	121, 135	ifcli 4512	. . . . . . 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 5062	. . . . . . . 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 5062	. . . . . . . 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 4535	. . . . . . 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 10791	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
151	126, 128, 143, 150	mulge0d 11211	. . . 4 ⊢ (𝜑 → 0 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
152	151	adantr 483	. . 3 ⊢ ((𝜑 ∧ ¬ (√‘𝐴) ∈ ℕ) → 0 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
153	1, 2, 133, 152	ifbothda 4503	. 2 ⊢ (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
154	60	nncnd 11648	. . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℂ)
155	60	nnne0d 11681	. . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ≠ 0)
156	17, 154, 155	divcan2d 11412	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 𝐴)
157	156	fveq2d 6668	. . 3 ⊢ (𝜑 → (𝐹‘((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))) = (𝐹‘𝐴))
158		pcndvds2 16198	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))
159	4, 16, 158	syl2anc 586	. . . . . 6 ⊢ (𝜑 → ¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))
160		coprm 16049	. . . . . . 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 586	. . . . . 6 ⊢ (𝜑 → (¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ↔ (𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1))
162	159, 161	mpbid 234	. . . . 5 ⊢ (𝜑 → (𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1)
163		prmz 16013	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
164	4, 163	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℤ)
165		rpexp1i 16059	. . . . . 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 1367	. . . . 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 26076	. . 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 2858	. 2 ⊢ (𝜑 → (𝐹‘𝐴) = ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
170	153, 169	breqtrrd 5086	1 ⊢ (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ (𝐹‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 {crab 3142 ifcif 4466 class class class wbr 5058 ↦ cmpt 5138 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 · cmul 10536 < clt 10669 ≤ cle 10670 / cdiv 11291 ℕcn 11632 2c2 11686 ℕ₀cn0 11891 ℤcz 11975 ℤ_≥cuz 12237 ℚcq 12342 ℝ⁺crp 12383 ..^cfzo 13027 ↑cexp 13423 √csqrt 14586 Σcsu 15036 ∥ cdvds 15601 gcd cgcd 15837 ℙcprime 16009 pCnt cpc 16167 Basecbs 16477 0_gc0g 16707 ℤRHomczrh 20641 ℤ/nℤczn 20644 DChrcdchr 25802

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-disj 5024 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-acn 9365 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-pi 15420 df-dvds 15602 df-gcd 15838 df-prm 16010 df-numer 16069 df-denom 16070 df-pc 16168 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-qus 16776 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-nsg 18271 df-eqg 18272 df-ghm 18350 df-cntz 18441 df-od 18650 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-rnghom 19461 df-drng 19498 df-subrg 19527 df-lmod 19630 df-lss 19698 df-lsp 19738 df-sra 19938 df-rgmod 19939 df-lidl 19940 df-rsp 19941 df-2idl 19999 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-zring 20612 df-zrh 20645 df-zn 20648 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 df-cxp 25135 df-dchr 25803

This theorem is referenced by: dchrisum0flb 26080

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