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Theorem dchrisum0flblem2 26755

Description: Lemma for dchrisum0flb 26756. 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 5092	. . 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 5092	. . 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 12228	. . . 4 ⊢ (1 · 1) = 1
4		dchrisum0flb.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℙ)
5	4	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℙ)
6		nnq 12795	. . . . . . . . . . . . . . 15 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ∈ ℚ)
7	6	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ∈ ℚ)
8		nnne0 12100	. . . . . . . . . . . . . . 15 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ≠ 0)
9	8	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ≠ 0)
10		2z 12445	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ
11	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℤ)
12		pcexp 16649	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ ((√‘𝐴) ∈ ℚ ∧ (√‘𝐴) ≠ 0) ∧ 2 ∈ ℤ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (2 · (𝑃 pCnt (√‘𝐴))))
13	5, 7, 9, 11, 12	syl121anc 1374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (2 · (𝑃 pCnt (√‘𝐴))))
14		dchrisum0flb.1	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘2))
15		eluz2nn 12717	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℕ)
16	14, 15	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℕ)
17	16	nncnd 12082	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℂ)
18	17	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝐴 ∈ ℂ)
19	18	sqsqrtd 15242	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((√‘𝐴)↑2) = 𝐴)
20	19	oveq2d 7345	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt ((√‘𝐴)↑2)) = (𝑃 pCnt 𝐴))
21		2cnd 12144	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℂ)
22		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘𝐴) ∈ ℕ)
23	5, 22	pccld 16640	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt (√‘𝐴)) ∈ ℕ₀)
24	23	nn0cnd 12388	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃 pCnt (√‘𝐴)) ∈ ℂ)
25	21, 24	mulcomd 11089	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (2 · (𝑃 pCnt (√‘𝐴))) = ((𝑃 pCnt (√‘𝐴)) · 2))
26	13, 20, 25	3eqtr3rd 2785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((𝑃 pCnt (√‘𝐴)) · 2) = (𝑃 pCnt 𝐴))
27	26	oveq2d 7345	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑((𝑃 pCnt (√‘𝐴)) · 2)) = (𝑃↑(𝑃 pCnt 𝐴)))
28		prmnn 16468	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
29	5, 28	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℕ)
30	29	nncnd 12082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝑃 ∈ ℂ)
31		2nn0 12343	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
32	31	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 2 ∈ ℕ₀)
33	30, 32, 23	expmuld 13960	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑((𝑃 pCnt (√‘𝐴)) · 2)) = ((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2))
34	27, 33	eqtr3d 2778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) = ((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2))
35	34	fveq2d 6823	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝑃↑(𝑃 pCnt 𝐴))) = (√‘((𝑃↑(𝑃 pCnt (√‘𝐴)))↑2)))
36	29, 23	nnexpcld 14053	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℕ)
37	36	nnrpd 12863	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℝ⁺)
38	37	rprege0d 12872	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → ((𝑃↑(𝑃 pCnt (√‘𝐴))) ∈ ℝ ∧ 0 ≤ (𝑃↑(𝑃 pCnt (√‘𝐴)))))
39		sqrtsq 15072	. . . . . . . . . 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 2776	. . . . . . . 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 4480	. . . . . 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 16640	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈ ℕ₀)
54	44, 45, 46, 47, 48, 49, 50, 51, 52, 4, 53	dchrisum0flblem1 26754	. . . . . . 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 5113	. . . . 5 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
57		pcdvds 16654	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
58	4, 16, 57	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
59	4, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ)
60	59, 53	nnexpcld 14053	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
61		nndivdvds 16063	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ) → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴 ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ))
62	16, 60, 61	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴 ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ))
63	58, 62	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ)
64	63	nnzd 12518	. . . . . . . . . 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 12863	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 𝐴 ∈ ℝ⁺)
68	67	rprege0d 12872	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
69	60	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
70	69	nnrpd 12863	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℝ⁺)
71		sqrtdiv 15068	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))))
73		nnz 12435	. . . . . . . . . . 11 ⊢ ((√‘𝐴) ∈ ℕ → (√‘𝐴) ∈ ℤ)
74		znq 12785	. . . . . . . . . . 11 ⊢ (((√‘𝐴) ∈ ℤ ∧ (√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ) → ((√‘𝐴) / (√‘(𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ)
75	73, 42, 74	syl2an2 683	. . . . . . . . . 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 16551	. . . . . . . . 9 ⊢ (((𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℤ ∧ (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℚ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ)
78	65, 76, 77	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ)
79	63	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ)
80	79	nnrpd 12863	. . . . . . . . 9 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ⁺)
81	80	sqrtgt0d 15215	. . . . . . . 8 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 0 < (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
82		elnnz 12422	. . . . . . . 8 ⊢ ((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ ↔ ((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℤ ∧ 0 < (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
83	78, 81, 82	sylanbrc 583	. . . . . . 7 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ)
84	83	iftrued 4480	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) = 1)
85		fveq2 6819	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → (√‘𝑦) = (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
86	85	eleq1d 2821	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → ((√‘𝑦) ∈ ℕ ↔ (√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ))
87	86	ifbid 4495	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → if((√‘𝑦) ∈ ℕ, 1, 0) = if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0))
88		fveq2 6819	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) → (𝐹‘𝑦) = (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
89	87, 88	breq12d 5102	. . . . . . . 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 12714	. . . . . . . . . 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 12081	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℝ)
95		dchrisum0flb.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∥ 𝐴)
96		pcelnn 16660	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑃 pCnt 𝐴) ∈ ℕ ↔ 𝑃 ∥ 𝐴))
97	4, 16, 96	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑃 pCnt 𝐴) ∈ ℕ ↔ 𝑃 ∥ 𝐴))
98	95, 97	mpbird 256	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈ ℕ)
99		prmuz2 16490	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
100		eluz2gt1 12753	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
101	4, 99, 100	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 < 𝑃)
102		expgt1 13914	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℝ ∧ (𝑃 pCnt 𝐴) ∈ ℕ ∧ 1 < 𝑃) → 1 < (𝑃↑(𝑃 pCnt 𝐴)))
103	94, 98, 101, 102	syl3anc 1370	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < (𝑃↑(𝑃 pCnt 𝐴)))
104		1red 11069	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
105		0lt1 11590	. . . . . . . . . . . . 13 ⊢ 0 < 1
106	105	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
107	60	nnred 12081	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℝ)
108	60	nngt0d 12115	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < (𝑃↑(𝑃 pCnt 𝐴)))
109	16	nnred 12081	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
110	16	nngt0d 12115	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝐴)
111		ltdiv2 11954	. . . . . . . . . . . 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 1385	. . . . . . . . . . 11 ⊢ (𝜑 → (1 < (𝑃↑(𝑃 pCnt 𝐴)) ↔ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < (𝐴 / 1)))
113	103, 112	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < (𝐴 / 1))
114	17	div1d 11836	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 / 1) = 𝐴)
115	113, 114	breqtrd 5115	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) < 𝐴)
116		elfzo2 13483	. . . . . . . . 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 1342	. . . . . . . 8 ⊢ (𝜑 → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ∈ (1..^𝐴))
118	89, 90, 117	rspcdva 3571	. . . . . . 7 ⊢ (𝜑 → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
119	118	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → if((√‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
120	84, 119	eqbrtrrd 5113	. . . . 5 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
121		1re 11068	. . . . . . . 8 ⊢ 1 ∈ ℝ
122		0le1 11591	. . . . . . . 8 ⊢ 0 ≤ 1
123	121, 122	pm3.2i 471	. . . . . . 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 26753	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
126	125, 60	ffvelcdmd 7012	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ)
127	126	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℝ)
128	125, 63	ffvelcdmd 7012	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℝ)
129	128	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) ∈ ℝ)
130		lemul12a 11926	. . . . . 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 696	. . . 4 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → (1 · 1) ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
133	3, 132	eqbrtrrid 5125	. . 3 ⊢ ((𝜑 ∧ (√‘𝐴) ∈ ℕ) → 1 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
134		0red 11071	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
135		0re 11070	. . . . . . . 8 ⊢ 0 ∈ ℝ
136	121, 135	ifcli 4519	. . . . . . 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 5093	. . . . . . . 8 ⊢ (1 = if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) → (0 ≤ 1 ↔ 0 ≤ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0)))
139		breq2 5093	. . . . . . . 8 ⊢ (0 = if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0) → (0 ≤ 0 ↔ 0 ≤ if((√‘(𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ, 1, 0)))
140		0le0 12167	. . . . . . . 8 ⊢ 0 ≤ 0
141	138, 139, 122, 140	keephyp 4543	. . . . . . 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 11225	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘(𝑃↑(𝑃 pCnt 𝐴))))
144	121, 135	ifcli 4519	. . . . . . 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 5093	. . . . . . . 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 5093	. . . . . . . 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 4543	. . . . . . 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 11225	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
151	126, 128, 143, 150	mulge0d 11645	. . . 4 ⊢ (𝜑 → 0 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
152	151	adantr 481	. . 3 ⊢ ((𝜑 ∧ ¬ (√‘𝐴) ∈ ℕ) → 0 ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
153	1, 2, 133, 152	ifbothda 4510	. 2 ⊢ (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
154	60	nncnd 12082	. . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℂ)
155	60	nnne0d 12116	. . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐴)) ≠ 0)
156	17, 154, 155	divcan2d 11846	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 𝐴)
157	156	fveq2d 6823	. . 3 ⊢ (𝜑 → (𝐹‘((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))) = (𝐹‘𝐴))
158		pcndvds2 16658	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))
159	4, 16, 158	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))
160		coprm 16505	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → (¬ 𝑃 ∥ (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) ↔ (𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1))
162	159, 161	mpbid 231	. . . . 5 ⊢ (𝜑 → (𝑃 gcd (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 1)
163		prmz 16469	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
164	4, 163	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℤ)
165		rpexp1i 16517	. . . . . 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 1370	. . . . 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 26752	. . 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 2778	. 2 ⊢ (𝜑 → (𝐹‘𝐴) = ((𝐹‘(𝑃↑(𝑃 pCnt 𝐴))) · (𝐹‘(𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))))
170	153, 169	breqtrrd 5117	1 ⊢ (𝜑 → if((√‘𝐴) ∈ ℕ, 1, 0) ≤ (𝐹‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 ∀wral 3061 {crab 3403 ifcif 4472 class class class wbr 5089 ↦ cmpt 5172 ⟶wf 6469 ‘cfv 6473 (class class class)co 7329 ℂcc 10962 ℝcr 10963 0cc0 10964 1c1 10965 · cmul 10969 < clt 11102 ≤ cle 11103 / cdiv 11725 ℕcn 12066 2c2 12121 ℕ₀cn0 12326 ℤcz 12412 ℤ_≥cuz 12675 ℚcq 12781 ℝ⁺crp 12823 ..^cfzo 13475 ↑cexp 13875 √csqrt 15035 Σcsu 15488 ∥ cdvds 16054 gcd cgcd 16292 ℙcprime 16465 pCnt cpc 16626 Basecbs 17001 0_gc0g 17239 ℤRHomczrh 20799 ℤ/nℤczn 20802 DChrcdchr 26478

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-inf2 9490 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-pre-sup 11042 ax-addf 11043 ax-mulf 11044

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-disj 5055 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-om 7773 df-1st 7891 df-2nd 7892 df-supp 8040 df-tpos 8104 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-2o 8360 df-oadd 8363 df-omul 8364 df-er 8561 df-ec 8563 df-qs 8567 df-map 8680 df-pm 8681 df-ixp 8749 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-fsupp 9219 df-fi 9260 df-sup 9291 df-inf 9292 df-oi 9359 df-card 9788 df-acn 9791 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-div 11726 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-z 12413 df-dec 12531 df-uz 12676 df-q 12782 df-rp 12824 df-xneg 12941 df-xadd 12942 df-xmul 12943 df-ioo 13176 df-ioc 13177 df-ico 13178 df-icc 13179 df-fz 13333 df-fzo 13476 df-fl 13605 df-mod 13683 df-seq 13815 df-exp 13876 df-fac 14081 df-bc 14110 df-hash 14138 df-shft 14869 df-cj 14901 df-re 14902 df-im 14903 df-sqrt 15037 df-abs 15038 df-limsup 15271 df-clim 15288 df-rlim 15289 df-sum 15489 df-ef 15868 df-sin 15870 df-cos 15871 df-pi 15873 df-dvds 16055 df-gcd 16293 df-prm 16466 df-numer 16528 df-denom 16529 df-pc 16627 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-starv 17066 df-sca 17067 df-vsca 17068 df-ip 17069 df-tset 17070 df-ple 17071 df-ds 17073 df-unif 17074 df-hom 17075 df-cco 17076 df-rest 17222 df-topn 17223 df-0g 17241 df-gsum 17242 df-topgen 17243 df-pt 17244 df-prds 17247 df-xrs 17302 df-qtop 17307 df-imas 17308 df-qus 17309 df-xps 17310 df-mre 17384 df-mrc 17385 df-acs 17387 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-mhm 18519 df-submnd 18520 df-grp 18668 df-minusg 18669 df-sbg 18670 df-mulg 18789 df-subg 18840 df-nsg 18841 df-eqg 18842 df-ghm 18920 df-cntz 19011 df-od 19224 df-cmn 19475 df-abl 19476 df-mgp 19808 df-ur 19825 df-ring 19872 df-cring 19873 df-oppr 19949 df-dvdsr 19970 df-unit 19971 df-invr 20001 df-dvr 20012 df-rnghom 20046 df-drng 20087 df-subrg 20119 df-lmod 20223 df-lss 20292 df-lsp 20332 df-sra 20532 df-rgmod 20533 df-lidl 20534 df-rsp 20535 df-2idl 20601 df-psmet 20687 df-xmet 20688 df-met 20689 df-bl 20690 df-mopn 20691 df-fbas 20692 df-fg 20693 df-cnfld 20696 df-zring 20769 df-zrh 20803 df-zn 20806 df-top 22141 df-topon 22158 df-topsp 22180 df-bases 22194 df-cld 22268 df-ntr 22269 df-cls 22270 df-nei 22347 df-lp 22385 df-perf 22386 df-cn 22476 df-cnp 22477 df-haus 22564 df-tx 22811 df-hmeo 23004 df-fil 23095 df-fm 23187 df-flim 23188 df-flf 23189 df-xms 23571 df-ms 23572 df-tms 23573 df-cncf 24139 df-limc 25128 df-dv 25129 df-log 25810 df-cxp 25811 df-dchr 26479

This theorem is referenced by: dchrisum0flb 26756

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