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Theorem dchrisum0flblem1 27459

Description: Lemma for dchrisum0flb 27461. Base case, prime power. (Contributed by Mario Carneiro, 5-May-2016.)

**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)
dchrisum0flblem1.1	⊢ (𝜑 → 𝑃 ∈ ℙ)
dchrisum0flblem1.2	⊢ (𝜑 → 𝐴 ∈ ℕ₀)

**Assertion**
Ref	Expression
dchrisum0flblem1	⊢ (𝜑 → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃↑𝐴)))

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

Proof of Theorem dchrisum0flblem1 Dummy variables 𝑘 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1red 11134	. . . . 5 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 1 ∈ ℝ)
2		0red 11136	. . . . 5 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ ¬ (√‘(𝑃↑𝐴)) ∈ ℕ) → 0 ∈ ℝ)
3	1, 2	ifclda 4492	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ∈ ℝ)
4		1red 11134	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → 1 ∈ ℝ)
5		fzfid 13924	. . . . . 6 ⊢ (𝜑 → (0...𝐴) ∈ Fin)
6		dchrisum0flb.r	. . . . . . . 8 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
7		rpvmasum.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnnn0d 12487	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
9		rpvmasum.z	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
10		eqid 2735	. . . . . . . . . . 11 ⊢ (Base‘𝑍) = (Base‘𝑍)
11		rpvmasum.l	. . . . . . . . . . 11 ⊢ 𝐿 = (ℤRHom‘𝑍)
12	9, 10, 11	znzrhfo 21516	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
13		fof 6741	. . . . . . . . . 10 ⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
14	8, 12, 13	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍))
15		dchrisum0flblem1.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℙ)
16		prmz 16633	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
17	15, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℤ)
18	14, 17	ffvelcdmd 7026	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝑃) ∈ (Base‘𝑍))
19	6, 18	ffvelcdmd 7026	. . . . . . 7 ⊢ (𝜑 → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
20		elfznn0 13563	. . . . . . 7 ⊢ (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℕ₀)
21		reexpcl 14029	. . . . . . 7 ⊢ (((𝑋‘(𝐿‘𝑃)) ∈ ℝ ∧ 𝑖 ∈ ℕ₀) → ((𝑋‘(𝐿‘𝑃))↑𝑖) ∈ ℝ)
22	19, 20, 21	syl2an 597	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ((𝑋‘(𝐿‘𝑃))↑𝑖) ∈ ℝ)
23	5, 22	fsumrecl 15685	. . . . 5 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) ∈ ℝ)
24	23	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) ∈ ℝ)
25		breq1 5077	. . . . . 6 ⊢ (1 = if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) → (1 ≤ 1 ↔ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ 1))
26		breq1 5077	. . . . . 6 ⊢ (0 = if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) → (0 ≤ 1 ↔ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ 1))
27		1le1 11767	. . . . . 6 ⊢ 1 ≤ 1
28		0le1 11662	. . . . . 6 ⊢ 0 ≤ 1
29	25, 26, 27, 28	keephyp 4528	. . . . 5 ⊢ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ 1
30	29	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ 1)
31		dchrisum0flblem1.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
32		nn0uz 12815	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
33	31, 32	eleqtrdi 2845	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘0))
34		fzn0 13481	. . . . . . . . 9 ⊢ ((0...𝐴) ≠ ∅ ↔ 𝐴 ∈ (ℤ_≥‘0))
35	33, 34	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → (0...𝐴) ≠ ∅)
36		hashnncl 14317	. . . . . . . . 9 ⊢ ((0...𝐴) ∈ Fin → ((♯‘(0...𝐴)) ∈ ℕ ↔ (0...𝐴) ≠ ∅))
37	5, 36	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(0...𝐴)) ∈ ℕ ↔ (0...𝐴) ≠ ∅))
38	35, 37	mpbird 257	. . . . . . 7 ⊢ (𝜑 → (♯‘(0...𝐴)) ∈ ℕ)
39	38	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (♯‘(0...𝐴)) ∈ ℕ)
40	39	nnge1d 12214	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → 1 ≤ (♯‘(0...𝐴)))
41		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (𝑋‘(𝐿‘𝑃)) = 1)
42	41	oveq1d 7371	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → ((𝑋‘(𝐿‘𝑃))↑𝑖) = (1↑𝑖))
43		elfzelz 13467	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℤ)
44		1exp 14042	. . . . . . . . 9 ⊢ (𝑖 ∈ ℤ → (1↑𝑖) = 1)
45	43, 44	syl 17	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝐴) → (1↑𝑖) = 1)
46	42, 45	sylan9eq 2790	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ 𝑖 ∈ (0...𝐴)) → ((𝑋‘(𝐿‘𝑃))↑𝑖) = 1)
47	46	sumeq2dv 15653	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = Σ𝑖 ∈ (0...𝐴)1)
48		fzfid 13924	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (0...𝐴) ∈ Fin)
49		ax-1cn 11085	. . . . . . 7 ⊢ 1 ∈ ℂ
50		fsumconst 15741	. . . . . . 7 ⊢ (((0...𝐴) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑖 ∈ (0...𝐴)1 = ((♯‘(0...𝐴)) · 1))
51	48, 49, 50	sylancl 587	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)1 = ((♯‘(0...𝐴)) · 1))
52	39	nncnd 12179	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (♯‘(0...𝐴)) ∈ ℂ)
53	52	mulridd 11151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → ((♯‘(0...𝐴)) · 1) = (♯‘(0...𝐴)))
54	47, 51, 53	3eqtrd 2774	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = (♯‘(0...𝐴)))
55	40, 54	breqtrrd 5102	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → 1 ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
56	3, 4, 24, 30, 55	letrd 11292	. . 3 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
57		oveq1 7363	. . . . . . 7 ⊢ (1 = if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) → (1 · (1 − (𝑋‘(𝐿‘𝑃)))) = (if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) · (1 − (𝑋‘(𝐿‘𝑃)))))
58	57	breq1d 5084	. . . . . 6 ⊢ (1 = if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) → ((1 · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ↔ (if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)))))
59		oveq1 7363	. . . . . . 7 ⊢ (0 = if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) → (0 · (1 − (𝑋‘(𝐿‘𝑃)))) = (if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) · (1 − (𝑋‘(𝐿‘𝑃)))))
60	59	breq1d 5084	. . . . . 6 ⊢ (0 = if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) → ((0 · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ↔ (if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)))))
61		1re 11133	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
62	19	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
63		resubcl 11447	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (𝑋‘(𝐿‘𝑃)) ∈ ℝ) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℝ)
64	61, 62, 63	sylancr 588	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℝ)
65	64	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℝ)
66	65	leidd 11705	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − (𝑋‘(𝐿‘𝑃))) ≤ (1 − (𝑋‘(𝐿‘𝑃))))
67	64	recnd 11162	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℂ)
68	67	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℂ)
69	68	mullidd 11152	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 · (1 − (𝑋‘(𝐿‘𝑃)))) = (1 − (𝑋‘(𝐿‘𝑃))))
70		nn0p1nn 12465	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ)
71	31, 70	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ)
72	71	ad3antrrr 731	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → (𝐴 + 1) ∈ ℕ)
73	72	0expd 14090	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → (0↑(𝐴 + 1)) = 0)
74		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → (𝑋‘(𝐿‘𝑃)) = 0)
75	74	oveq1d 7371	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (0↑(𝐴 + 1)))
76	73, 75, 74	3eqtr4d 2780	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (𝑋‘(𝐿‘𝑃)))
77		neg1cn 12133	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℂ
78	31	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 ∈ ℕ₀)
79		expp1 14019	. . . . . . . . . . . . 13 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℕ₀) → (-1↑(𝐴 + 1)) = ((-1↑𝐴) · -1))
80	77, 78, 79	sylancr 588	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑(𝐴 + 1)) = ((-1↑𝐴) · -1))
81		prmnn 16632	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
82	15, 81	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑃 ∈ ℕ)
83	82, 31	nnexpcld 14196	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑃↑𝐴) ∈ ℕ)
84	83	nncnd 12179	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑃↑𝐴) ∈ ℂ)
85	84	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃↑𝐴) ∈ ℂ)
86	85	sqsqrtd 15393	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((√‘(𝑃↑𝐴))↑2) = (𝑃↑𝐴))
87	86	oveq2d 7372	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt ((√‘(𝑃↑𝐴))↑2)) = (𝑃 pCnt (𝑃↑𝐴)))
88	15	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝑃 ∈ ℙ)
89		nnq 12901	. . . . . . . . . . . . . . . . . . 19 ⊢ ((√‘(𝑃↑𝐴)) ∈ ℕ → (√‘(𝑃↑𝐴)) ∈ ℚ)
90	89	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (√‘(𝑃↑𝐴)) ∈ ℚ)
91		nnne0 12200	. . . . . . . . . . . . . . . . . . 19 ⊢ ((√‘(𝑃↑𝐴)) ∈ ℕ → (√‘(𝑃↑𝐴)) ≠ 0)
92	91	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (√‘(𝑃↑𝐴)) ≠ 0)
93		2z 12548	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℤ
94	93	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 2 ∈ ℤ)
95		pcexp 16819	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ ℙ ∧ ((√‘(𝑃↑𝐴)) ∈ ℚ ∧ (√‘(𝑃↑𝐴)) ≠ 0) ∧ 2 ∈ ℤ) → (𝑃 pCnt ((√‘(𝑃↑𝐴))↑2)) = (2 · (𝑃 pCnt (√‘(𝑃↑𝐴)))))
96	88, 90, 92, 94, 95	syl121anc 1378	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt ((√‘(𝑃↑𝐴))↑2)) = (2 · (𝑃 pCnt (√‘(𝑃↑𝐴)))))
97	78	nn0zd 12538	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 ∈ ℤ)
98		pcid 16833	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴)
99	88, 97, 98	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴)
100	87, 96, 99	3eqtr3rd 2779	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 = (2 · (𝑃 pCnt (√‘(𝑃↑𝐴)))))
101	100	oveq2d 7372	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑𝐴) = (-1↑(2 · (𝑃 pCnt (√‘(𝑃↑𝐴))))))
102	77	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → -1 ∈ ℂ)
103		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (√‘(𝑃↑𝐴)) ∈ ℕ)
104	88, 103	pccld 16810	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt (√‘(𝑃↑𝐴))) ∈ ℕ₀)
105		2nn0 12443	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ₀
106	105	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 2 ∈ ℕ₀)
107	102, 104, 106	expmuld 14100	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑(2 · (𝑃 pCnt (√‘(𝑃↑𝐴))))) = ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))))
108		neg1sqe1 14147	. . . . . . . . . . . . . . . . 17 ⊢ (-1↑2) = 1
109	108	oveq1i 7366	. . . . . . . . . . . . . . . 16 ⊢ ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = (1↑(𝑃 pCnt (√‘(𝑃↑𝐴))))
110	104	nn0zd 12538	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt (√‘(𝑃↑𝐴))) ∈ ℤ)
111		1exp 14042	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 pCnt (√‘(𝑃↑𝐴))) ∈ ℤ → (1↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = 1)
112	110, 111	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = 1)
113	109, 112	eqtrid 2782	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = 1)
114	101, 107, 113	3eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑𝐴) = 1)
115	114	oveq1d 7371	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑𝐴) · -1) = (1 · -1))
116	77	mullidi 11139	. . . . . . . . . . . . 13 ⊢ (1 · -1) = -1
117	115, 116	eqtrdi 2786	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑𝐴) · -1) = -1)
118	80, 117	eqtrd 2770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑(𝐴 + 1)) = -1)
119	118	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (-1↑(𝐴 + 1)) = -1)
120	19	recnd 11162	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
121	120	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
122	121	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
123	122	negnegd 11485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → --(𝑋‘(𝐿‘𝑃)) = (𝑋‘(𝐿‘𝑃)))
124		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ≠ 1)
125	124	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) ≠ 1)
126		rpvmasum2.g	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐺 = (DChr‘𝑁)
127		rpvmasum2.d	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 = (Base‘𝐺)
128		dchrisum0f.x	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
129	128	ad3antrrr 731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → 𝑋 ∈ 𝐷)
130		eqid 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
131	126, 9, 127, 10, 130, 128, 18	dchrn0 27201	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑋‘(𝐿‘𝑃)) ≠ 0 ↔ (𝐿‘𝑃) ∈ (Unit‘𝑍)))
132	131	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((𝑋‘(𝐿‘𝑃)) ≠ 0 ↔ (𝐿‘𝑃) ∈ (Unit‘𝑍)))
133	132	biimpa 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝐿‘𝑃) ∈ (Unit‘𝑍))
134	126, 127, 129, 9, 130, 133	dchrabs 27211	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (abs‘(𝑋‘(𝐿‘𝑃))) = 1)
135		eqeq1 2739	. . . . . . . . . . . . . . . . . 18 ⊢ ((abs‘(𝑋‘(𝐿‘𝑃))) = (𝑋‘(𝐿‘𝑃)) → ((abs‘(𝑋‘(𝐿‘𝑃))) = 1 ↔ (𝑋‘(𝐿‘𝑃)) = 1))
136	134, 135	syl5ibcom 245	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → ((abs‘(𝑋‘(𝐿‘𝑃))) = (𝑋‘(𝐿‘𝑃)) → (𝑋‘(𝐿‘𝑃)) = 1))
137	136	necon3ad 2943	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → ((𝑋‘(𝐿‘𝑃)) ≠ 1 → ¬ (abs‘(𝑋‘(𝐿‘𝑃))) = (𝑋‘(𝐿‘𝑃))))
138	125, 137	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → ¬ (abs‘(𝑋‘(𝐿‘𝑃))) = (𝑋‘(𝐿‘𝑃)))
139	62	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
140	139	absord 15367	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → ((abs‘(𝑋‘(𝐿‘𝑃))) = (𝑋‘(𝐿‘𝑃)) ∨ (abs‘(𝑋‘(𝐿‘𝑃))) = -(𝑋‘(𝐿‘𝑃))))
141	140	ord 865	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (¬ (abs‘(𝑋‘(𝐿‘𝑃))) = (𝑋‘(𝐿‘𝑃)) → (abs‘(𝑋‘(𝐿‘𝑃))) = -(𝑋‘(𝐿‘𝑃))))
142	138, 141	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (abs‘(𝑋‘(𝐿‘𝑃))) = -(𝑋‘(𝐿‘𝑃)))
143	142, 134	eqtr3d 2772	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → -(𝑋‘(𝐿‘𝑃)) = 1)
144	143	negeqd 11376	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → --(𝑋‘(𝐿‘𝑃)) = -1)
145	123, 144	eqtr3d 2772	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) = -1)
146	145	oveq1d 7371	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (-1↑(𝐴 + 1)))
147	119, 146, 145	3eqtr4d 2780	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (𝑋‘(𝐿‘𝑃)))
148	76, 147	pm2.61dane 3017	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (𝑋‘(𝐿‘𝑃)))
149	148	oveq2d 7372	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = (1 − (𝑋‘(𝐿‘𝑃))))
150	66, 69, 149	3brtr4d 5106	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
151	67	mul02d 11333	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (0 · (1 − (𝑋‘(𝐿‘𝑃)))) = 0)
152		peano2nn0 12466	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ₀)
153	31, 152	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ₀)
154	19, 153	reexpcld 14114	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℝ)
155	154	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℝ)
156	155	recnd 11162	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℂ)
157	156	abscld 15390	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ∈ ℝ)
158		1red 11134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 1 ∈ ℝ)
159	155	leabsd 15366	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ≤ (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
160	153	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝐴 + 1) ∈ ℕ₀)
161	121, 160	absexpd 15406	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = ((abs‘(𝑋‘(𝐿‘𝑃)))↑(𝐴 + 1)))
162	121	abscld 15390	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘(𝑋‘(𝐿‘𝑃))) ∈ ℝ)
163	121	absge0d 15398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 ≤ (abs‘(𝑋‘(𝐿‘𝑃))))
164	126, 127, 9, 10, 128, 18	dchrabs2 27213	. . . . . . . . . . . . 13 ⊢ (𝜑 → (abs‘(𝑋‘(𝐿‘𝑃))) ≤ 1)
165	164	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘(𝑋‘(𝐿‘𝑃))) ≤ 1)
166		exple1 14128	. . . . . . . . . . . 12 ⊢ ((((abs‘(𝑋‘(𝐿‘𝑃))) ∈ ℝ ∧ 0 ≤ (abs‘(𝑋‘(𝐿‘𝑃))) ∧ (abs‘(𝑋‘(𝐿‘𝑃))) ≤ 1) ∧ (𝐴 + 1) ∈ ℕ₀) → ((abs‘(𝑋‘(𝐿‘𝑃)))↑(𝐴 + 1)) ≤ 1)
167	162, 163, 165, 160, 166	syl31anc 1376	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((abs‘(𝑋‘(𝐿‘𝑃)))↑(𝐴 + 1)) ≤ 1)
168	161, 167	eqbrtrd 5096	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ≤ 1)
169	155, 157, 158, 159, 168	letrd 11292	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ≤ 1)
170		subge0 11652	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℝ) → (0 ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ↔ ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ≤ 1))
171	61, 155, 170	sylancr 588	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (0 ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ↔ ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ≤ 1))
172	169, 171	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
173	151, 172	eqbrtrd 5096	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (0 · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
174	173	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ ¬ (√‘(𝑃↑𝐴)) ∈ ℕ) → (0 · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
175	58, 60, 150, 174	ifbothda 4495	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
176		0re 11135	. . . . . . . 8 ⊢ 0 ∈ ℝ
177	61, 176	ifcli 4504	. . . . . . 7 ⊢ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ∈ ℝ
178	177	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ∈ ℝ)
179		resubcl 11447	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℝ) → (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ∈ ℝ)
180	61, 155, 179	sylancr 588	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ∈ ℝ)
181	62	leabsd 15366	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ≤ (abs‘(𝑋‘(𝐿‘𝑃))))
182	62, 162, 158, 181, 165	letrd 11292	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ≤ 1)
183	124	necomd 2985	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 1 ≠ (𝑋‘(𝐿‘𝑃)))
184	62, 158, 182, 183	leneltd 11289	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) < 1)
185		posdif 11632	. . . . . . . 8 ⊢ (((𝑋‘(𝐿‘𝑃)) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑋‘(𝐿‘𝑃)) < 1 ↔ 0 < (1 − (𝑋‘(𝐿‘𝑃)))))
186	62, 61, 185	sylancl 587	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃)) < 1 ↔ 0 < (1 − (𝑋‘(𝐿‘𝑃)))))
187	184, 186	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 < (1 − (𝑋‘(𝐿‘𝑃))))
188		lemuldiv 12025	. . . . . 6 ⊢ ((if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ∈ ℝ ∧ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ∈ ℝ ∧ ((1 − (𝑋‘(𝐿‘𝑃))) ∈ ℝ ∧ 0 < (1 − (𝑋‘(𝐿‘𝑃))))) → ((if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ↔ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ ((1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) / (1 − (𝑋‘(𝐿‘𝑃))))))
189	178, 180, 64, 187, 188	syl112anc 1377	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ↔ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ ((1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) / (1 − (𝑋‘(𝐿‘𝑃))))))
190	175, 189	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ ((1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) / (1 − (𝑋‘(𝐿‘𝑃)))))
191	31	nn0zd 12538	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
192		fzval3 13678	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (0...𝐴) = (0..^(𝐴 + 1)))
193	191, 192	syl 17	. . . . . . 7 ⊢ (𝜑 → (0...𝐴) = (0..^(𝐴 + 1)))
194	193	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (0...𝐴) = (0..^(𝐴 + 1)))
195	194	sumeq1d 15651	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = Σ𝑖 ∈ (0..^(𝐴 + 1))((𝑋‘(𝐿‘𝑃))↑𝑖))
196		0nn0 12441	. . . . . . 7 ⊢ 0 ∈ ℕ₀
197	196	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 ∈ ℕ₀)
198	153, 32	eleqtrdi 2845	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ (ℤ_≥‘0))
199	198	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝐴 + 1) ∈ (ℤ_≥‘0))
200	121, 124, 197, 199	geoserg 15820	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → Σ𝑖 ∈ (0..^(𝐴 + 1))((𝑋‘(𝐿‘𝑃))↑𝑖) = ((((𝑋‘(𝐿‘𝑃))↑0) − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) / (1 − (𝑋‘(𝐿‘𝑃)))))
201	121	exp0d 14091	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑0) = 1)
202	201	oveq1d 7371	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (((𝑋‘(𝐿‘𝑃))↑0) − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
203	202	oveq1d 7371	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((((𝑋‘(𝐿‘𝑃))↑0) − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) / (1 − (𝑋‘(𝐿‘𝑃)))) = ((1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) / (1 − (𝑋‘(𝐿‘𝑃)))))
204	195, 200, 203	3eqtrd 2774	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = ((1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) / (1 − (𝑋‘(𝐿‘𝑃)))))
205	190, 204	breqtrrd 5102	. . 3 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
206	56, 205	pm2.61dane 3017	. 2 ⊢ (𝜑 → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
207		rpvmasum2.1	. . . . 5 ⊢ 1 = (0_g‘𝐺)
208		dchrisum0f.f	. . . . 5 ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)))
209	9, 11, 7, 126, 127, 207, 208	dchrisum0fval 27456	. . . 4 ⊢ ((𝑃↑𝐴) ∈ ℕ → (𝐹‘(𝑃↑𝐴)) = Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} (𝑋‘(𝐿‘𝑘)))
210	83, 209	syl 17	. . 3 ⊢ (𝜑 → (𝐹‘(𝑃↑𝐴)) = Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} (𝑋‘(𝐿‘𝑘)))
211		2fveq3 6834	. . . 4 ⊢ (𝑘 = (𝑃↑𝑖) → (𝑋‘(𝐿‘𝑘)) = (𝑋‘(𝐿‘(𝑃↑𝑖))))
212		eqid 2735	. . . . . 6 ⊢ (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏)) = (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏))
213	212	dvdsppwf1o 27137	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ₀) → (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏)):(0...𝐴)–1-1-onto→{𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)})
214	15, 31, 213	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏)):(0...𝐴)–1-1-onto→{𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)})
215		oveq2 7364	. . . . . 6 ⊢ (𝑏 = 𝑖 → (𝑃↑𝑏) = (𝑃↑𝑖))
216		ovex 7389	. . . . . 6 ⊢ (𝑃↑𝑏) ∈ V
217	215, 212, 216	fvmpt3i 6942	. . . . 5 ⊢ (𝑖 ∈ (0...𝐴) → ((𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏))‘𝑖) = (𝑃↑𝑖))
218	217	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ((𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏))‘𝑖) = (𝑃↑𝑖))
219	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → 𝑋:(Base‘𝑍)⟶ℝ)
220		elrabi 3627	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} → 𝑘 ∈ ℕ)
221	220	nnzd 12539	. . . . . . 7 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} → 𝑘 ∈ ℤ)
222		ffvelcdm 7022	. . . . . . 7 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑘 ∈ ℤ) → (𝐿‘𝑘) ∈ (Base‘𝑍))
223	14, 221, 222	syl2an 597	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → (𝐿‘𝑘) ∈ (Base‘𝑍))
224	219, 223	ffvelcdmd 7026	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → (𝑋‘(𝐿‘𝑘)) ∈ ℝ)
225	224	recnd 11162	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
226	211, 5, 214, 218, 225	fsumf1o 15674	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} (𝑋‘(𝐿‘𝑘)) = Σ𝑖 ∈ (0...𝐴)(𝑋‘(𝐿‘(𝑃↑𝑖))))
227		zsubrg 21389	. . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
228		eqid 2735	. . . . . . . . . . . 12 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
229	228	subrgsubm 20551	. . . . . . . . . . 11 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
230	227, 229	mp1i 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
231	20	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑖 ∈ ℕ₀)
232	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑃 ∈ ℤ)
233		eqid 2735	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
234		zringmpg 21440	. . . . . . . . . . . 12 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = (mulGrp‘ℤ_ring)
235	234	eqcomi 2744	. . . . . . . . . . 11 ⊢ (mulGrp‘ℤ_ring) = ((mulGrp‘ℂ_fld) ↾_s ℤ)
236		eqid 2735	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘ℤ_ring)) = (._g‘(mulGrp‘ℤ_ring))
237	233, 235, 236	submmulg 19083	. . . . . . . . . 10 ⊢ ((ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)) ∧ 𝑖 ∈ ℕ₀ ∧ 𝑃 ∈ ℤ) → (𝑖(._g‘(mulGrp‘ℂ_fld))𝑃) = (𝑖(._g‘(mulGrp‘ℤ_ring))𝑃))
238	230, 231, 232, 237	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℂ_fld))𝑃) = (𝑖(._g‘(mulGrp‘ℤ_ring))𝑃))
239	82	nncnd 12179	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℂ)
240		cnfldexp 21374	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℂ ∧ 𝑖 ∈ ℕ₀) → (𝑖(._g‘(mulGrp‘ℂ_fld))𝑃) = (𝑃↑𝑖))
241	239, 20, 240	syl2an 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℂ_fld))𝑃) = (𝑃↑𝑖))
242	238, 241	eqtr3d 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℤ_ring))𝑃) = (𝑃↑𝑖))
243	242	fveq2d 6833	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑖(._g‘(mulGrp‘ℤ_ring))𝑃)) = (𝐿‘(𝑃↑𝑖)))
244	9	zncrng 21513	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
245		crngring 20215	. . . . . . . . . . 11 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
246	8, 244, 245	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Ring)
247	11	zrhrhm 21480	. . . . . . . . . 10 ⊢ (𝑍 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑍))
248		eqid 2735	. . . . . . . . . . 11 ⊢ (mulGrp‘ℤ_ring) = (mulGrp‘ℤ_ring)
249		eqid 2735	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
250	248, 249	rhmmhm 20448	. . . . . . . . . 10 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑍) → 𝐿 ∈ ((mulGrp‘ℤ_ring) MndHom (mulGrp‘𝑍)))
251	246, 247, 250	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ((mulGrp‘ℤ_ring) MndHom (mulGrp‘𝑍)))
252	251	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝐿 ∈ ((mulGrp‘ℤ_ring) MndHom (mulGrp‘𝑍)))
253		zringbas 21422	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤ_ring)
254	248, 253	mgpbas 20115	. . . . . . . . 9 ⊢ ℤ = (Base‘(mulGrp‘ℤ_ring))
255		eqid 2735	. . . . . . . . 9 ⊢ (._g‘(mulGrp‘𝑍)) = (._g‘(mulGrp‘𝑍))
256	254, 236, 255	mhmmulg 19080	. . . . . . . 8 ⊢ ((𝐿 ∈ ((mulGrp‘ℤ_ring) MndHom (mulGrp‘𝑍)) ∧ 𝑖 ∈ ℕ₀ ∧ 𝑃 ∈ ℤ) → (𝐿‘(𝑖(._g‘(mulGrp‘ℤ_ring))𝑃)) = (𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃)))
257	252, 231, 232, 256	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑖(._g‘(mulGrp‘ℤ_ring))𝑃)) = (𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃)))
258	243, 257	eqtr3d 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑃↑𝑖)) = (𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃)))
259	258	fveq2d 6833	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝐿‘(𝑃↑𝑖))) = (𝑋‘(𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃))))
260	126, 9, 127	dchrmhm 27192	. . . . . . . 8 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
261	260, 128	sselid 3915	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
262	261	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
263	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘𝑃) ∈ (Base‘𝑍))
264	249, 10	mgpbas 20115	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
265	264, 255, 233	mhmmulg 19080	. . . . . 6 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ 𝑖 ∈ ℕ₀ ∧ (𝐿‘𝑃) ∈ (Base‘𝑍)) → (𝑋‘(𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃))) = (𝑖(._g‘(mulGrp‘ℂ_fld))(𝑋‘(𝐿‘𝑃))))
266	262, 231, 263, 265	syl3anc 1374	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃))) = (𝑖(._g‘(mulGrp‘ℂ_fld))(𝑋‘(𝐿‘𝑃))))
267		cnfldexp 21374	. . . . . 6 ⊢ (((𝑋‘(𝐿‘𝑃)) ∈ ℂ ∧ 𝑖 ∈ ℕ₀) → (𝑖(._g‘(mulGrp‘ℂ_fld))(𝑋‘(𝐿‘𝑃))) = ((𝑋‘(𝐿‘𝑃))↑𝑖))
268	120, 20, 267	syl2an 597	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℂ_fld))(𝑋‘(𝐿‘𝑃))) = ((𝑋‘(𝐿‘𝑃))↑𝑖))
269	259, 266, 268	3eqtrd 2774	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝐿‘(𝑃↑𝑖))) = ((𝑋‘(𝐿‘𝑃))↑𝑖))
270	269	sumeq2dv 15653	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐴)(𝑋‘(𝐿‘(𝑃↑𝑖))) = Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
271	210, 226, 270	3eqtrd 2774	. 2 ⊢ (𝜑 → (𝐹‘(𝑃↑𝐴)) = Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
272	206, 271	breqtrrd 5102	1 ⊢ (𝜑 → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃↑𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2930 {crab 3387 ∅c0 4263 ifcif 4456 class class class wbr 5074 ↦ cmpt 5155 ⟶wf 6483 –onto→wfo 6485 –1-1-onto→wf1o 6486 ‘cfv 6487 (class class class)co 7356 Fincfn 8882 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 · cmul 11032 < clt 11168 ≤ cle 11169 − cmin 11366 -cneg 11367 / cdiv 11796 ℕcn 12163 2c2 12225 ℕ₀cn0 12426 ℤcz 12513 ℤ_≥cuz 12777 ℚcq 12887 ...cfz 13450 ..^cfzo 13597 ↑cexp 14012 ♯chash 14281 √csqrt 15184 abscabs 15185 Σcsu 15637 ∥ cdvds 16210 ℙcprime 16629 pCnt cpc 16796 Basecbs 17168 ↾_s cress 17189 0_gc0g 17391 MndHom cmhm 18738 SubMndcsubmnd 18739 ._gcmg 19032 mulGrpcmgp 20110 Ringcrg 20203 CRingccrg 20204 Unitcui 20324 RingHom crh 20438 SubRingcsubrg 20535 ℂ_fldccnfld 21341 ℤ_ringczring 21415 ℤRHomczrh 21468 ℤ/nℤczn 21471 DChrcdchr 27183

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 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106 ax-mulf 11107

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-disj 5042 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8100 df-tpos 8165 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-oadd 8398 df-omul 8399 df-er 8632 df-ec 8634 df-qs 8638 df-map 8764 df-pm 8765 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9264 df-fi 9313 df-sup 9344 df-inf 9345 df-oi 9414 df-card 9852 df-acn 9855 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ioc 13292 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15638 df-ef 16021 df-sin 16023 df-cos 16024 df-pi 16026 df-dvds 16211 df-gcd 16453 df-prm 16630 df-pc 16797 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17455 df-qtop 17460 df-imas 17461 df-qus 17462 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19033 df-subg 19088 df-nsg 19089 df-eqg 19090 df-ghm 19177 df-cntz 19281 df-od 19492 df-cmn 19746 df-abl 19747 df-mgp 20111 df-rng 20123 df-ur 20152 df-ring 20205 df-cring 20206 df-oppr 20306 df-dvdsr 20326 df-unit 20327 df-invr 20357 df-dvr 20370 df-rhm 20441 df-subrng 20512 df-subrg 20536 df-drng 20697 df-lmod 20846 df-lss 20916 df-lsp 20956 df-sra 21157 df-rgmod 21158 df-lidl 21195 df-rsp 21196 df-2idl 21237 df-psmet 21333 df-xmet 21334 df-met 21335 df-bl 21336 df-mopn 21337 df-fbas 21338 df-fg 21339 df-cnfld 21342 df-zring 21416 df-zrh 21472 df-zn 21475 df-top 22847 df-topon 22864 df-topsp 22886 df-bases 22899 df-cld 22972 df-ntr 22973 df-cls 22974 df-nei 23051 df-lp 23089 df-perf 23090 df-cn 23180 df-cnp 23181 df-haus 23268 df-tx 23515 df-hmeo 23708 df-fil 23799 df-fm 23891 df-flim 23892 df-flf 23893 df-xms 24273 df-ms 24274 df-tms 24275 df-cncf 24833 df-limc 25821 df-dv 25822 df-log 26508 df-cxp 26509 df-dchr 27184

This theorem is referenced by: dchrisum0flblem2 27460 dchrisum0flb 27461

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