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Theorem dchrisum0flblem1 27683

Description: Lemma for dchrisum0flb 27685. 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 11215	. . . . 5 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 1 ∈ ℝ)
2		0red 11217	. . . . 5 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ ¬ (√‘(𝑃↑𝐴)) ∈ ℕ) → 0 ∈ ℝ)
3	1, 2	ifclda 4522	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ∈ ℝ)
4		1red 11215	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → 1 ∈ ℝ)
5		fzfid 14016	. . . . . 6 ⊢ (𝜑 → (0...𝐴) ∈ Fin)
6		dchrisum0flb.r	. . . . . . . 8 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
7		rpvmasum.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnnn0d 12571	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
9		rpvmasum.z	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
10		eqid 2762	. . . . . . . . . . 11 ⊢ (Base‘𝑍) = (Base‘𝑍)
11		rpvmasum.l	. . . . . . . . . . 11 ⊢ 𝐿 = (ℤRHom‘𝑍)
12	9, 10, 11	znzrhfo 21708	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
13		fof 6792	. . . . . . . . . 10 ⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
14	8, 12, 13	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍))
15		dchrisum0flblem1.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℙ)
16		prmz 16739	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
17	15, 16	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℤ)
18	14, 17	ffvelcdmd 7080	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝑃) ∈ (Base‘𝑍))
19	6, 18	ffvelcdmd 7080	. . . . . . 7 ⊢ (𝜑 → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
20		elfznn0 13655	. . . . . . 7 ⊢ (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℕ₀)
21		reexpcl 14121	. . . . . . 7 ⊢ (((𝑋‘(𝐿‘𝑃)) ∈ ℝ ∧ 𝑖 ∈ ℕ₀) → ((𝑋‘(𝐿‘𝑃))↑𝑖) ∈ ℝ)
22	19, 20, 21	syl2an 607	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ((𝑋‘(𝐿‘𝑃))↑𝑖) ∈ ℝ)
23	5, 22	fsumrecl 15792	. . . . 5 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) ∈ ℝ)
24	23	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) ∈ ℝ)
25		breq1 5111	. . . . . 6 ⊢ (1 = if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) → (1 ≤ 1 ↔ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ 1))
26		breq1 5111	. . . . . 6 ⊢ (0 = if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) → (0 ≤ 1 ↔ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ 1))
27		1le1 11848	. . . . . 6 ⊢ 1 ≤ 1
28		0le1 11743	. . . . . 6 ⊢ 0 ≤ 1
29	25, 26, 27, 28	keephyp 4558	. . . . 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 12906	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
33	31, 32	eleqtrdi 2872	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘0))
34		fzn0 13572	. . . . . . . . 9 ⊢ ((0...𝐴) ≠ ∅ ↔ 𝐴 ∈ (ℤ_≥‘0))
35	33, 34	sylibr 237	. . . . . . . 8 ⊢ (𝜑 → (0...𝐴) ≠ ∅)
36		hashnncl 14409	. . . . . . . . 9 ⊢ ((0...𝐴) ∈ Fin → ((♯‘(0...𝐴)) ∈ ℕ ↔ (0...𝐴) ≠ ∅))
37	5, 36	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(0...𝐴)) ∈ ℕ ↔ (0...𝐴) ≠ ∅))
38	35, 37	mpbird 260	. . . . . . 7 ⊢ (𝜑 → (♯‘(0...𝐴)) ∈ ℕ)
39	38	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (♯‘(0...𝐴)) ∈ ℕ)
40	39	nnge1d 12290	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → 1 ≤ (♯‘(0...𝐴)))
41		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (𝑋‘(𝐿‘𝑃)) = 1)
42	41	oveq1d 7427	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → ((𝑋‘(𝐿‘𝑃))↑𝑖) = (1↑𝑖))
43		elfzelz 13558	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℤ)
44		1exp 14134	. . . . . . . . 9 ⊢ (𝑖 ∈ ℤ → (1↑𝑖) = 1)
45	43, 44	syl 18	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝐴) → (1↑𝑖) = 1)
46	42, 45	sylan9eq 2817	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ 𝑖 ∈ (0...𝐴)) → ((𝑋‘(𝐿‘𝑃))↑𝑖) = 1)
47	46	sumeq2dv 15760	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = Σ𝑖 ∈ (0...𝐴)1)
48		fzfid 14016	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (0...𝐴) ∈ Fin)
49		ax-1cn 11164	. . . . . . 7 ⊢ 1 ∈ ℂ
50		fsumconst 15848	. . . . . . 7 ⊢ (((0...𝐴) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑖 ∈ (0...𝐴)1 = ((♯‘(0...𝐴)) · 1))
51	48, 49, 50	sylancl 597	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)1 = ((♯‘(0...𝐴)) · 1))
52	39	nncnd 12255	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (♯‘(0...𝐴)) ∈ ℂ)
53	52	mulridd 11232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → ((♯‘(0...𝐴)) · 1) = (♯‘(0...𝐴)))
54	47, 51, 53	3eqtrd 2801	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = (♯‘(0...𝐴)))
55	40, 54	breqtrrd 5138	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → 1 ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
56	3, 4, 24, 30, 55	letrd 11373	. . 3 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
57		oveq1 7419	. . . . . . 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 5118	. . . . . 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 7419	. . . . . . 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 5118	. . . . . 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 11214	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
62	19	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
63		resubcl 11528	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (𝑋‘(𝐿‘𝑃)) ∈ ℝ) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℝ)
64	61, 62, 63	sylancr 598	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℝ)
65	64	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℝ)
66	65	leidd 11786	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − (𝑋‘(𝐿‘𝑃))) ≤ (1 − (𝑋‘(𝐿‘𝑃))))
67	64	recnd 11243	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℂ)
68	67	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℂ)
69	68	mullidd 11233	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 · (1 − (𝑋‘(𝐿‘𝑃)))) = (1 − (𝑋‘(𝐿‘𝑃))))
70		nn0p1nn 12549	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ)
71	31, 70	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ)
72	71	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → (𝐴 + 1) ∈ ℕ)
73	72	0expd 14182	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → (0↑(𝐴 + 1)) = 0)
74		simpr 489	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → (𝑋‘(𝐿‘𝑃)) = 0)
75	74	oveq1d 7427	. . . . . . . . . 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 2807	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (𝑋‘(𝐿‘𝑃)))
77		neg1cn 12209	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℂ
78	31	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 ∈ ℕ₀)
79		expp1 14111	. . . . . . . . . . . . 13 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℕ₀) → (-1↑(𝐴 + 1)) = ((-1↑𝐴) · -1))
80	77, 78, 79	sylancr 598	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑(𝐴 + 1)) = ((-1↑𝐴) · -1))
81		prmnn 16738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
82	15, 81	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑃 ∈ ℕ)
83	82, 31	nnexpcld 14288	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑃↑𝐴) ∈ ℕ)
84	83	nncnd 12255	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑃↑𝐴) ∈ ℂ)
85	84	ad2antrr 738	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃↑𝐴) ∈ ℂ)
86	85	sqsqrtd 15500	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((√‘(𝑃↑𝐴))↑2) = (𝑃↑𝐴))
87	86	oveq2d 7428	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt ((√‘(𝑃↑𝐴))↑2)) = (𝑃 pCnt (𝑃↑𝐴)))
88	15	ad2antrr 738	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝑃 ∈ ℙ)
89		nnq 12992	. . . . . . . . . . . . . . . . . . 19 ⊢ ((√‘(𝑃↑𝐴)) ∈ ℕ → (√‘(𝑃↑𝐴)) ∈ ℚ)
90	89	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (√‘(𝑃↑𝐴)) ∈ ℚ)
91		nnne0 12276	. . . . . . . . . . . . . . . . . . 19 ⊢ ((√‘(𝑃↑𝐴)) ∈ ℕ → (√‘(𝑃↑𝐴)) ≠ 0)
92	91	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (√‘(𝑃↑𝐴)) ≠ 0)
93		2z 12632	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℤ
94	93	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 2 ∈ ℤ)
95		pcexp 16925	. . . . . . . . . . . . . . . . . 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 1401	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt ((√‘(𝑃↑𝐴))↑2)) = (2 · (𝑃 pCnt (√‘(𝑃↑𝐴)))))
97	78	nn0zd 12622	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 ∈ ℤ)
98		pcid 16939	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴)
99	88, 97, 98	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴)
100	87, 96, 99	3eqtr3rd 2806	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 = (2 · (𝑃 pCnt (√‘(𝑃↑𝐴)))))
101	100	oveq2d 7428	. . . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (√‘(𝑃↑𝐴)) ∈ ℕ)
104	88, 103	pccld 16916	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt (√‘(𝑃↑𝐴))) ∈ ℕ₀)
105		2nn0 12527	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ₀
106	105	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 2 ∈ ℕ₀)
107	102, 104, 106	expmuld 14192	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑(2 · (𝑃 pCnt (√‘(𝑃↑𝐴))))) = ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))))
108		neg1sqe1 14239	. . . . . . . . . . . . . . . . 17 ⊢ (-1↑2) = 1
109	108	oveq1i 7422	. . . . . . . . . . . . . . . 16 ⊢ ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = (1↑(𝑃 pCnt (√‘(𝑃↑𝐴))))
110	104	nn0zd 12622	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt (√‘(𝑃↑𝐴))) ∈ ℤ)
111		1exp 14134	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 pCnt (√‘(𝑃↑𝐴))) ∈ ℤ → (1↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = 1)
112	110, 111	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = 1)
113	109, 112	eqtrid 2809	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = 1)
114	101, 107, 113	3eqtrd 2801	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑𝐴) = 1)
115	114	oveq1d 7427	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑𝐴) · -1) = (1 · -1))
116	77	mullidi 11220	. . . . . . . . . . . . 13 ⊢ (1 · -1) = -1
117	115, 116	eqtrdi 2813	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑𝐴) · -1) = -1)
118	80, 117	eqtrd 2797	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑(𝐴 + 1)) = -1)
119	118	adantr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (-1↑(𝐴 + 1)) = -1)
120	19	recnd 11243	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
121	120	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
122	121	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
123	122	negnegd 11566	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → --(𝑋‘(𝐿‘𝑃)) = (𝑋‘(𝐿‘𝑃)))
124		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ≠ 1)
125	124	ad2antrr 738	. . . . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → 𝑋 ∈ 𝐷)
130		eqid 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
131	126, 9, 127, 10, 130, 128, 18	dchrn0 27425	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑋‘(𝐿‘𝑃)) ≠ 0 ↔ (𝐿‘𝑃) ∈ (Unit‘𝑍)))
132	131	ad2antrr 738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((𝑋‘(𝐿‘𝑃)) ≠ 0 ↔ (𝐿‘𝑃) ∈ (Unit‘𝑍)))
133	132	biimpa 481	. . . . . . . . . . . . . . . . . . 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 27435	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (abs‘(𝑋‘(𝐿‘𝑃))) = 1)
135		eqeq1 2766	. . . . . . . . . . . . . . . . . 18 ⊢ ((abs‘(𝑋‘(𝐿‘𝑃))) = (𝑋‘(𝐿‘𝑃)) → ((abs‘(𝑋‘(𝐿‘𝑃))) = 1 ↔ (𝑋‘(𝐿‘𝑃)) = 1))
136	134, 135	syl5ibcom 248	. . . . . . . . . . . . . . . . 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 2970	. . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → ¬ (abs‘(𝑋‘(𝐿‘𝑃))) = (𝑋‘(𝐿‘𝑃)))
139	62	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
140	139	absord 15474	. . . . . . . . . . . . . . . 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 877	. . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (abs‘(𝑋‘(𝐿‘𝑃))) = -(𝑋‘(𝐿‘𝑃)))
143	142, 134	eqtr3d 2799	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → -(𝑋‘(𝐿‘𝑃)) = 1)
144	143	negeqd 11457	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → --(𝑋‘(𝐿‘𝑃)) = -1)
145	123, 144	eqtr3d 2799	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) = -1)
146	145	oveq1d 7427	. . . . . . . . . 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 2807	. . . . . . . . 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 3044	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (𝑋‘(𝐿‘𝑃)))
149	148	oveq2d 7428	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = (1 − (𝑋‘(𝐿‘𝑃))))
150	66, 69, 149	3brtr4d 5142	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
151	67	mul02d 11414	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (0 · (1 − (𝑋‘(𝐿‘𝑃)))) = 0)
152		peano2nn0 12550	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ₀)
153	31, 152	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ₀)
154	19, 153	reexpcld 14206	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℝ)
155	154	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℝ)
156	155	recnd 11243	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℂ)
157	156	abscld 15497	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ∈ ℝ)
158		1red 11215	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 1 ∈ ℝ)
159	155	leabsd 15473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ≤ (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
160	153	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝐴 + 1) ∈ ℕ₀)
161	121, 160	absexpd 15513	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = ((abs‘(𝑋‘(𝐿‘𝑃)))↑(𝐴 + 1)))
162	121	abscld 15497	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘(𝑋‘(𝐿‘𝑃))) ∈ ℝ)
163	121	absge0d 15505	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 ≤ (abs‘(𝑋‘(𝐿‘𝑃))))
164	126, 127, 9, 10, 128, 18	dchrabs2 27437	. . . . . . . . . . . . 13 ⊢ (𝜑 → (abs‘(𝑋‘(𝐿‘𝑃))) ≤ 1)
165	164	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘(𝑋‘(𝐿‘𝑃))) ≤ 1)
166		exple1 14220	. . . . . . . . . . . 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 1399	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((abs‘(𝑋‘(𝐿‘𝑃)))↑(𝐴 + 1)) ≤ 1)
168	161, 167	eqbrtrd 5132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ≤ 1)
169	155, 157, 158, 159, 168	letrd 11373	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ≤ 1)
170		subge0 11733	. . . . . . . . . 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 598	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (0 ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ↔ ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ≤ 1))
172	169, 171	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
173	151, 172	eqbrtrd 5132	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (0 · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
174	173	adantr 485	. . . . . 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 4525	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
176		0re 11216	. . . . . . . 8 ⊢ 0 ∈ ℝ
177	61, 176	ifcli 4534	. . . . . . 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 11528	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℝ) → (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ∈ ℝ)
180	61, 155, 179	sylancr 598	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ∈ ℝ)
181	62	leabsd 15473	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ≤ (abs‘(𝑋‘(𝐿‘𝑃))))
182	62, 162, 158, 181, 165	letrd 11373	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ≤ 1)
183	124	necomd 3012	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 1 ≠ (𝑋‘(𝐿‘𝑃)))
184	62, 158, 182, 183	leneltd 11370	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) < 1)
185		posdif 11713	. . . . . . . 8 ⊢ (((𝑋‘(𝐿‘𝑃)) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑋‘(𝐿‘𝑃)) < 1 ↔ 0 < (1 − (𝑋‘(𝐿‘𝑃)))))
186	62, 61, 185	sylancl 597	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃)) < 1 ↔ 0 < (1 − (𝑋‘(𝐿‘𝑃)))))
187	184, 186	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 < (1 − (𝑋‘(𝐿‘𝑃))))
188		lemuldiv 12101	. . . . . 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 1400	. . . . 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 235	. . . 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 12622	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
192		fzval3 13770	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (0...𝐴) = (0..^(𝐴 + 1)))
193	191, 192	syl 18	. . . . . . 7 ⊢ (𝜑 → (0...𝐴) = (0..^(𝐴 + 1)))
194	193	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (0...𝐴) = (0..^(𝐴 + 1)))
195	194	sumeq1d 15758	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = Σ𝑖 ∈ (0..^(𝐴 + 1))((𝑋‘(𝐿‘𝑃))↑𝑖))
196		0nn0 12525	. . . . . . 7 ⊢ 0 ∈ ℕ₀
197	196	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 ∈ ℕ₀)
198	153, 32	eleqtrdi 2872	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ (ℤ_≥‘0))
199	198	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝐴 + 1) ∈ (ℤ_≥‘0))
200	121, 124, 197, 199	geoserg 15927	. . . . 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 14183	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑0) = 1)
202	201	oveq1d 7427	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (((𝑋‘(𝐿‘𝑃))↑0) − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
203	202	oveq1d 7427	. . . . 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 2801	. . . 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 5138	. . 3 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
206	56, 205	pm2.61dane 3044	. 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 27680	. . . 4 ⊢ ((𝑃↑𝐴) ∈ ℕ → (𝐹‘(𝑃↑𝐴)) = Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} (𝑋‘(𝐿‘𝑘)))
210	83, 209	syl 18	. . 3 ⊢ (𝜑 → (𝐹‘(𝑃↑𝐴)) = Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} (𝑋‘(𝐿‘𝑘)))
211		2fveq3 6886	. . . 4 ⊢ (𝑘 = (𝑃↑𝑖) → (𝑋‘(𝐿‘𝑘)) = (𝑋‘(𝐿‘(𝑃↑𝑖))))
212		eqid 2762	. . . . . 6 ⊢ (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏)) = (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏))
213	212	dvdsppwf1o 27361	. . . . 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 595	. . . 4 ⊢ (𝜑 → (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏)):(0...𝐴)–1-1-onto→{𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)})
215		oveq2 7420	. . . . . 6 ⊢ (𝑏 = 𝑖 → (𝑃↑𝑏) = (𝑃↑𝑖))
216		ovex 7445	. . . . . 6 ⊢ (𝑃↑𝑏) ∈ V
217	215, 212, 216	fvmpt3i 6995	. . . . 5 ⊢ (𝑖 ∈ (0...𝐴) → ((𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏))‘𝑖) = (𝑃↑𝑖))
218	217	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ((𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏))‘𝑖) = (𝑃↑𝑖))
219	6	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → 𝑋:(Base‘𝑍)⟶ℝ)
220		elrabi 3645	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} → 𝑘 ∈ ℕ)
221	220	nnzd 12623	. . . . . . 7 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} → 𝑘 ∈ ℤ)
222		ffvelcdm 7076	. . . . . . 7 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑘 ∈ ℤ) → (𝐿‘𝑘) ∈ (Base‘𝑍))
223	14, 221, 222	syl2an 607	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → (𝐿‘𝑘) ∈ (Base‘𝑍))
224	219, 223	ffvelcdmd 7080	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → (𝑋‘(𝐿‘𝑘)) ∈ ℝ)
225	224	recnd 11243	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
226	211, 5, 214, 218, 225	fsumf1o 15781	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} (𝑋‘(𝐿‘𝑘)) = Σ𝑖 ∈ (0...𝐴)(𝑋‘(𝐿‘(𝑃↑𝑖))))
227		zsubrg 21581	. . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
228		eqid 2762	. . . . . . . . . . . 12 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
229	228	subrgsubm 20695	. . . . . . . . . . 11 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
230	227, 229	mp1i 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
231	20	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑖 ∈ ℕ₀)
232	17	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑃 ∈ ℤ)
233		eqid 2762	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
234		zringmpg 21632	. . . . . . . . . . . 12 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = (mulGrp‘ℤ_ring)
235	234	eqcomi 2771	. . . . . . . . . . 11 ⊢ (mulGrp‘ℤ_ring) = ((mulGrp‘ℂ_fld) ↾_s ℤ)
236		eqid 2762	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘ℤ_ring)) = (._g‘(mulGrp‘ℤ_ring))
237	233, 235, 236	submmulg 19190	. . . . . . . . . 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 1397	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℂ_fld))𝑃) = (𝑖(._g‘(mulGrp‘ℤ_ring))𝑃))
239	82	nncnd 12255	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℂ)
240		cnfldexp 21566	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℂ ∧ 𝑖 ∈ ℕ₀) → (𝑖(._g‘(mulGrp‘ℂ_fld))𝑃) = (𝑃↑𝑖))
241	239, 20, 240	syl2an 607	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℂ_fld))𝑃) = (𝑃↑𝑖))
242	238, 241	eqtr3d 2799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℤ_ring))𝑃) = (𝑃↑𝑖))
243	242	fveq2d 6885	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑖(._g‘(mulGrp‘ℤ_ring))𝑃)) = (𝐿‘(𝑃↑𝑖)))
244	9	zncrng 21705	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
245		crngring 20333	. . . . . . . . . . 11 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
246	8, 244, 245	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Ring)
247	11	zrhrhm 21672	. . . . . . . . . 10 ⊢ (𝑍 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑍))
248		eqid 2762	. . . . . . . . . . 11 ⊢ (mulGrp‘ℤ_ring) = (mulGrp‘ℤ_ring)
249		eqid 2762	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
250	248, 249	rhmmhm 20569	. . . . . . . . . 10 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑍) → 𝐿 ∈ ((mulGrp‘ℤ_ring) MndHom (mulGrp‘𝑍)))
251	246, 247, 250	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ((mulGrp‘ℤ_ring) MndHom (mulGrp‘𝑍)))
252	251	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝐿 ∈ ((mulGrp‘ℤ_ring) MndHom (mulGrp‘𝑍)))
253		zringbas 21614	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤ_ring)
254	248, 253	mgpbas 20227	. . . . . . . . 9 ⊢ ℤ = (Base‘(mulGrp‘ℤ_ring))
255		eqid 2762	. . . . . . . . 9 ⊢ (._g‘(mulGrp‘𝑍)) = (._g‘(mulGrp‘𝑍))
256	254, 236, 255	mhmmulg 19187	. . . . . . . 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 1397	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑖(._g‘(mulGrp‘ℤ_ring))𝑃)) = (𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃)))
258	243, 257	eqtr3d 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑃↑𝑖)) = (𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃)))
259	258	fveq2d 6885	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝐿‘(𝑃↑𝑖))) = (𝑋‘(𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃))))
260	126, 9, 127	dchrmhm 27416	. . . . . . . 8 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
261	260, 128	sselid 3934	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
262	261	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
263	18	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘𝑃) ∈ (Base‘𝑍))
264	249, 10	mgpbas 20227	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
265	264, 255, 233	mhmmulg 19187	. . . . . 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 1397	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃))) = (𝑖(._g‘(mulGrp‘ℂ_fld))(𝑋‘(𝐿‘𝑃))))
267		cnfldexp 21566	. . . . . 6 ⊢ (((𝑋‘(𝐿‘𝑃)) ∈ ℂ ∧ 𝑖 ∈ ℕ₀) → (𝑖(._g‘(mulGrp‘ℂ_fld))(𝑋‘(𝐿‘𝑃))) = ((𝑋‘(𝐿‘𝑃))↑𝑖))
268	120, 20, 267	syl2an 607	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℂ_fld))(𝑋‘(𝐿‘𝑃))) = ((𝑋‘(𝐿‘𝑃))↑𝑖))
269	259, 266, 268	3eqtrd 2801	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝐿‘(𝑃↑𝑖))) = ((𝑋‘(𝐿‘𝑃))↑𝑖))
270	269	sumeq2dv 15760	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐴)(𝑋‘(𝐿‘(𝑃↑𝑖))) = Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
271	210, 226, 270	3eqtrd 2801	. 2 ⊢ (𝜑 → (𝐹‘(𝑃↑𝐴)) = Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
272	206, 271	breqtrrd 5138	1 ⊢ (𝜑 → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃↑𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 {crab 3415 ∅c0 4285 ifcif 4486 class class class wbr 5108 ↦ cmpt 5191 ⟶wf 6532 –onto→wfo 6534 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7412 Fincfn 8941 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11249 ≤ cle 11250 − cmin 11447 -cneg 11448 / cdiv 11877 ℕcn 12239 2c2 12301 ℕ₀cn0 12510 ℤcz 12597 ℤ_≥cuz 12868 ℚcq 12978 ...cfz 13541 ..^cfzo 13689 ↑cexp 14104 ♯chash 14373 √csqrt 15291 abscabs 15292 Σcsu 15744 ∥ cdvds 16316 ℙcprime 16735 pCnt cpc 16902 Basecbs 17275 ↾_s cress 17296 0_gc0g 17498 MndHom cmhm 18845 SubMndcsubmnd 18846 ._gcmg 19139 mulGrpcmgp 20222 Ringcrg 20321 CRingccrg 20322 Unitcui 20444 RingHom crh 20558 SubRingcsubrg 20679 ℂ_fldccnfld 21533 ℤ_ringczring 21607 ℤRHomczrh 21660 ℤ/nℤczn 21663 DChrcdchr 27407

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-disj 5076 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-tpos 8220 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-oadd 8455 df-omul 8456 df-er 8692 df-ec 8694 df-qs 8698 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9932 df-acn 9935 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-fac 14317 df-bc 14346 df-hash 14374 df-shft 15111 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-limsup 15529 df-clim 15546 df-rlim 15547 df-sum 15745 df-ef 16127 df-sin 16129 df-cos 16130 df-pi 16132 df-dvds 16317 df-gcd 16559 df-prm 16736 df-pc 16903 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-xrs 17562 df-qtop 17567 df-imas 17568 df-qus 17569 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-mhm 18847 df-submnd 18848 df-grp 19009 df-minusg 19010 df-sbg 19011 df-mulg 19140 df-subg 19195 df-nsg 19196 df-eqg 19197 df-ghm 19290 df-cntz 19393 df-od 19604 df-cmn 19858 df-abl 19859 df-mgp 20223 df-rng 20237 df-ur 20270 df-ring 20323 df-cring 20324 df-oppr 20426 df-dvdsr 20446 df-unit 20447 df-invr 20477 df-dvr 20490 df-rhm 20561 df-subrng 20656 df-subrg 20680 df-drng 20840 df-lmod 20994 df-lss 21064 df-lsp 21104 df-sra 21305 df-rgmod 21306 df-lidl 21343 df-rsp 21344 df-2idl 21400 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-zring 21608 df-zrh 21664 df-zn 21667 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-lp 23304 df-perf 23305 df-cn 23395 df-cnp 23396 df-haus 23483 df-tx 23730 df-hmeo 23923 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-xms 24488 df-ms 24489 df-tms 24490 df-cncf 25048 df-limc 26036 df-dv 26037 df-log 26732 df-cxp 26733 df-dchr 27408

This theorem is used by: dchrisum0flblem2 27684 dchrisum0flb 27685

W3C validator