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Theorem dchrisum0flblem1 26561

Description: Lemma for dchrisum0flb 26563. 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 10907	. . . . 5 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 1 ∈ ℝ)
2		0red 10909	. . . . 5 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ ¬ (√‘(𝑃↑𝐴)) ∈ ℕ) → 0 ∈ ℝ)
3	1, 2	ifclda 4491	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ∈ ℝ)
4		1red 10907	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → 1 ∈ ℝ)
5		fzfid 13621	. . . . . 6 ⊢ (𝜑 → (0...𝐴) ∈ Fin)
6		dchrisum0flb.r	. . . . . . . 8 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
7		rpvmasum.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnnn0d 12223	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
9		rpvmasum.z	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
10		eqid 2738	. . . . . . . . . . 11 ⊢ (Base‘𝑍) = (Base‘𝑍)
11		rpvmasum.l	. . . . . . . . . . 11 ⊢ 𝐿 = (ℤRHom‘𝑍)
12	9, 10, 11	znzrhfo 20667	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
13		fof 6672	. . . . . . . . . 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 16308	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
17	15, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℤ)
18	14, 17	ffvelrnd 6944	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝑃) ∈ (Base‘𝑍))
19	6, 18	ffvelrnd 6944	. . . . . . 7 ⊢ (𝜑 → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
20		elfznn0 13278	. . . . . . 7 ⊢ (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℕ₀)
21		reexpcl 13727	. . . . . . 7 ⊢ (((𝑋‘(𝐿‘𝑃)) ∈ ℝ ∧ 𝑖 ∈ ℕ₀) → ((𝑋‘(𝐿‘𝑃))↑𝑖) ∈ ℝ)
22	19, 20, 21	syl2an 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ((𝑋‘(𝐿‘𝑃))↑𝑖) ∈ ℝ)
23	5, 22	fsumrecl 15374	. . . . 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 5073	. . . . . 6 ⊢ (1 = if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) → (1 ≤ 1 ↔ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ 1))
26		breq1 5073	. . . . . 6 ⊢ (0 = if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) → (0 ≤ 1 ↔ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ 1))
27		1le1 11533	. . . . . 6 ⊢ 1 ≤ 1
28		0le1 11428	. . . . . 6 ⊢ 0 ≤ 1
29	25, 26, 27, 28	keephyp 4527	. . . . 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 12549	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
33	31, 32	eleqtrdi 2849	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘0))
34		fzn0 13199	. . . . . . . . 9 ⊢ ((0...𝐴) ≠ ∅ ↔ 𝐴 ∈ (ℤ_≥‘0))
35	33, 34	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → (0...𝐴) ≠ ∅)
36		hashnncl 14009	. . . . . . . . 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 256	. . . . . . 7 ⊢ (𝜑 → (♯‘(0...𝐴)) ∈ ℕ)
39	38	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (♯‘(0...𝐴)) ∈ ℕ)
40	39	nnge1d 11951	. . . . 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 7270	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → ((𝑋‘(𝐿‘𝑃))↑𝑖) = (1↑𝑖))
43		elfzelz 13185	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℤ)
44		1exp 13740	. . . . . . . . 9 ⊢ (𝑖 ∈ ℤ → (1↑𝑖) = 1)
45	43, 44	syl 17	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝐴) → (1↑𝑖) = 1)
46	42, 45	sylan9eq 2799	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ 𝑖 ∈ (0...𝐴)) → ((𝑋‘(𝐿‘𝑃))↑𝑖) = 1)
47	46	sumeq2dv 15343	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = Σ𝑖 ∈ (0...𝐴)1)
48		fzfid 13621	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (0...𝐴) ∈ Fin)
49		ax-1cn 10860	. . . . . . 7 ⊢ 1 ∈ ℂ
50		fsumconst 15430	. . . . . . 7 ⊢ (((0...𝐴) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑖 ∈ (0...𝐴)1 = ((♯‘(0...𝐴)) · 1))
51	48, 49, 50	sylancl 585	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)1 = ((♯‘(0...𝐴)) · 1))
52	39	nncnd 11919	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (♯‘(0...𝐴)) ∈ ℂ)
53	52	mulid1d 10923	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → ((♯‘(0...𝐴)) · 1) = (♯‘(0...𝐴)))
54	47, 51, 53	3eqtrd 2782	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = (♯‘(0...𝐴)))
55	40, 54	breqtrrd 5098	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → 1 ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
56	3, 4, 24, 30, 55	letrd 11062	. . 3 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
57		oveq1 7262	. . . . . . 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 5080	. . . . . 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 7262	. . . . . . 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 5080	. . . . . 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 10906	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
62	19	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
63		resubcl 11215	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (𝑋‘(𝐿‘𝑃)) ∈ ℝ) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℝ)
64	61, 62, 63	sylancr 586	. . . . . . . . 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 11471	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − (𝑋‘(𝐿‘𝑃))) ≤ (1 − (𝑋‘(𝐿‘𝑃))))
67	64	recnd 10934	. . . . . . . . 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	mulid2d 10924	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 · (1 − (𝑋‘(𝐿‘𝑃)))) = (1 − (𝑋‘(𝐿‘𝑃))))
70		nn0p1nn 12202	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ)
71	31, 70	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ)
72	71	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → (𝐴 + 1) ∈ ℕ)
73	72	0expd 13785	. . . . . . . . . 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 7270	. . . . . . . . . 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 2788	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (𝑋‘(𝐿‘𝑃)))
77		neg1cn 12017	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℂ
78	31	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 ∈ ℕ₀)
79		expp1 13717	. . . . . . . . . . . . 13 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℕ₀) → (-1↑(𝐴 + 1)) = ((-1↑𝐴) · -1))
80	77, 78, 79	sylancr 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑(𝐴 + 1)) = ((-1↑𝐴) · -1))
81		prmnn 16307	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
82	15, 81	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑃 ∈ ℕ)
83	82, 31	nnexpcld 13888	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑃↑𝐴) ∈ ℕ)
84	83	nncnd 11919	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑃↑𝐴) ∈ ℂ)
85	84	ad2antrr 722	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃↑𝐴) ∈ ℂ)
86	85	sqsqrtd 15079	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((√‘(𝑃↑𝐴))↑2) = (𝑃↑𝐴))
87	86	oveq2d 7271	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt ((√‘(𝑃↑𝐴))↑2)) = (𝑃 pCnt (𝑃↑𝐴)))
88	15	ad2antrr 722	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝑃 ∈ ℙ)
89		nnq 12631	. . . . . . . . . . . . . . . . . . 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 11937	. . . . . . . . . . . . . . . . . . 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 12282	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℤ
94	93	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 2 ∈ ℤ)
95		pcexp 16488	. . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt ((√‘(𝑃↑𝐴))↑2)) = (2 · (𝑃 pCnt (√‘(𝑃↑𝐴)))))
97	78	nn0zd 12353	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 ∈ ℤ)
98		pcid 16502	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴)
99	88, 97, 98	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴)
100	87, 96, 99	3eqtr3rd 2787	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 = (2 · (𝑃 pCnt (√‘(𝑃↑𝐴)))))
101	100	oveq2d 7271	. . . . . . . . . . . . . . 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 16479	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt (√‘(𝑃↑𝐴))) ∈ ℕ₀)
105		2nn0 12180	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ₀
106	105	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 2 ∈ ℕ₀)
107	102, 104, 106	expmuld 13795	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑(2 · (𝑃 pCnt (√‘(𝑃↑𝐴))))) = ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))))
108		neg1sqe1 13841	. . . . . . . . . . . . . . . . 17 ⊢ (-1↑2) = 1
109	108	oveq1i 7265	. . . . . . . . . . . . . . . 16 ⊢ ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = (1↑(𝑃 pCnt (√‘(𝑃↑𝐴))))
110	104	nn0zd 12353	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt (√‘(𝑃↑𝐴))) ∈ ℤ)
111		1exp 13740	. . . . . . . . . . . . . . . . 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	syl5eq 2791	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = 1)
114	101, 107, 113	3eqtrd 2782	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑𝐴) = 1)
115	114	oveq1d 7270	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑𝐴) · -1) = (1 · -1))
116	77	mulid2i 10911	. . . . . . . . . . . . 13 ⊢ (1 · -1) = -1
117	115, 116	eqtrdi 2795	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑𝐴) · -1) = -1)
118	80, 117	eqtrd 2778	. . . . . . . . . . 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 10934	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
121	120	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
122	121	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
123	122	negnegd 11253	. . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → 𝑋 ∈ 𝐷)
130		eqid 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
131	126, 9, 127, 10, 130, 128, 18	dchrn0 26303	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑋‘(𝐿‘𝑃)) ≠ 0 ↔ (𝐿‘𝑃) ∈ (Unit‘𝑍)))
132	131	ad2antrr 722	. . . . . . . . . . . . . . . . . . . 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 26313	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (abs‘(𝑋‘(𝐿‘𝑃))) = 1)
135		eqeq1 2742	. . . . . . . . . . . . . . . . . 18 ⊢ ((abs‘(𝑋‘(𝐿‘𝑃))) = (𝑋‘(𝐿‘𝑃)) → ((abs‘(𝑋‘(𝐿‘𝑃))) = 1 ↔ (𝑋‘(𝐿‘𝑃)) = 1))
136	134, 135	syl5ibcom 244	. . . . . . . . . . . . . . . . 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 2955	. . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
140	139	absord 15055	. . . . . . . . . . . . . . . 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 860	. . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → -(𝑋‘(𝐿‘𝑃)) = 1)
144	143	negeqd 11145	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → --(𝑋‘(𝐿‘𝑃)) = -1)
145	123, 144	eqtr3d 2780	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) = -1)
146	145	oveq1d 7270	. . . . . . . . . 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 2788	. . . . . . . . 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 3031	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (𝑋‘(𝐿‘𝑃)))
149	148	oveq2d 7271	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = (1 − (𝑋‘(𝐿‘𝑃))))
150	66, 69, 149	3brtr4d 5102	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
151	67	mul02d 11103	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (0 · (1 − (𝑋‘(𝐿‘𝑃)))) = 0)
152		peano2nn0 12203	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ₀)
153	31, 152	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ₀)
154	19, 153	reexpcld 13809	. . . . . . . . . . 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 10934	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℂ)
157	156	abscld 15076	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ∈ ℝ)
158		1red 10907	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 1 ∈ ℝ)
159	155	leabsd 15054	. . . . . . . . . 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 15092	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = ((abs‘(𝑋‘(𝐿‘𝑃)))↑(𝐴 + 1)))
162	121	abscld 15076	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘(𝑋‘(𝐿‘𝑃))) ∈ ℝ)
163	121	absge0d 15084	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 ≤ (abs‘(𝑋‘(𝐿‘𝑃))))
164	126, 127, 9, 10, 128, 18	dchrabs2 26315	. . . . . . . . . . . . 13 ⊢ (𝜑 → (abs‘(𝑋‘(𝐿‘𝑃))) ≤ 1)
165	164	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘(𝑋‘(𝐿‘𝑃))) ≤ 1)
166		exple1 13822	. . . . . . . . . . . 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 1371	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((abs‘(𝑋‘(𝐿‘𝑃)))↑(𝐴 + 1)) ≤ 1)
168	161, 167	eqbrtrd 5092	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ≤ 1)
169	155, 157, 158, 159, 168	letrd 11062	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ≤ 1)
170		subge0 11418	. . . . . . . . . 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 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (0 ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ↔ ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ≤ 1))
172	169, 171	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
173	151, 172	eqbrtrd 5092	. . . . . . 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 4494	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
176		0re 10908	. . . . . . . 8 ⊢ 0 ∈ ℝ
177	61, 176	ifcli 4503	. . . . . . 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 11215	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℝ) → (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ∈ ℝ)
180	61, 155, 179	sylancr 586	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ∈ ℝ)
181	62	leabsd 15054	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ≤ (abs‘(𝑋‘(𝐿‘𝑃))))
182	62, 162, 158, 181, 165	letrd 11062	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ≤ 1)
183	124	necomd 2998	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 1 ≠ (𝑋‘(𝐿‘𝑃)))
184	62, 158, 182, 183	leneltd 11059	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) < 1)
185		posdif 11398	. . . . . . . 8 ⊢ (((𝑋‘(𝐿‘𝑃)) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑋‘(𝐿‘𝑃)) < 1 ↔ 0 < (1 − (𝑋‘(𝐿‘𝑃)))))
186	62, 61, 185	sylancl 585	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃)) < 1 ↔ 0 < (1 − (𝑋‘(𝐿‘𝑃)))))
187	184, 186	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 < (1 − (𝑋‘(𝐿‘𝑃))))
188		lemuldiv 11785	. . . . . 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 1372	. . . . 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 231	. . . 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 12353	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
192		fzval3 13384	. . . . . . . 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 15341	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = Σ𝑖 ∈ (0..^(𝐴 + 1))((𝑋‘(𝐿‘𝑃))↑𝑖))
196		0nn0 12178	. . . . . . 7 ⊢ 0 ∈ ℕ₀
197	196	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 ∈ ℕ₀)
198	153, 32	eleqtrdi 2849	. . . . . . 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 15506	. . . . 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 13786	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑0) = 1)
202	201	oveq1d 7270	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (((𝑋‘(𝐿‘𝑃))↑0) − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
203	202	oveq1d 7270	. . . . 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 2782	. . . 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 5098	. . 3 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
206	56, 205	pm2.61dane 3031	. 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 26558	. . . 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 6761	. . . 4 ⊢ (𝑘 = (𝑃↑𝑖) → (𝑋‘(𝐿‘𝑘)) = (𝑋‘(𝐿‘(𝑃↑𝑖))))
212		eqid 2738	. . . . . 6 ⊢ (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏)) = (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏))
213	212	dvdsppwf1o 26240	. . . . 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 583	. . . 4 ⊢ (𝜑 → (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏)):(0...𝐴)–1-1-onto→{𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)})
215		oveq2 7263	. . . . . 6 ⊢ (𝑏 = 𝑖 → (𝑃↑𝑏) = (𝑃↑𝑖))
216		ovex 7288	. . . . . 6 ⊢ (𝑃↑𝑏) ∈ V
217	215, 212, 216	fvmpt3i 6862	. . . . 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 3611	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} → 𝑘 ∈ ℕ)
221	220	nnzd 12354	. . . . . . 7 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} → 𝑘 ∈ ℤ)
222		ffvelrn 6941	. . . . . . 7 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑘 ∈ ℤ) → (𝐿‘𝑘) ∈ (Base‘𝑍))
223	14, 221, 222	syl2an 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → (𝐿‘𝑘) ∈ (Base‘𝑍))
224	219, 223	ffvelrnd 6944	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → (𝑋‘(𝐿‘𝑘)) ∈ ℝ)
225	224	recnd 10934	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
226	211, 5, 214, 218, 225	fsumf1o 15363	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} (𝑋‘(𝐿‘𝑘)) = Σ𝑖 ∈ (0...𝐴)(𝑋‘(𝐿‘(𝑃↑𝑖))))
227		zsubrg 20563	. . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
228		eqid 2738	. . . . . . . . . . . 12 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
229	228	subrgsubm 19952	. . . . . . . . . . 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 2738	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
234		zringmpg 20605	. . . . . . . . . . . 12 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = (mulGrp‘ℤ_ring)
235	234	eqcomi 2747	. . . . . . . . . . 11 ⊢ (mulGrp‘ℤ_ring) = ((mulGrp‘ℂ_fld) ↾_s ℤ)
236		eqid 2738	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘ℤ_ring)) = (._g‘(mulGrp‘ℤ_ring))
237	233, 235, 236	submmulg 18662	. . . . . . . . . 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 1369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℂ_fld))𝑃) = (𝑖(._g‘(mulGrp‘ℤ_ring))𝑃))
239	82	nncnd 11919	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℂ)
240		cnfldexp 20543	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℂ ∧ 𝑖 ∈ ℕ₀) → (𝑖(._g‘(mulGrp‘ℂ_fld))𝑃) = (𝑃↑𝑖))
241	239, 20, 240	syl2an 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℂ_fld))𝑃) = (𝑃↑𝑖))
242	238, 241	eqtr3d 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℤ_ring))𝑃) = (𝑃↑𝑖))
243	242	fveq2d 6760	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑖(._g‘(mulGrp‘ℤ_ring))𝑃)) = (𝐿‘(𝑃↑𝑖)))
244	9	zncrng 20664	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
245		crngring 19710	. . . . . . . . . . 11 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
246	8, 244, 245	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Ring)
247	11	zrhrhm 20625	. . . . . . . . . 10 ⊢ (𝑍 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑍))
248		eqid 2738	. . . . . . . . . . 11 ⊢ (mulGrp‘ℤ_ring) = (mulGrp‘ℤ_ring)
249		eqid 2738	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
250	248, 249	rhmmhm 19881	. . . . . . . . . 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 20588	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤ_ring)
254	248, 253	mgpbas 19641	. . . . . . . . 9 ⊢ ℤ = (Base‘(mulGrp‘ℤ_ring))
255		eqid 2738	. . . . . . . . 9 ⊢ (._g‘(mulGrp‘𝑍)) = (._g‘(mulGrp‘𝑍))
256	254, 236, 255	mhmmulg 18659	. . . . . . . 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 1369	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑖(._g‘(mulGrp‘ℤ_ring))𝑃)) = (𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃)))
258	243, 257	eqtr3d 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑃↑𝑖)) = (𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃)))
259	258	fveq2d 6760	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝐿‘(𝑃↑𝑖))) = (𝑋‘(𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃))))
260	126, 9, 127	dchrmhm 26294	. . . . . . . 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 19641	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
265	264, 255, 233	mhmmulg 18659	. . . . . 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 1369	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃))) = (𝑖(._g‘(mulGrp‘ℂ_fld))(𝑋‘(𝐿‘𝑃))))
267		cnfldexp 20543	. . . . . 6 ⊢ (((𝑋‘(𝐿‘𝑃)) ∈ ℂ ∧ 𝑖 ∈ ℕ₀) → (𝑖(._g‘(mulGrp‘ℂ_fld))(𝑋‘(𝐿‘𝑃))) = ((𝑋‘(𝐿‘𝑃))↑𝑖))
268	120, 20, 267	syl2an 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℂ_fld))(𝑋‘(𝐿‘𝑃))) = ((𝑋‘(𝐿‘𝑃))↑𝑖))
269	259, 266, 268	3eqtrd 2782	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝐿‘(𝑃↑𝑖))) = ((𝑋‘(𝐿‘𝑃))↑𝑖))
270	269	sumeq2dv 15343	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐴)(𝑋‘(𝐿‘(𝑃↑𝑖))) = Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
271	210, 226, 270	3eqtrd 2782	. 2 ⊢ (𝜑 → (𝐹‘(𝑃↑𝐴)) = Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
272	206, 271	breqtrrd 5098	1 ⊢ (𝜑 → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃↑𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 {crab 3067 ∅c0 4253 ifcif 4456 class class class wbr 5070 ↦ cmpt 5153 ⟶wf 6414 –onto→wfo 6416 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ℚcq 12617 ...cfz 13168 ..^cfzo 13311 ↑cexp 13710 ♯chash 13972 √csqrt 14872 abscabs 14873 Σcsu 15325 ∥ cdvds 15891 ℙcprime 16304 pCnt cpc 16465 Basecbs 16840 ↾_s cress 16867 0_gc0g 17067 MndHom cmhm 18343 SubMndcsubmnd 18344 ._gcmg 18615 mulGrpcmgp 19635 Ringcrg 19698 CRingccrg 19699 Unitcui 19796 RingHom crh 19871 SubRingcsubrg 19935 ℂ_fldccnfld 20510 ℤ_ringzring 20582 ℤRHomczrh 20613 ℤ/nℤczn 20616 DChrcdchr 26285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-omul 8272 df-er 8456 df-ec 8458 df-qs 8462 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-acn 9631 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-dvds 15892 df-gcd 16130 df-prm 16305 df-pc 16466 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-qus 17137 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-nsg 18668 df-eqg 18669 df-ghm 18747 df-cntz 18838 df-od 19051 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-subrg 19937 df-lmod 20040 df-lss 20109 df-lsp 20149 df-sra 20349 df-rgmod 20350 df-lidl 20351 df-rsp 20352 df-2idl 20416 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-zn 20620 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617 df-cxp 25618 df-dchr 26286

This theorem is referenced by: dchrisum0flblem2 26562 dchrisum0flb 26563

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