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Theorem dchrisum0flblem1 26856

Description: Lemma for dchrisum0flb 26858. 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 11156	. . . . 5 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 1 ∈ ℝ)
2		0red 11158	. . . . 5 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ ¬ (√‘(𝑃↑𝐴)) ∈ ℕ) → 0 ∈ ℝ)
3	1, 2	ifclda 4521	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ∈ ℝ)
4		1red 11156	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → 1 ∈ ℝ)
5		fzfid 13878	. . . . . 6 ⊢ (𝜑 → (0...𝐴) ∈ Fin)
6		dchrisum0flb.r	. . . . . . . 8 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
7		rpvmasum.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnnn0d 12473	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
9		rpvmasum.z	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
10		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝑍) = (Base‘𝑍)
11		rpvmasum.l	. . . . . . . . . . 11 ⊢ 𝐿 = (ℤRHom‘𝑍)
12	9, 10, 11	znzrhfo 20954	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
13		fof 6756	. . . . . . . . . 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 16551	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
17	15, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℤ)
18	14, 17	ffvelcdmd 7036	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝑃) ∈ (Base‘𝑍))
19	6, 18	ffvelcdmd 7036	. . . . . . 7 ⊢ (𝜑 → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
20		elfznn0 13534	. . . . . . 7 ⊢ (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℕ₀)
21		reexpcl 13984	. . . . . . 7 ⊢ (((𝑋‘(𝐿‘𝑃)) ∈ ℝ ∧ 𝑖 ∈ ℕ₀) → ((𝑋‘(𝐿‘𝑃))↑𝑖) ∈ ℝ)
22	19, 20, 21	syl2an 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ((𝑋‘(𝐿‘𝑃))↑𝑖) ∈ ℝ)
23	5, 22	fsumrecl 15619	. . . . 5 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) ∈ ℝ)
24	23	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) ∈ ℝ)
25		breq1 5108	. . . . . 6 ⊢ (1 = if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) → (1 ≤ 1 ↔ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ 1))
26		breq1 5108	. . . . . 6 ⊢ (0 = if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) → (0 ≤ 1 ↔ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ 1))
27		1le1 11783	. . . . . 6 ⊢ 1 ≤ 1
28		0le1 11678	. . . . . 6 ⊢ 0 ≤ 1
29	25, 26, 27, 28	keephyp 4557	. . . . 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 12805	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
33	31, 32	eleqtrdi 2848	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘0))
34		fzn0 13455	. . . . . . . . 9 ⊢ ((0...𝐴) ≠ ∅ ↔ 𝐴 ∈ (ℤ_≥‘0))
35	33, 34	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → (0...𝐴) ≠ ∅)
36		hashnncl 14266	. . . . . . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (♯‘(0...𝐴)) ∈ ℕ)
40	39	nnge1d 12201	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → 1 ≤ (♯‘(0...𝐴)))
41		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (𝑋‘(𝐿‘𝑃)) = 1)
42	41	oveq1d 7372	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → ((𝑋‘(𝐿‘𝑃))↑𝑖) = (1↑𝑖))
43		elfzelz 13441	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℤ)
44		1exp 13997	. . . . . . . . 9 ⊢ (𝑖 ∈ ℤ → (1↑𝑖) = 1)
45	43, 44	syl 17	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝐴) → (1↑𝑖) = 1)
46	42, 45	sylan9eq 2796	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ 𝑖 ∈ (0...𝐴)) → ((𝑋‘(𝐿‘𝑃))↑𝑖) = 1)
47	46	sumeq2dv 15588	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = Σ𝑖 ∈ (0...𝐴)1)
48		fzfid 13878	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (0...𝐴) ∈ Fin)
49		ax-1cn 11109	. . . . . . 7 ⊢ 1 ∈ ℂ
50		fsumconst 15675	. . . . . . 7 ⊢ (((0...𝐴) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑖 ∈ (0...𝐴)1 = ((♯‘(0...𝐴)) · 1))
51	48, 49, 50	sylancl 586	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)1 = ((♯‘(0...𝐴)) · 1))
52	39	nncnd 12169	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (♯‘(0...𝐴)) ∈ ℂ)
53	52	mulid1d 11172	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → ((♯‘(0...𝐴)) · 1) = (♯‘(0...𝐴)))
54	47, 51, 53	3eqtrd 2780	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = (♯‘(0...𝐴)))
55	40, 54	breqtrrd 5133	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → 1 ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
56	3, 4, 24, 30, 55	letrd 11312	. . 3 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
57		oveq1 7364	. . . . . . 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 5115	. . . . . 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 7364	. . . . . . 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 5115	. . . . . 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 11155	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
62	19	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
63		resubcl 11465	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (𝑋‘(𝐿‘𝑃)) ∈ ℝ) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℝ)
64	61, 62, 63	sylancr 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℝ)
65	64	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℝ)
66	65	leidd 11721	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − (𝑋‘(𝐿‘𝑃))) ≤ (1 − (𝑋‘(𝐿‘𝑃))))
67	64	recnd 11183	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℂ)
68	67	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℂ)
69	68	mulid2d 11173	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 · (1 − (𝑋‘(𝐿‘𝑃)))) = (1 − (𝑋‘(𝐿‘𝑃))))
70		nn0p1nn 12452	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ)
71	31, 70	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ)
72	71	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → (𝐴 + 1) ∈ ℕ)
73	72	0expd 14044	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → (0↑(𝐴 + 1)) = 0)
74		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → (𝑋‘(𝐿‘𝑃)) = 0)
75	74	oveq1d 7372	. . . . . . . . . 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 2786	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (𝑋‘(𝐿‘𝑃)))
77		neg1cn 12267	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℂ
78	31	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 ∈ ℕ₀)
79		expp1 13974	. . . . . . . . . . . . 13 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℕ₀) → (-1↑(𝐴 + 1)) = ((-1↑𝐴) · -1))
80	77, 78, 79	sylancr 587	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑(𝐴 + 1)) = ((-1↑𝐴) · -1))
81		prmnn 16550	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
82	15, 81	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑃 ∈ ℕ)
83	82, 31	nnexpcld 14148	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑃↑𝐴) ∈ ℕ)
84	83	nncnd 12169	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑃↑𝐴) ∈ ℂ)
85	84	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃↑𝐴) ∈ ℂ)
86	85	sqsqrtd 15324	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((√‘(𝑃↑𝐴))↑2) = (𝑃↑𝐴))
87	86	oveq2d 7373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt ((√‘(𝑃↑𝐴))↑2)) = (𝑃 pCnt (𝑃↑𝐴)))
88	15	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝑃 ∈ ℙ)
89		nnq 12887	. . . . . . . . . . . . . . . . . . 19 ⊢ ((√‘(𝑃↑𝐴)) ∈ ℕ → (√‘(𝑃↑𝐴)) ∈ ℚ)
90	89	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (√‘(𝑃↑𝐴)) ∈ ℚ)
91		nnne0 12187	. . . . . . . . . . . . . . . . . . 19 ⊢ ((√‘(𝑃↑𝐴)) ∈ ℕ → (√‘(𝑃↑𝐴)) ≠ 0)
92	91	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (√‘(𝑃↑𝐴)) ≠ 0)
93		2z 12535	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℤ
94	93	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 2 ∈ ℤ)
95		pcexp 16731	. . . . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt ((√‘(𝑃↑𝐴))↑2)) = (2 · (𝑃 pCnt (√‘(𝑃↑𝐴)))))
97	78	nn0zd 12525	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 ∈ ℤ)
98		pcid 16745	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴)
99	88, 97, 98	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴)
100	87, 96, 99	3eqtr3rd 2785	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 = (2 · (𝑃 pCnt (√‘(𝑃↑𝐴)))))
101	100	oveq2d 7373	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (√‘(𝑃↑𝐴)) ∈ ℕ)
104	88, 103	pccld 16722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt (√‘(𝑃↑𝐴))) ∈ ℕ₀)
105		2nn0 12430	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ₀
106	105	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 2 ∈ ℕ₀)
107	102, 104, 106	expmuld 14054	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑(2 · (𝑃 pCnt (√‘(𝑃↑𝐴))))) = ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))))
108		neg1sqe1 14100	. . . . . . . . . . . . . . . . 17 ⊢ (-1↑2) = 1
109	108	oveq1i 7367	. . . . . . . . . . . . . . . 16 ⊢ ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = (1↑(𝑃 pCnt (√‘(𝑃↑𝐴))))
110	104	nn0zd 12525	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt (√‘(𝑃↑𝐴))) ∈ ℤ)
111		1exp 13997	. . . . . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = 1)
114	101, 107, 113	3eqtrd 2780	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑𝐴) = 1)
115	114	oveq1d 7372	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑𝐴) · -1) = (1 · -1))
116	77	mulid2i 11160	. . . . . . . . . . . . 13 ⊢ (1 · -1) = -1
117	115, 116	eqtrdi 2792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑𝐴) · -1) = -1)
118	80, 117	eqtrd 2776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑(𝐴 + 1)) = -1)
119	118	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (-1↑(𝐴 + 1)) = -1)
120	19	recnd 11183	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
121	120	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
122	121	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
123	122	negnegd 11503	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → --(𝑋‘(𝐿‘𝑃)) = (𝑋‘(𝐿‘𝑃)))
124		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ≠ 1)
125	124	ad2antrr 724	. . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → 𝑋 ∈ 𝐷)
130		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
131	126, 9, 127, 10, 130, 128, 18	dchrn0 26598	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑋‘(𝐿‘𝑃)) ≠ 0 ↔ (𝐿‘𝑃) ∈ (Unit‘𝑍)))
132	131	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((𝑋‘(𝐿‘𝑃)) ≠ 0 ↔ (𝐿‘𝑃) ∈ (Unit‘𝑍)))
133	132	biimpa 477	. . . . . . . . . . . . . . . . . . 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 26608	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (abs‘(𝑋‘(𝐿‘𝑃))) = 1)
135		eqeq1 2740	. . . . . . . . . . . . . . . . . 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 2956	. . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
140	139	absord 15300	. . . . . . . . . . . . . . . 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 862	. . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → -(𝑋‘(𝐿‘𝑃)) = 1)
144	143	negeqd 11395	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → --(𝑋‘(𝐿‘𝑃)) = -1)
145	123, 144	eqtr3d 2778	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) = -1)
146	145	oveq1d 7372	. . . . . . . . . 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 2786	. . . . . . . . 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 3032	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (𝑋‘(𝐿‘𝑃)))
149	148	oveq2d 7373	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = (1 − (𝑋‘(𝐿‘𝑃))))
150	66, 69, 149	3brtr4d 5137	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
151	67	mul02d 11353	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (0 · (1 − (𝑋‘(𝐿‘𝑃)))) = 0)
152		peano2nn0 12453	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ₀)
153	31, 152	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ₀)
154	19, 153	reexpcld 14068	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℝ)
155	154	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℝ)
156	155	recnd 11183	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℂ)
157	156	abscld 15321	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ∈ ℝ)
158		1red 11156	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 1 ∈ ℝ)
159	155	leabsd 15299	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ≤ (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
160	153	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝐴 + 1) ∈ ℕ₀)
161	121, 160	absexpd 15337	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = ((abs‘(𝑋‘(𝐿‘𝑃)))↑(𝐴 + 1)))
162	121	abscld 15321	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘(𝑋‘(𝐿‘𝑃))) ∈ ℝ)
163	121	absge0d 15329	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 ≤ (abs‘(𝑋‘(𝐿‘𝑃))))
164	126, 127, 9, 10, 128, 18	dchrabs2 26610	. . . . . . . . . . . . 13 ⊢ (𝜑 → (abs‘(𝑋‘(𝐿‘𝑃))) ≤ 1)
165	164	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘(𝑋‘(𝐿‘𝑃))) ≤ 1)
166		exple1 14081	. . . . . . . . . . . 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 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((abs‘(𝑋‘(𝐿‘𝑃)))↑(𝐴 + 1)) ≤ 1)
168	161, 167	eqbrtrd 5127	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ≤ 1)
169	155, 157, 158, 159, 168	letrd 11312	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ≤ 1)
170		subge0 11668	. . . . . . . . . 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 587	. . . . . . . . 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 5127	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (0 · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
174	173	adantr 481	. . . . . 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 4524	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
176		0re 11157	. . . . . . . 8 ⊢ 0 ∈ ℝ
177	61, 176	ifcli 4533	. . . . . . 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 11465	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℝ) → (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ∈ ℝ)
180	61, 155, 179	sylancr 587	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ∈ ℝ)
181	62	leabsd 15299	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ≤ (abs‘(𝑋‘(𝐿‘𝑃))))
182	62, 162, 158, 181, 165	letrd 11312	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ≤ 1)
183	124	necomd 2999	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 1 ≠ (𝑋‘(𝐿‘𝑃)))
184	62, 158, 182, 183	leneltd 11309	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) < 1)
185		posdif 11648	. . . . . . . 8 ⊢ (((𝑋‘(𝐿‘𝑃)) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑋‘(𝐿‘𝑃)) < 1 ↔ 0 < (1 − (𝑋‘(𝐿‘𝑃)))))
186	62, 61, 185	sylancl 586	. . . . . . 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 12035	. . . . . 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 1374	. . . . 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 12525	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
192		fzval3 13641	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (0...𝐴) = (0..^(𝐴 + 1)))
193	191, 192	syl 17	. . . . . . 7 ⊢ (𝜑 → (0...𝐴) = (0..^(𝐴 + 1)))
194	193	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (0...𝐴) = (0..^(𝐴 + 1)))
195	194	sumeq1d 15586	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = Σ𝑖 ∈ (0..^(𝐴 + 1))((𝑋‘(𝐿‘𝑃))↑𝑖))
196		0nn0 12428	. . . . . . 7 ⊢ 0 ∈ ℕ₀
197	196	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 ∈ ℕ₀)
198	153, 32	eleqtrdi 2848	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ (ℤ_≥‘0))
199	198	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝐴 + 1) ∈ (ℤ_≥‘0))
200	121, 124, 197, 199	geoserg 15751	. . . . 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 14045	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑0) = 1)
202	201	oveq1d 7372	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (((𝑋‘(𝐿‘𝑃))↑0) − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
203	202	oveq1d 7372	. . . . 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 2780	. . . 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 5133	. . 3 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
206	56, 205	pm2.61dane 3032	. 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 26853	. . . 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 6847	. . . 4 ⊢ (𝑘 = (𝑃↑𝑖) → (𝑋‘(𝐿‘𝑘)) = (𝑋‘(𝐿‘(𝑃↑𝑖))))
212		eqid 2736	. . . . . 6 ⊢ (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏)) = (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏))
213	212	dvdsppwf1o 26535	. . . . 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 584	. . . 4 ⊢ (𝜑 → (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏)):(0...𝐴)–1-1-onto→{𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)})
215		oveq2 7365	. . . . . 6 ⊢ (𝑏 = 𝑖 → (𝑃↑𝑏) = (𝑃↑𝑖))
216		ovex 7390	. . . . . 6 ⊢ (𝑃↑𝑏) ∈ V
217	215, 212, 216	fvmpt3i 6953	. . . . 5 ⊢ (𝑖 ∈ (0...𝐴) → ((𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏))‘𝑖) = (𝑃↑𝑖))
218	217	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ((𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏))‘𝑖) = (𝑃↑𝑖))
219	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → 𝑋:(Base‘𝑍)⟶ℝ)
220		elrabi 3639	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} → 𝑘 ∈ ℕ)
221	220	nnzd 12526	. . . . . . 7 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} → 𝑘 ∈ ℤ)
222		ffvelcdm 7032	. . . . . . 7 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑘 ∈ ℤ) → (𝐿‘𝑘) ∈ (Base‘𝑍))
223	14, 221, 222	syl2an 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → (𝐿‘𝑘) ∈ (Base‘𝑍))
224	219, 223	ffvelcdmd 7036	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → (𝑋‘(𝐿‘𝑘)) ∈ ℝ)
225	224	recnd 11183	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
226	211, 5, 214, 218, 225	fsumf1o 15608	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} (𝑋‘(𝐿‘𝑘)) = Σ𝑖 ∈ (0...𝐴)(𝑋‘(𝐿‘(𝑃↑𝑖))))
227		zsubrg 20850	. . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
228		eqid 2736	. . . . . . . . . . . 12 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
229	228	subrgsubm 20235	. . . . . . . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑖 ∈ ℕ₀)
232	17	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑃 ∈ ℤ)
233		eqid 2736	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
234		zringmpg 20892	. . . . . . . . . . . 12 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = (mulGrp‘ℤ_ring)
235	234	eqcomi 2745	. . . . . . . . . . 11 ⊢ (mulGrp‘ℤ_ring) = ((mulGrp‘ℂ_fld) ↾_s ℤ)
236		eqid 2736	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘ℤ_ring)) = (._g‘(mulGrp‘ℤ_ring))
237	233, 235, 236	submmulg 18920	. . . . . . . . . 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 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℂ_fld))𝑃) = (𝑖(._g‘(mulGrp‘ℤ_ring))𝑃))
239	82	nncnd 12169	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℂ)
240		cnfldexp 20830	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℂ ∧ 𝑖 ∈ ℕ₀) → (𝑖(._g‘(mulGrp‘ℂ_fld))𝑃) = (𝑃↑𝑖))
241	239, 20, 240	syl2an 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℂ_fld))𝑃) = (𝑃↑𝑖))
242	238, 241	eqtr3d 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℤ_ring))𝑃) = (𝑃↑𝑖))
243	242	fveq2d 6846	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑖(._g‘(mulGrp‘ℤ_ring))𝑃)) = (𝐿‘(𝑃↑𝑖)))
244	9	zncrng 20951	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
245		crngring 19976	. . . . . . . . . . 11 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
246	8, 244, 245	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Ring)
247	11	zrhrhm 20912	. . . . . . . . . 10 ⊢ (𝑍 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑍))
248		eqid 2736	. . . . . . . . . . 11 ⊢ (mulGrp‘ℤ_ring) = (mulGrp‘ℤ_ring)
249		eqid 2736	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
250	248, 249	rhmmhm 20153	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝐿 ∈ ((mulGrp‘ℤ_ring) MndHom (mulGrp‘𝑍)))
253		zringbas 20875	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤ_ring)
254	248, 253	mgpbas 19902	. . . . . . . . 9 ⊢ ℤ = (Base‘(mulGrp‘ℤ_ring))
255		eqid 2736	. . . . . . . . 9 ⊢ (._g‘(mulGrp‘𝑍)) = (._g‘(mulGrp‘𝑍))
256	254, 236, 255	mhmmulg 18917	. . . . . . . 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 1371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑖(._g‘(mulGrp‘ℤ_ring))𝑃)) = (𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃)))
258	243, 257	eqtr3d 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑃↑𝑖)) = (𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃)))
259	258	fveq2d 6846	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝐿‘(𝑃↑𝑖))) = (𝑋‘(𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃))))
260	126, 9, 127	dchrmhm 26589	. . . . . . . 8 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
261	260, 128	sselid 3942	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
262	261	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
263	18	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘𝑃) ∈ (Base‘𝑍))
264	249, 10	mgpbas 19902	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
265	264, 255, 233	mhmmulg 18917	. . . . . 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 1371	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃))) = (𝑖(._g‘(mulGrp‘ℂ_fld))(𝑋‘(𝐿‘𝑃))))
267		cnfldexp 20830	. . . . . 6 ⊢ (((𝑋‘(𝐿‘𝑃)) ∈ ℂ ∧ 𝑖 ∈ ℕ₀) → (𝑖(._g‘(mulGrp‘ℂ_fld))(𝑋‘(𝐿‘𝑃))) = ((𝑋‘(𝐿‘𝑃))↑𝑖))
268	120, 20, 267	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℂ_fld))(𝑋‘(𝐿‘𝑃))) = ((𝑋‘(𝐿‘𝑃))↑𝑖))
269	259, 266, 268	3eqtrd 2780	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝐿‘(𝑃↑𝑖))) = ((𝑋‘(𝐿‘𝑃))↑𝑖))
270	269	sumeq2dv 15588	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐴)(𝑋‘(𝐿‘(𝑃↑𝑖))) = Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
271	210, 226, 270	3eqtrd 2780	. 2 ⊢ (𝜑 → (𝐹‘(𝑃↑𝐴)) = Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
272	206, 271	breqtrrd 5133	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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 {crab 3407 ∅c0 4282 ifcif 4486 class class class wbr 5105 ↦ cmpt 5188 ⟶wf 6492 –onto→wfo 6494 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 < clt 11189 ≤ cle 11190 − cmin 11385 -cneg 11386 / cdiv 11812 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 ℚcq 12873 ...cfz 13424 ..^cfzo 13567 ↑cexp 13967 ♯chash 14230 √csqrt 15118 abscabs 15119 Σcsu 15570 ∥ cdvds 16136 ℙcprime 16547 pCnt cpc 16708 Basecbs 17083 ↾_s cress 17112 0_gc0g 17321 MndHom cmhm 18599 SubMndcsubmnd 18600 ._gcmg 18872 mulGrpcmgp 19896 Ringcrg 19964 CRingccrg 19965 Unitcui 20068 RingHom crh 20143 SubRingcsubrg 20218 ℂ_fldccnfld 20796 ℤ_ringczring 20869 ℤRHomczrh 20900 ℤ/nℤczn 20903 DChrcdchr 26580

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-ec 8650 df-qs 8654 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-dvds 16137 df-gcd 16375 df-prm 16548 df-pc 16709 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-qus 17391 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-nsg 18926 df-eqg 18927 df-ghm 19006 df-cntz 19097 df-od 19310 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-rnghom 20146 df-drng 20187 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-sra 20633 df-rgmod 20634 df-lidl 20635 df-rsp 20636 df-2idl 20702 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-zring 20870 df-zrh 20904 df-zn 20907 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-log 25912 df-cxp 25913 df-dchr 26581

This theorem is referenced by: dchrisum0flblem2 26857 dchrisum0flb 26858

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