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Theorem dchrisum0flblem1 27471

Description: Lemma for dchrisum0flb 27473. 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 11236	. . . . 5 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 1 ∈ ℝ)
2		0red 11238	. . . . 5 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ ¬ (√‘(𝑃↑𝐴)) ∈ ℕ) → 0 ∈ ℝ)
3	1, 2	ifclda 4536	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ∈ ℝ)
4		1red 11236	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → 1 ∈ ℝ)
5		fzfid 13991	. . . . . 6 ⊢ (𝜑 → (0...𝐴) ∈ Fin)
6		dchrisum0flb.r	. . . . . . . 8 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
7		rpvmasum.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnnn0d 12562	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
9		rpvmasum.z	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
10		eqid 2735	. . . . . . . . . . 11 ⊢ (Base‘𝑍) = (Base‘𝑍)
11		rpvmasum.l	. . . . . . . . . . 11 ⊢ 𝐿 = (ℤRHom‘𝑍)
12	9, 10, 11	znzrhfo 21508	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
13		fof 6790	. . . . . . . . . 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 16694	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
17	15, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℤ)
18	14, 17	ffvelcdmd 7075	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝑃) ∈ (Base‘𝑍))
19	6, 18	ffvelcdmd 7075	. . . . . . 7 ⊢ (𝜑 → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
20		elfznn0 13637	. . . . . . 7 ⊢ (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℕ₀)
21		reexpcl 14096	. . . . . . 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 15750	. . . . 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 5122	. . . . . 6 ⊢ (1 = if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) → (1 ≤ 1 ↔ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ 1))
26		breq1 5122	. . . . . 6 ⊢ (0 = if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) → (0 ≤ 1 ↔ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ 1))
27		1le1 11865	. . . . . 6 ⊢ 1 ≤ 1
28		0le1 11760	. . . . . 6 ⊢ 0 ≤ 1
29	25, 26, 27, 28	keephyp 4572	. . . . 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 12894	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
33	31, 32	eleqtrdi 2844	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘0))
34		fzn0 13555	. . . . . . . . 9 ⊢ ((0...𝐴) ≠ ∅ ↔ 𝐴 ∈ (ℤ_≥‘0))
35	33, 34	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → (0...𝐴) ≠ ∅)
36		hashnncl 14384	. . . . . . . . 9 ⊢ ((0...𝐴) ∈ Fin → ((♯‘(0...𝐴)) ∈ ℕ ↔ (0...𝐴) ≠ ∅))
37	5, 36	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(0...𝐴)) ∈ ℕ ↔ (0...𝐴) ≠ ∅))
38	35, 37	mpbird 257	. . . . . . 7 ⊢ (𝜑 → (♯‘(0...𝐴)) ∈ ℕ)
39	38	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (♯‘(0...𝐴)) ∈ ℕ)
40	39	nnge1d 12288	. . . . 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 7420	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → ((𝑋‘(𝐿‘𝑃))↑𝑖) = (1↑𝑖))
43		elfzelz 13541	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℤ)
44		1exp 14109	. . . . . . . . 9 ⊢ (𝑖 ∈ ℤ → (1↑𝑖) = 1)
45	43, 44	syl 17	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝐴) → (1↑𝑖) = 1)
46	42, 45	sylan9eq 2790	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) ∧ 𝑖 ∈ (0...𝐴)) → ((𝑋‘(𝐿‘𝑃))↑𝑖) = 1)
47	46	sumeq2dv 15718	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = Σ𝑖 ∈ (0...𝐴)1)
48		fzfid 13991	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (0...𝐴) ∈ Fin)
49		ax-1cn 11187	. . . . . . 7 ⊢ 1 ∈ ℂ
50		fsumconst 15806	. . . . . . 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 12256	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → (♯‘(0...𝐴)) ∈ ℂ)
53	52	mulridd 11252	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → ((♯‘(0...𝐴)) · 1) = (♯‘(0...𝐴)))
54	47, 51, 53	3eqtrd 2774	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = (♯‘(0...𝐴)))
55	40, 54	breqtrrd 5147	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → 1 ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
56	3, 4, 24, 30, 55	letrd 11392	. . 3 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) = 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
57		oveq1 7412	. . . . . . 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 5129	. . . . . 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 7412	. . . . . . 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 5129	. . . . . 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 11235	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
62	19	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
63		resubcl 11547	. . . . . . . . . 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 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℝ)
66	65	leidd 11803	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − (𝑋‘(𝐿‘𝑃))) ≤ (1 − (𝑋‘(𝐿‘𝑃))))
67	64	recnd 11263	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℂ)
68	67	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − (𝑋‘(𝐿‘𝑃))) ∈ ℂ)
69	68	mullidd 11253	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 · (1 − (𝑋‘(𝐿‘𝑃)))) = (1 − (𝑋‘(𝐿‘𝑃))))
70		nn0p1nn 12540	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ)
71	31, 70	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ)
72	71	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → (𝐴 + 1) ∈ ℕ)
73	72	0expd 14157	. . . . . . . . . 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 7420	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (0↑(𝐴 + 1)))
76	73, 75, 74	3eqtr4d 2780	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) = 0) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (𝑋‘(𝐿‘𝑃)))
77		neg1cn 12354	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℂ
78	31	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 ∈ ℕ₀)
79		expp1 14086	. . . . . . . . . . . . 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 16693	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
82	15, 81	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑃 ∈ ℕ)
83	82, 31	nnexpcld 14263	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑃↑𝐴) ∈ ℕ)
84	83	nncnd 12256	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑃↑𝐴) ∈ ℂ)
85	84	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃↑𝐴) ∈ ℂ)
86	85	sqsqrtd 15458	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((√‘(𝑃↑𝐴))↑2) = (𝑃↑𝐴))
87	86	oveq2d 7421	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt ((√‘(𝑃↑𝐴))↑2)) = (𝑃 pCnt (𝑃↑𝐴)))
88	15	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝑃 ∈ ℙ)
89		nnq 12978	. . . . . . . . . . . . . . . . . . 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 12274	. . . . . . . . . . . . . . . . . . 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 12624	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℤ
94	93	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 2 ∈ ℤ)
95		pcexp 16879	. . . . . . . . . . . . . . . . . 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 1377	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt ((√‘(𝑃↑𝐴))↑2)) = (2 · (𝑃 pCnt (√‘(𝑃↑𝐴)))))
97	78	nn0zd 12614	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 ∈ ℤ)
98		pcid 16893	. . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 𝐴 = (2 · (𝑃 pCnt (√‘(𝑃↑𝐴)))))
101	100	oveq2d 7421	. . . . . . . . . . . . . . 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 16870	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt (√‘(𝑃↑𝐴))) ∈ ℕ₀)
105		2nn0 12518	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ₀
106	105	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → 2 ∈ ℕ₀)
107	102, 104, 106	expmuld 14167	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑(2 · (𝑃 pCnt (√‘(𝑃↑𝐴))))) = ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))))
108		neg1sqe1 14214	. . . . . . . . . . . . . . . . 17 ⊢ (-1↑2) = 1
109	108	oveq1i 7415	. . . . . . . . . . . . . . . 16 ⊢ ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = (1↑(𝑃 pCnt (√‘(𝑃↑𝐴))))
110	104	nn0zd 12614	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (𝑃 pCnt (√‘(𝑃↑𝐴))) ∈ ℤ)
111		1exp 14109	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 pCnt (√‘(𝑃↑𝐴))) ∈ ℤ → (1↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = 1)
112	110, 111	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = 1)
113	109, 112	eqtrid 2782	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑2)↑(𝑃 pCnt (√‘(𝑃↑𝐴)))) = 1)
114	101, 107, 113	3eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑𝐴) = 1)
115	114	oveq1d 7420	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑𝐴) · -1) = (1 · -1))
116	77	mullidi 11240	. . . . . . . . . . . . 13 ⊢ (1 · -1) = -1
117	115, 116	eqtrdi 2786	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((-1↑𝐴) · -1) = -1)
118	80, 117	eqtrd 2770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (-1↑(𝐴 + 1)) = -1)
119	118	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (-1↑(𝐴 + 1)) = -1)
120	19	recnd 11263	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
121	120	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
122	121	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) ∈ ℂ)
123	122	negnegd 11585	. . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → 𝑋 ∈ 𝐷)
130		eqid 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
131	126, 9, 127, 10, 130, 128, 18	dchrn0 27213	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑋‘(𝐿‘𝑃)) ≠ 0 ↔ (𝐿‘𝑃) ∈ (Unit‘𝑍)))
132	131	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 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 27223	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (abs‘(𝑋‘(𝐿‘𝑃))) = 1)
135		eqeq1 2739	. . . . . . . . . . . . . . . . . 18 ⊢ ((abs‘(𝑋‘(𝐿‘𝑃))) = (𝑋‘(𝐿‘𝑃)) → ((abs‘(𝑋‘(𝐿‘𝑃))) = 1 ↔ (𝑋‘(𝐿‘𝑃)) = 1))
136	134, 135	syl5ibcom 245	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → ((abs‘(𝑋‘(𝐿‘𝑃))) = (𝑋‘(𝐿‘𝑃)) → (𝑋‘(𝐿‘𝑃)) = 1))
137	136	necon3ad 2945	. . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) ∈ ℝ)
140	139	absord 15434	. . . . . . . . . . . . . . . 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 864	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (¬ (abs‘(𝑋‘(𝐿‘𝑃))) = (𝑋‘(𝐿‘𝑃)) → (abs‘(𝑋‘(𝐿‘𝑃))) = -(𝑋‘(𝐿‘𝑃))))
142	138, 141	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (abs‘(𝑋‘(𝐿‘𝑃))) = -(𝑋‘(𝐿‘𝑃)))
143	142, 134	eqtr3d 2772	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → -(𝑋‘(𝐿‘𝑃)) = 1)
144	143	negeqd 11476	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → --(𝑋‘(𝐿‘𝑃)) = -1)
145	123, 144	eqtr3d 2772	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → (𝑋‘(𝐿‘𝑃)) = -1)
146	145	oveq1d 7420	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (-1↑(𝐴 + 1)))
147	119, 146, 145	3eqtr4d 2780	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) ∧ (𝑋‘(𝐿‘𝑃)) ≠ 0) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (𝑋‘(𝐿‘𝑃)))
148	76, 147	pm2.61dane 3019	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) = (𝑋‘(𝐿‘𝑃)))
149	148	oveq2d 7421	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = (1 − (𝑋‘(𝐿‘𝑃))))
150	66, 69, 149	3brtr4d 5151	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) ∧ (√‘(𝑃↑𝐴)) ∈ ℕ) → (1 · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
151	67	mul02d 11433	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (0 · (1 − (𝑋‘(𝐿‘𝑃)))) = 0)
152		peano2nn0 12541	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ₀)
153	31, 152	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ₀)
154	19, 153	reexpcld 14181	. . . . . . . . . . 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 11263	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ∈ ℂ)
157	156	abscld 15455	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ∈ ℝ)
158		1red 11236	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 1 ∈ ℝ)
159	155	leabsd 15433	. . . . . . . . . 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 15471	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = ((abs‘(𝑋‘(𝐿‘𝑃)))↑(𝐴 + 1)))
162	121	abscld 15455	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘(𝑋‘(𝐿‘𝑃))) ∈ ℝ)
163	121	absge0d 15463	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 ≤ (abs‘(𝑋‘(𝐿‘𝑃))))
164	126, 127, 9, 10, 128, 18	dchrabs2 27225	. . . . . . . . . . . . 13 ⊢ (𝜑 → (abs‘(𝑋‘(𝐿‘𝑃))) ≤ 1)
165	164	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘(𝑋‘(𝐿‘𝑃))) ≤ 1)
166		exple1 14195	. . . . . . . . . . . 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 1375	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((abs‘(𝑋‘(𝐿‘𝑃)))↑(𝐴 + 1)) ≤ 1)
168	161, 167	eqbrtrd 5141	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (abs‘((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ≤ 1)
169	155, 157, 158, 159, 168	letrd 11392	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1)) ≤ 1)
170		subge0 11750	. . . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
173	151, 172	eqbrtrd 5141	. . . . . . 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 4539	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
176		0re 11237	. . . . . . . 8 ⊢ 0 ∈ ℝ
177	61, 176	ifcli 4548	. . . . . . 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 11547	. . . . . . 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 15433	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ≤ (abs‘(𝑋‘(𝐿‘𝑃))))
182	62, 162, 158, 181, 165	letrd 11392	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) ≤ 1)
183	124	necomd 2987	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 1 ≠ (𝑋‘(𝐿‘𝑃)))
184	62, 158, 182, 183	leneltd 11389	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (𝑋‘(𝐿‘𝑃)) < 1)
185		posdif 11730	. . . . . . . 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 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 < (1 − (𝑋‘(𝐿‘𝑃))))
188		lemuldiv 12122	. . . . . 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 1376	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) · (1 − (𝑋‘(𝐿‘𝑃)))) ≤ (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) ↔ if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ ((1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) / (1 − (𝑋‘(𝐿‘𝑃))))))
190	175, 189	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ ((1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) / (1 − (𝑋‘(𝐿‘𝑃)))))
191	31	nn0zd 12614	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
192		fzval3 13750	. . . . . . . 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 15716	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = Σ𝑖 ∈ (0..^(𝐴 + 1))((𝑋‘(𝐿‘𝑃))↑𝑖))
196		0nn0 12516	. . . . . . 7 ⊢ 0 ∈ ℕ₀
197	196	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → 0 ∈ ℕ₀)
198	153, 32	eleqtrdi 2844	. . . . . . 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 15882	. . . . 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 14158	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((𝑋‘(𝐿‘𝑃))↑0) = 1)
202	201	oveq1d 7420	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → (((𝑋‘(𝐿‘𝑃))↑0) − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) = (1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))))
203	202	oveq1d 7420	. . . . 5 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → ((((𝑋‘(𝐿‘𝑃))↑0) − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) / (1 − (𝑋‘(𝐿‘𝑃)))) = ((1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) / (1 − (𝑋‘(𝐿‘𝑃)))))
204	195, 200, 203	3eqtrd 2774	. . . 4 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖) = ((1 − ((𝑋‘(𝐿‘𝑃))↑(𝐴 + 1))) / (1 − (𝑋‘(𝐿‘𝑃)))))
205	190, 204	breqtrrd 5147	. . 3 ⊢ ((𝜑 ∧ (𝑋‘(𝐿‘𝑃)) ≠ 1) → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
206	56, 205	pm2.61dane 3019	. 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 27468	. . . 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 6881	. . . 4 ⊢ (𝑘 = (𝑃↑𝑖) → (𝑋‘(𝐿‘𝑘)) = (𝑋‘(𝐿‘(𝑃↑𝑖))))
212		eqid 2735	. . . . . 6 ⊢ (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏)) = (𝑏 ∈ (0...𝐴) ↦ (𝑃↑𝑏))
213	212	dvdsppwf1o 27148	. . . . 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 7413	. . . . . 6 ⊢ (𝑏 = 𝑖 → (𝑃↑𝑏) = (𝑃↑𝑖))
216		ovex 7438	. . . . . 6 ⊢ (𝑃↑𝑏) ∈ V
217	215, 212, 216	fvmpt3i 6991	. . . . 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 3666	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} → 𝑘 ∈ ℕ)
221	220	nnzd 12615	. . . . . . 7 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} → 𝑘 ∈ ℤ)
222		ffvelcdm 7071	. . . . . . 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 7075	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → (𝑋‘(𝐿‘𝑘)) ∈ ℝ)
225	224	recnd 11263	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)}) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
226	211, 5, 214, 218, 225	fsumf1o 15739	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝑃↑𝐴)} (𝑋‘(𝐿‘𝑘)) = Σ𝑖 ∈ (0...𝐴)(𝑋‘(𝐿‘(𝑃↑𝑖))))
227		zsubrg 21388	. . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
228		eqid 2735	. . . . . . . . . . . 12 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
229	228	subrgsubm 20545	. . . . . . . . . . 11 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
230	227, 229	mp1i 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
231	20	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑖 ∈ ℕ₀)
232	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑃 ∈ ℤ)
233		eqid 2735	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
234		zringmpg 21432	. . . . . . . . . . . 12 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = (mulGrp‘ℤ_ring)
235	234	eqcomi 2744	. . . . . . . . . . 11 ⊢ (mulGrp‘ℤ_ring) = ((mulGrp‘ℂ_fld) ↾_s ℤ)
236		eqid 2735	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘ℤ_ring)) = (._g‘(mulGrp‘ℤ_ring))
237	233, 235, 236	submmulg 19101	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℂ_fld))𝑃) = (𝑖(._g‘(mulGrp‘ℤ_ring))𝑃))
239	82	nncnd 12256	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℂ)
240		cnfldexp 21367	. . . . . . . . . 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 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑖(._g‘(mulGrp‘ℤ_ring))𝑃) = (𝑃↑𝑖))
243	242	fveq2d 6880	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑖(._g‘(mulGrp‘ℤ_ring))𝑃)) = (𝐿‘(𝑃↑𝑖)))
244	9	zncrng 21505	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
245		crngring 20205	. . . . . . . . . . 11 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
246	8, 244, 245	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Ring)
247	11	zrhrhm 21472	. . . . . . . . . 10 ⊢ (𝑍 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑍))
248		eqid 2735	. . . . . . . . . . 11 ⊢ (mulGrp‘ℤ_ring) = (mulGrp‘ℤ_ring)
249		eqid 2735	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
250	248, 249	rhmmhm 20439	. . . . . . . . . 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 21414	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤ_ring)
254	248, 253	mgpbas 20105	. . . . . . . . 9 ⊢ ℤ = (Base‘(mulGrp‘ℤ_ring))
255		eqid 2735	. . . . . . . . 9 ⊢ (._g‘(mulGrp‘𝑍)) = (._g‘(mulGrp‘𝑍))
256	254, 236, 255	mhmmulg 19098	. . . . . . . 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 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑖(._g‘(mulGrp‘ℤ_ring))𝑃)) = (𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃)))
258	243, 257	eqtr3d 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝐿‘(𝑃↑𝑖)) = (𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃)))
259	258	fveq2d 6880	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝐿‘(𝑃↑𝑖))) = (𝑋‘(𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃))))
260	126, 9, 127	dchrmhm 27204	. . . . . . . 8 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
261	260, 128	sselid 3956	. . . . . . 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 20105	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
265	264, 255, 233	mhmmulg 19098	. . . . . 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 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝑖(._g‘(mulGrp‘𝑍))(𝐿‘𝑃))) = (𝑖(._g‘(mulGrp‘ℂ_fld))(𝑋‘(𝐿‘𝑃))))
267		cnfldexp 21367	. . . . . 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 2774	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑋‘(𝐿‘(𝑃↑𝑖))) = ((𝑋‘(𝐿‘𝑃))↑𝑖))
270	269	sumeq2dv 15718	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐴)(𝑋‘(𝐿‘(𝑃↑𝑖))) = Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
271	210, 226, 270	3eqtrd 2774	. 2 ⊢ (𝜑 → (𝐹‘(𝑃↑𝐴)) = Σ𝑖 ∈ (0...𝐴)((𝑋‘(𝐿‘𝑃))↑𝑖))
272	206, 271	breqtrrd 5147	1 ⊢ (𝜑 → if((√‘(𝑃↑𝐴)) ∈ ℕ, 1, 0) ≤ (𝐹‘(𝑃↑𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 {crab 3415 ∅c0 4308 ifcif 4500 class class class wbr 5119 ↦ cmpt 5201 ⟶wf 6527 –onto→wfo 6529 –1-1-onto→wf1o 6530 ‘cfv 6531 (class class class)co 7405 Fincfn 8959 ℂcc 11127 ℝcr 11128 0cc0 11129 1c1 11130 + caddc 11132 · cmul 11134 < clt 11269 ≤ cle 11270 − cmin 11466 -cneg 11467 / cdiv 11894 ℕcn 12240 2c2 12295 ℕ₀cn0 12501 ℤcz 12588 ℤ_≥cuz 12852 ℚcq 12964 ...cfz 13524 ..^cfzo 13671 ↑cexp 14079 ♯chash 14348 √csqrt 15252 abscabs 15253 Σcsu 15702 ∥ cdvds 16272 ℙcprime 16690 pCnt cpc 16856 Basecbs 17228 ↾_s cress 17251 0_gc0g 17453 MndHom cmhm 18759 SubMndcsubmnd 18760 ._gcmg 19050 mulGrpcmgp 20100 Ringcrg 20193 CRingccrg 20194 Unitcui 20315 RingHom crh 20429 SubRingcsubrg 20529 ℂ_fldccnfld 21315 ℤ_ringczring 21407 ℤRHomczrh 21460 ℤ/nℤczn 21463 DChrcdchr 27195

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208 ax-mulf 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-disj 5087 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8719 df-ec 8721 df-qs 8725 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-card 9953 df-acn 9956 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-shft 15086 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-limsup 15487 df-clim 15504 df-rlim 15505 df-sum 15703 df-ef 16083 df-sin 16085 df-cos 16086 df-pi 16088 df-dvds 16273 df-gcd 16514 df-prm 16691 df-pc 16857 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-qus 17523 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-mulg 19051 df-subg 19106 df-nsg 19107 df-eqg 19108 df-ghm 19196 df-cntz 19300 df-od 19509 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-cring 20196 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-dvr 20361 df-rhm 20432 df-subrng 20506 df-subrg 20530 df-drng 20691 df-lmod 20819 df-lss 20889 df-lsp 20929 df-sra 21131 df-rgmod 21132 df-lidl 21169 df-rsp 21170 df-2idl 21211 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-zring 21408 df-zrh 21464 df-zn 21467 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-lp 23074 df-perf 23075 df-cn 23165 df-cnp 23166 df-haus 23253 df-tx 23500 df-hmeo 23693 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-xms 24259 df-ms 24260 df-tms 24261 df-cncf 24822 df-limc 25819 df-dv 25820 df-log 26517 df-cxp 26518 df-dchr 27196

This theorem is referenced by: dchrisum0flblem2 27472 dchrisum0flb 27473

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