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Theorem dchrisum0re 26089

Description: Suppose 𝑋 is a non-principal Dirichlet character with Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 = 0. Then 𝑋 is a real character. Part of Lemma 9.4.4 of [Shapiro], p. 382. (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)
rpvmasum2.w	⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
dchrisum0.b	⊢ (𝜑 → 𝑋 ∈ 𝑊)

**Assertion**
Ref	Expression
dchrisum0re	⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)

Distinct variable groups: 𝑦,𝑚, 1 𝑚,𝑁,𝑦 𝜑,𝑚 𝑚,𝑍,𝑦 𝐷,𝑚,𝑦 𝑚,𝐿,𝑦 𝑚,𝑋,𝑦 Allowed substitution hints: 𝜑(𝑦) 𝐺(𝑦,𝑚) 𝑊(𝑦,𝑚)

Proof of Theorem dchrisum0re Dummy variables 𝑘 𝑛 𝑥 𝑓 𝑐 𝑡 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpvmasum2.g	. . . 4 ⊢ 𝐺 = (DChr‘𝑁)
2		rpvmasum.z	. . . 4 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
3		rpvmasum2.d	. . . 4 ⊢ 𝐷 = (Base‘𝐺)
4		eqid 2821	. . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍)
5		rpvmasum2.w	. . . . . . 7 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
6	5	ssrab3 4057	. . . . . 6 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
7		dchrisum0.b	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
8	6, 7	sseldi 3965	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
9	8	eldifad 3948	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
10	1, 2, 3, 4, 9	dchrf 25818	. . 3 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
11	10	ffnd 6515	. 2 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍))
12	10	ffvelrnda 6851	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ)
13		fvco3 6760	. . . . . 6 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑥 ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘𝑥) = (∗‘(𝑋‘𝑥)))
14	10, 13	sylan 582	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘𝑥) = (∗‘(𝑋‘𝑥)))
15		logno1 25219	. . . . . . . 8 ⊢ ¬ (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1)
16		1red 10642	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) → 1 ∈ ℝ)
17		rpvmasum.l	. . . . . . . . . . . . 13 ⊢ 𝐿 = (ℤRHom‘𝑍)
18		rpvmasum.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
19		rpvmasum2.1	. . . . . . . . . . . . 13 ⊢ 1 = (0_g‘𝐺)
20		eqid 2821	. . . . . . . . . . . . 13 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
21	18	nnnn0d 11956	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
22	2	zncrng 20691	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
23	21, 22	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑍 ∈ CRing)
24		crngring 19308	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
25	23, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ Ring)
26		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
27	20, 26	1unit 19408	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ Ring → (1_r‘𝑍) ∈ (Unit‘𝑍))
28	25, 27	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1_r‘𝑍) ∈ (Unit‘𝑍))
29		eqid 2821	. . . . . . . . . . . . 13 ⊢ (^◡𝐿 “ {(1_r‘𝑍)}) = (^◡𝐿 “ {(1_r‘𝑍)})
30		eqidd 2822	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑊) → (1_r‘𝑍) = (1_r‘𝑍))
31	2, 17, 18, 1, 3, 19, 5, 20, 28, 29, 30	rpvmasum2 26088	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))) ∈ 𝑂(1))
32	31	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) → (𝑥 ∈ ℝ⁺ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))) ∈ 𝑂(1))
33	18	phicld 16109	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
34	33	nnnn0d 11956	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ₀)
35	34	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (ϕ‘𝑁) ∈ ℕ₀)
36	35	nn0red 11957	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (ϕ‘𝑁) ∈ ℝ)
37		fzfid 13342	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
38		inss1 4205	. . . . . . . . . . . . . . . . 17 ⊢ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) ⊆ (1...(⌊‘𝑥))
39		ssfi 8738	. . . . . . . . . . . . . . . . 17 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) ⊆ (1...(⌊‘𝑥))) → ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) ∈ Fin)
40	37, 38, 39	sylancl 588	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) ∈ Fin)
41		elinel1 4172	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) → 𝑛 ∈ (1...(⌊‘𝑥)))
42		elfznn 12937	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
43	42	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
44	41, 43	sylan2 594	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 𝑛 ∈ ℕ)
45		vmacl 25695	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
46		nndivre 11679	. . . . . . . . . . . . . . . . . 18 ⊢ (((Λ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
47	45, 46	mpancom 686	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
48	44, 47	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
49	40, 48	fsumrecl 15091	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛) ∈ ℝ)
50	36, 49	remulcld 10671	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ∈ ℝ)
51		relogcl 25159	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
52	51	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
53		1re 10641	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
54	1, 3	dchrfi 25831	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)
55	18, 54	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐷 ∈ Fin)
56		difss 4108	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∖ { 1 }) ⊆ 𝐷
57	6, 56	sstri 3976	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑊 ⊆ 𝐷
58		ssfi 8738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ Fin ∧ 𝑊 ⊆ 𝐷) → 𝑊 ∈ Fin)
59	55, 57, 58	sylancl 588	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑊 ∈ Fin)
60		hashcl 13718	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ₀)
61	59, 60	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
62	61	nn0red 11957	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ)
63		resubcl 10950	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ (♯‘𝑊) ∈ ℝ) → (1 − (♯‘𝑊)) ∈ ℝ)
64	53, 62, 63	sylancr 589	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 − (♯‘𝑊)) ∈ ℝ)
65	64	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 − (♯‘𝑊)) ∈ ℝ)
66	52, 65	remulcld 10671	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · (1 − (♯‘𝑊))) ∈ ℝ)
67	50, 66	resubcld 11068	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈ ℝ)
68	67	recnd 10669	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈ ℂ)
69	68	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ 𝑥 ∈ ℝ⁺) → (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈ ℂ)
70	51	adantl 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
71	70	recnd 10669	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
72	51	ad2antrl 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
73	66	ad2ant2r 745	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (1 − (♯‘𝑊))) ∈ ℝ)
74	72, 73	readdcld 10670	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈ ℝ)
75		0red 10644	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ∈ ℝ)
76	50	ad2ant2r 745	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ∈ ℝ)
77		2re 11712	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
78	77	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 2 ∈ ℝ)
79	62	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (♯‘𝑊) ∈ ℝ)
80	78, 79	resubcld 11068	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (2 − (♯‘𝑊)) ∈ ℝ)
81		log1 25169	. . . . . . . . . . . . . . . . 17 ⊢ (log‘1) = 0
82		simprr 771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
83		1rp 12394	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ⁺
84		simprl 769	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
85		logleb 25186	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
86	83, 84, 85	sylancr 589	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
87	82, 86	mpbid 234	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘1) ≤ (log‘𝑥))
88	81, 87	eqbrtrrid 5102	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ (log‘𝑥))
89	59	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑊 ∈ Fin)
90		eqid 2821	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
91	1, 3, 9, 90	dchrinv 25837	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((inv_g‘𝐺)‘𝑋) = (∗ ∘ 𝑋))
92	1	dchrabl 25830	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
93	18, 92	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐺 ∈ Abel)
94		ablgrp 18911	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
95	93, 94	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐺 ∈ Grp)
96	3, 90	grpinvcl 18151	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐷) → ((inv_g‘𝐺)‘𝑋) ∈ 𝐷)
97	95, 9, 96	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((inv_g‘𝐺)‘𝑋) ∈ 𝐷)
98	91, 97	eqeltrrd 2914	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (∗ ∘ 𝑋) ∈ 𝐷)
99		eldifsni 4722	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → 𝑋 ≠ 1 )
100	8, 99	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑋 ≠ 1 )
101	3, 19	grpidcl 18131	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐺 ∈ Grp → 1 ∈ 𝐷)
102	95, 101	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 1 ∈ 𝐷)
103	3, 90, 95, 9, 102	grpinv11 18168	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (((inv_g‘𝐺)‘𝑋) = ((inv_g‘𝐺)‘ 1 ) ↔ 𝑋 = 1 ))
104	103	necon3bid 3060	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((inv_g‘𝐺)‘𝑋) ≠ ((inv_g‘𝐺)‘ 1 ) ↔ 𝑋 ≠ 1 ))
105	100, 104	mpbird 259	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((inv_g‘𝐺)‘𝑋) ≠ ((inv_g‘𝐺)‘ 1 ))
106	19, 90	grpinvid 18160	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐺 ∈ Grp → ((inv_g‘𝐺)‘ 1 ) = 1 )
107	95, 106	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((inv_g‘𝐺)‘ 1 ) = 1 )
108	105, 91, 107	3netr3d 3092	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (∗ ∘ 𝑋) ≠ 1 )
109		eldifsn 4719	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∗ ∘ 𝑋) ∈ (𝐷 ∖ { 1 }) ↔ ((∗ ∘ 𝑋) ∈ 𝐷 ∧ (∗ ∘ 𝑋) ≠ 1 ))
110	98, 108, 109	sylanbrc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (∗ ∘ 𝑋) ∈ (𝐷 ∖ { 1 }))
111		nnuz 12282	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ = (ℤ_≥‘1)
112		1zzd 12014	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 1 ∈ ℤ)
113		2fveq3 6675	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘𝑚)))
114		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → 𝑛 = 𝑚)
115	113, 114	oveq12d 7174	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑚 → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
116	115	fveq2d 6674	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
117		eqid 2821	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛))) = (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))
118		fvex 6683	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ V
119	116, 117, 118	fvmpt 6768	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
120	119	adantl 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
121		nnre 11645	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
122	121	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
123	122	cjred 14585	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘𝑚) = 𝑚)
124	123	oveq2d 7172	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((∗‘(𝑋‘(𝐿‘𝑚))) / (∗‘𝑚)) = ((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚))
125	10	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋:(Base‘𝑍)⟶ℂ)
126	2, 4, 17	znzrhfo 20694	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
127	21, 126	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑍))
128		fof 6590	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
129	127, 128	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍))
130		nnz 12005	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
131		ffvelrn 6849	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ (Base‘𝑍))
132	129, 130, 131	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘𝑚) ∈ (Base‘𝑍))
133	125, 132	ffvelrnd 6852	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
134		nncn 11646	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
135	134	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
136		nnne0 11672	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
137	136	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
138	133, 135, 137	cjdivd 14582	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = ((∗‘(𝑋‘(𝐿‘𝑚))) / (∗‘𝑚)))
139		fvco3 6760	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ (𝐿‘𝑚) ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘(𝐿‘𝑚)) = (∗‘(𝑋‘(𝐿‘𝑚))))
140	125, 132, 139	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((∗ ∘ 𝑋)‘(𝐿‘𝑚)) = (∗‘(𝑋‘(𝐿‘𝑚))))
141	140	oveq1d 7171	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = ((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚))
142	124, 138, 141	3eqtr4d 2866	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
143	120, 142	eqtrd 2856	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
144	133	cjcld 14555	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘(𝑋‘(𝐿‘𝑚))) ∈ ℂ)
145	144, 135, 137	divcld 11416	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚) ∈ ℂ)
146	141, 145	eqeltrd 2913	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
147		eqid 2821	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
148	2, 17, 18, 1, 3, 19, 9, 100, 147	dchrmusumlema 26069	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))
149		simprrl 779	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡)
150	7	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑋 ∈ 𝑊)
151	18	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑁 ∈ ℕ)
152	9	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑋 ∈ 𝐷)
153	100	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑋 ≠ 1 )
154		simprl 769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑐 ∈ (0[,)+∞))
155		simprrr 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦))
156	2, 17, 151, 1, 3, 19, 152, 153, 147, 154, 149, 155, 5	dchrvmaeq0 26080	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → (𝑋 ∈ 𝑊 ↔ 𝑡 = 0))
157	150, 156	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑡 = 0)
158	149, 157	breqtrd 5092	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 0)
159	158	rexlimdvaa 3285	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 0))
160	159	exlimdv 1934	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 0))
161	148, 160	mpd 15	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 0)
162		seqex 13372	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))) ∈ V
163	162	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))) ∈ V)
164		2fveq3 6675	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
165		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚)
166	164, 165	oveq12d 7174	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
167		ovex 7189	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V
168	166, 147, 167	fvmpt 6768	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ℕ → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
169	168	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
170	133, 135, 137	divcld 11416	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
171	169, 170	eqeltrd 2913	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) ∈ ℂ)
172	111, 112, 171	serf 13399	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))):ℕ⟶ℂ)
173	172	ffvelrnda 6851	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘) ∈ ℂ)
174		fzfid 13342	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
175		simpl 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝜑)
176		elfznn 12937	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
177	175, 176, 170	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
178	174, 177	fsumcj 15165	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∗‘Σ𝑚 ∈ (1...𝑘)((𝑋‘(𝐿‘𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...𝑘)(∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
179	175, 176, 169	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
180		simpr 487	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
181	180, 111	eleqtrdi 2923	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ_≥‘1))
182	179, 181, 177	fsumser 15087	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((𝑋‘(𝐿‘𝑚)) / 𝑚) = (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘))
183	182	fveq2d 6674	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∗‘Σ𝑚 ∈ (1...𝑘)((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (∗‘(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘)))
184	175, 176, 120	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
185	170	cjcld 14555	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ)
186	175, 176, 185	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ)
187	184, 181, 186	fsumser 15087	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛))))‘𝑘))
188	178, 183, 187	3eqtr3rd 2865	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛))))‘𝑘) = (∗‘(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘)))
189	111, 161, 163, 112, 173, 188	climcj 14961	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))) ⇝ (∗‘0))
190		cj0 14517	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∗‘0) = 0
191	189, 190	breqtrdi 5107	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))) ⇝ 0)
192	111, 112, 143, 146, 191	isumclim 15112	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = 0)
193		fveq1 6669	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = (∗ ∘ 𝑋) → (𝑦‘(𝐿‘𝑚)) = ((∗ ∘ 𝑋)‘(𝐿‘𝑚)))
194	193	oveq1d 7171	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = (∗ ∘ 𝑋) → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
195	194	sumeq2sdv 15061	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (∗ ∘ 𝑋) → Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
196	195	eqeq1d 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (∗ ∘ 𝑋) → (Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = 0))
197	196, 5	elrab2 3683	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∗ ∘ 𝑋) ∈ 𝑊 ↔ ((∗ ∘ 𝑋) ∈ (𝐷 ∖ { 1 }) ∧ Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = 0))
198	110, 192, 197	sylanbrc 585	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (∗ ∘ 𝑋) ∈ 𝑊)
199	198	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (∗ ∘ 𝑋) ∈ 𝑊)
200	7	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑋 ∈ 𝑊)
201		simplr 767	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (∗ ∘ 𝑋) ≠ 𝑋)
202	89, 199, 200, 201	nehash2 13833	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 2 ≤ (♯‘𝑊))
203		suble0 11154	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℝ ∧ (♯‘𝑊) ∈ ℝ) → ((2 − (♯‘𝑊)) ≤ 0 ↔ 2 ≤ (♯‘𝑊)))
204	77, 79, 203	sylancr 589	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((2 − (♯‘𝑊)) ≤ 0 ↔ 2 ≤ (♯‘𝑊)))
205	202, 204	mpbird 259	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (2 − (♯‘𝑊)) ≤ 0)
206	80, 75, 72, 88, 205	lemul2ad 11580	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (2 − (♯‘𝑊))) ≤ ((log‘𝑥) · 0))
207		df-2 11701	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 = (1 + 1)
208	207	oveq1i 7166	. . . . . . . . . . . . . . . . . 18 ⊢ (2 − (♯‘𝑊)) = ((1 + 1) − (♯‘𝑊))
209		1cnd 10636	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ∈ ℂ)
210	79	recnd 10669	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (♯‘𝑊) ∈ ℂ)
211	209, 209, 210	addsubassd 11017	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((1 + 1) − (♯‘𝑊)) = (1 + (1 − (♯‘𝑊))))
212	208, 211	syl5eq 2868	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (2 − (♯‘𝑊)) = (1 + (1 − (♯‘𝑊))))
213	212	oveq2d 7172	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (2 − (♯‘𝑊))) = ((log‘𝑥) · (1 + (1 − (♯‘𝑊)))))
214	71	adantrr 715	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℂ)
215	64	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 − (♯‘𝑊)) ∈ ℝ)
216	215	recnd 10669	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 − (♯‘𝑊)) ∈ ℂ)
217	214, 209, 216	adddid 10665	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (1 + (1 − (♯‘𝑊)))) = (((log‘𝑥) · 1) + ((log‘𝑥) · (1 − (♯‘𝑊)))))
218	214	mulid1d 10658	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · 1) = (log‘𝑥))
219	218	oveq1d 7171	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((log‘𝑥) · 1) + ((log‘𝑥) · (1 − (♯‘𝑊)))) = ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))))
220	213, 217, 219	3eqtrd 2860	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (2 − (♯‘𝑊))) = ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))))
221	214	mul01d 10839	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · 0) = 0)
222	206, 220, 221	3brtr3d 5097	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))) ≤ 0)
223	33	nnred 11653	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℝ)
224	223	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (ϕ‘𝑁) ∈ ℝ)
225	49	ad2ant2r 745	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛) ∈ ℝ)
226	34	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (ϕ‘𝑁) ∈ ℕ₀)
227	226	nn0ge0d 11959	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ (ϕ‘𝑁))
228	44, 45	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → (Λ‘𝑛) ∈ ℝ)
229		vmage0 25698	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
230	44, 229	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 0 ≤ (Λ‘𝑛))
231	44	nnred 11653	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 𝑛 ∈ ℝ)
232	44	nngt0d 11687	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 0 < 𝑛)
233		divge0 11509	. . . . . . . . . . . . . . . . . 18 ⊢ ((((Λ‘𝑛) ∈ ℝ ∧ 0 ≤ (Λ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((Λ‘𝑛) / 𝑛))
234	228, 230, 231, 232, 233	syl22anc 836	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 0 ≤ ((Λ‘𝑛) / 𝑛))
235	40, 48, 234	fsumge0 15150	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛))
236	235	ad2ant2r 745	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛))
237	224, 225, 227, 236	mulge0d 11217	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)))
238	74, 75, 76, 222, 237	letrd 10797	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))) ≤ ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)))
239		leaddsub 11116	. . . . . . . . . . . . . 14 ⊢ (((log‘𝑥) ∈ ℝ ∧ ((log‘𝑥) · (1 − (♯‘𝑊))) ∈ ℝ ∧ ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ∈ ℝ) → (((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))) ≤ ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ↔ (log‘𝑥) ≤ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))))
240	72, 73, 76, 239	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))) ≤ ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ↔ (log‘𝑥) ≤ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))))
241	238, 240	mpbid 234	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ≤ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))))
242	72, 88	absidd 14782	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(log‘𝑥)) = (log‘𝑥))
243	67	ad2ant2r 745	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈ ℝ)
244	75, 72, 243, 88, 241	letrd 10797	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))))
245	243, 244	absidd 14782	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))) = (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))))
246	241, 242, 245	3brtr4d 5098	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(log‘𝑥)) ≤ (abs‘(((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊))))))
247	16, 32, 69, 71, 246	o1le 15009	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1))
248	247	ex 415	. . . . . . . . 9 ⊢ (𝜑 → ((∗ ∘ 𝑋) ≠ 𝑋 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1)))
249	248	necon1bd 3034	. . . . . . . 8 ⊢ (𝜑 → (¬ (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1) → (∗ ∘ 𝑋) = 𝑋))
250	15, 249	mpi 20	. . . . . . 7 ⊢ (𝜑 → (∗ ∘ 𝑋) = 𝑋)
251	250	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (∗ ∘ 𝑋) = 𝑋)
252	251	fveq1d 6672	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘𝑥) = (𝑋‘𝑥))
253	14, 252	eqtr3d 2858	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (∗‘(𝑋‘𝑥)) = (𝑋‘𝑥))
254	12, 253	cjrebd 14561	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℝ)
255	254	ralrimiva 3182	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(𝑋‘𝑥) ∈ ℝ)
256		ffnfv 6882	. 2 ⊢ (𝑋:(Base‘𝑍)⟶ℝ ↔ (𝑋 Fn (Base‘𝑍) ∧ ∀𝑥 ∈ (Base‘𝑍)(𝑋‘𝑥) ∈ ℝ))
257	11, 255, 256	sylanbrc 585	1 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 {csn 4567 class class class wbr 5066 ↦ cmpt 5146 ^◡ccnv 5554 “ cima 5558 ∘ ccom 5559 Fn wfn 6350 ⟶wf 6351 –onto→wfo 6353 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 +∞cpnf 10672 < clt 10675 ≤ cle 10676 − cmin 10870 / cdiv 11297 ℕcn 11638 2c2 11693 ℕ₀cn0 11898 ℤcz 11982 ℤ_≥cuz 12244 ℝ⁺crp 12390 [,)cico 12741 ...cfz 12893 ⌊cfl 13161 seqcseq 13370 ♯chash 13691 ∗ccj 14455 abscabs 14593 ⇝ cli 14841 𝑂(1)co1 14843 Σcsu 15042 ϕcphi 16101 Basecbs 16483 0_gc0g 16713 Grpcgrp 18103 inv_gcminusg 18104 Abelcabl 18907 1_rcur 19251 Ringcrg 19297 CRingccrg 19298 Unitcui 19389 ℤRHomczrh 20647 ℤ/nℤczn 20650 logclog 25138 Λcvma 25669 DChrcdchr 25808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-rpss 7449 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-omul 8107 df-er 8289 df-ec 8291 df-qs 8295 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-o1 14847 df-lo1 14848 df-sum 15043 df-ef 15421 df-e 15422 df-sin 15423 df-cos 15424 df-tan 15425 df-pi 15426 df-dvds 15608 df-gcd 15844 df-prm 16016 df-phi 16103 df-pc 16174 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-qus 16782 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-nsg 18277 df-eqg 18278 df-ghm 18356 df-gim 18399 df-ga 18420 df-cntz 18447 df-oppg 18474 df-od 18656 df-gex 18657 df-pgp 18658 df-lsm 18761 df-pj1 18762 df-cmn 18908 df-abl 18909 df-cyg 18997 df-dprd 19117 df-dpj 19118 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-rnghom 19467 df-drng 19504 df-subrg 19533 df-lmod 19636 df-lss 19704 df-lsp 19744 df-sra 19944 df-rgmod 19945 df-lidl 19946 df-rsp 19947 df-2idl 20005 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-zring 20618 df-zrh 20651 df-zn 20654 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-0p 24271 df-limc 24464 df-dv 24465 df-ply 24778 df-idp 24779 df-coe 24780 df-dgr 24781 df-quot 24880 df-ulm 24965 df-log 25140 df-cxp 25141 df-atan 25445 df-em 25570 df-cht 25674 df-vma 25675 df-chp 25676 df-ppi 25677 df-mu 25678 df-dchr 25809

This theorem is referenced by: dchrisum0 26096

W3C validator