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Theorem dchrisum0re 27747

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 2762	. . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍)
5		rpvmasum2.w	. . . . . . 7 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
6	5	ssrab3 4033	. . . . . 6 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
7		dchrisum0.b	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
8	6, 7	sselid 3932	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
9	8	eldifad 3914	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
10	1, 2, 3, 4, 9	dchrf 27476	. . 3 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
11	10	ffnd 6707	. 2 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍))
12	10	ffvelcdmda 7080	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ)
13		fvco3 6982	. . . . . 6 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑥 ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘𝑥) = (∗‘(𝑋‘𝑥)))
14	10, 13	sylan 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘𝑥) = (∗‘(𝑋‘𝑥)))
15		logno1 26871	. . . . . . . 8 ⊢ ¬ (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1)
16		1red 11236	. . . . . . . . . . 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 2762	. . . . . . . . . . . . 13 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
21	18	nnnn0d 12592	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
22	2	zncrng 21758	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
23	21, 22	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑍 ∈ CRing)
24		crngring 20385	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
25	23, 24	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ Ring)
26		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
27	20, 26	1unit 20516	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ Ring → (1_r‘𝑍) ∈ (Unit‘𝑍))
28	25, 27	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1_r‘𝑍) ∈ (Unit‘𝑍))
29		eqid 2762	. . . . . . . . . . . . 13 ⊢ (^◡𝐿 “ {(1_r‘𝑍)}) = (^◡𝐿 “ {(1_r‘𝑍)})
30		eqidd 2763	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑊) → (1_r‘𝑍) = (1_r‘𝑍))
31	2, 17, 18, 1, 3, 19, 5, 20, 28, 29, 30	rpvmasum2 27746	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 16867	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
34	33	nnnn0d 12592	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ₀)
35	34	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (ϕ‘𝑁) ∈ ℕ₀)
36	35	nn0red 12593	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (ϕ‘𝑁) ∈ ℝ)
37		fzfid 14039	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
38		inss1 4185	. . . . . . . . . . . . . . . . 17 ⊢ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) ⊆ (1...(⌊‘𝑥))
39		ssfi 9170	. . . . . . . . . . . . . . . . 17 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) ⊆ (1...(⌊‘𝑥))) → ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) ∈ Fin)
40	37, 38, 39	sylancl 598	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) ∈ Fin)
41		elinel1 4150	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) → 𝑛 ∈ (1...(⌊‘𝑥)))
42		elfznn 13610	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
43	42	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
44	41, 43	sylan2 605	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 𝑛 ∈ ℕ)
45		vmacl 27352	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
46		nndivre 12304	. . . . . . . . . . . . . . . . . 18 ⊢ (((Λ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
47	45, 46	mpancom 701	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
48	44, 47	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
49	40, 48	fsumrecl 15822	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛) ∈ ℝ)
50	36, 49	remulcld 11266	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ∈ ℝ)
51		relogcl 26810	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
52	51	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
53		1re 11235	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
54	1, 3	dchrfi 27489	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)
55	18, 54	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐷 ∈ Fin)
56		difss 4086	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∖ { 1 }) ⊆ 𝐷
57	6, 56	sstri 3943	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑊 ⊆ 𝐷
58		ssfi 9170	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ Fin ∧ 𝑊 ⊆ 𝐷) → 𝑊 ∈ Fin)
59	55, 57, 58	sylancl 598	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑊 ∈ Fin)
60		hashcl 14422	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ₀)
61	59, 60	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
62	61	nn0red 12593	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ)
63		resubcl 11549	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ (♯‘𝑊) ∈ ℝ) → (1 − (♯‘𝑊)) ∈ ℝ)
64	53, 62, 63	sylancr 599	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 − (♯‘𝑊)) ∈ ℝ)
65	64	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 − (♯‘𝑊)) ∈ ℝ)
66	52, 65	remulcld 11266	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · (1 − (♯‘𝑊))) ∈ ℝ)
67	50, 66	resubcld 11669	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈ ℝ)
68	67	recnd 11264	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈ ℂ)
69	68	adantlr 728	. . . . . . . . . . 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 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
71	70	recnd 11264	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
72	51	ad2antrl 741	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
73	66	ad2ant2r 760	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (1 − (♯‘𝑊))) ∈ ℝ)
74	72, 73	readdcld 11265	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈ ℝ)
75		0red 11238	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ∈ ℝ)
76	50	ad2ant2r 760	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ∈ ℝ)
77		2re 12342	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
78	77	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 2 ∈ ℝ)
79	62	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (♯‘𝑊) ∈ ℝ)
80	78, 79	resubcld 11669	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (2 − (♯‘𝑊)) ∈ ℝ)
81		log1 26820	. . . . . . . . . . . . . . . . 17 ⊢ (log‘1) = 0
82		simprr 785	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
83		1rp 13048	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ⁺
84		simprl 783	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
85		logleb 26838	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
86	83, 84, 85	sylancr 599	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
87	82, 86	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘1) ≤ (log‘𝑥))
88	81, 87	eqbrtrrid 5145	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ (log‘𝑥))
89	59	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑊 ∈ Fin)
90		eqid 2762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
91	1, 3, 9, 90	dchrinv 27495	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((inv_g‘𝐺)‘𝑋) = (∗ ∘ 𝑋))
92	1	dchrabl 27488	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
93	18, 92	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐺 ∈ Abel)
94		ablgrp 19913	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
95	93, 94	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐺 ∈ Grp)
96	3, 90	grpinvcl 19112	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐷) → ((inv_g‘𝐺)‘𝑋) ∈ 𝐷)
97	95, 9, 96	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((inv_g‘𝐺)‘𝑋) ∈ 𝐷)
98	91, 97	eqeltrrd 2863	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (∗ ∘ 𝑋) ∈ 𝐷)
99		eldifsni 4756	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → 𝑋 ≠ 1 )
100	8, 99	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑋 ≠ 1 )
101	3, 19	grpidcl 19090	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐺 ∈ Grp → 1 ∈ 𝐷)
102	95, 101	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 1 ∈ 𝐷)
103	3, 90, 95, 9, 102	grpinv11 19132	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (((inv_g‘𝐺)‘𝑋) = ((inv_g‘𝐺)‘ 1 ) ↔ 𝑋 = 1 ))
104	103	necon3bid 3001	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((inv_g‘𝐺)‘𝑋) ≠ ((inv_g‘𝐺)‘ 1 ) ↔ 𝑋 ≠ 1 ))
105	100, 104	mpbird 260	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((inv_g‘𝐺)‘𝑋) ≠ ((inv_g‘𝐺)‘ 1 ))
106	19, 90	grpinvid 19124	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐺 ∈ Grp → ((inv_g‘𝐺)‘ 1 ) = 1 )
107	95, 106	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((inv_g‘𝐺)‘ 1 ) = 1 )
108	105, 91, 107	3netr3d 3033	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (∗ ∘ 𝑋) ≠ 1 )
109		eldifsn 4751	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∗ ∘ 𝑋) ∈ (𝐷 ∖ { 1 }) ↔ ((∗ ∘ 𝑋) ∈ 𝐷 ∧ (∗ ∘ 𝑋) ≠ 1 ))
110	98, 108, 109	sylanbrc 595	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (∗ ∘ 𝑋) ∈ (𝐷 ∖ { 1 }))
111		nnuz 12929	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ = (ℤ_≥‘1)
112		1zzd 12652	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 1 ∈ ℤ)
113		2fveq3 6887	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘𝑚)))
114		id 23	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → 𝑛 = 𝑚)
115	113, 114	oveq12d 7434	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑚 → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
116	115	fveq2d 6886	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
117		eqid 2762	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛))) = (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))
118		fvex 6895	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ V
119	116, 117, 118	fvmpt 6990	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
120	119	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
121		nnre 12267	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
122	121	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
123	122	cjred 15315	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘𝑚) = 𝑚)
124	123	oveq2d 7432	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((∗‘(𝑋‘(𝐿‘𝑚))) / (∗‘𝑚)) = ((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚))
125	10	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋:(Base‘𝑍)⟶ℂ)
126	2, 4, 17	znzrhfo 21761	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
127	21, 126	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑍))
128		fof 6793	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
129	127, 128	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍))
130		nnz 12639	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
131		ffvelcdm 7077	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ (Base‘𝑍))
132	129, 130, 131	syl2an 608	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘𝑚) ∈ (Base‘𝑍))
133	125, 132	ffvelcdmd 7081	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
134		nncn 12268	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
135	134	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
136		nnne0 12297	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
137	136	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
138	133, 135, 137	cjdivd 15312	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = ((∗‘(𝑋‘(𝐿‘𝑚))) / (∗‘𝑚)))
139		fvco3 6982	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ (𝐿‘𝑚) ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘(𝐿‘𝑚)) = (∗‘(𝑋‘(𝐿‘𝑚))))
140	125, 132, 139	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((∗ ∘ 𝑋)‘(𝐿‘𝑚)) = (∗‘(𝑋‘(𝐿‘𝑚))))
141	140	oveq1d 7431	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = ((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚))
142	124, 138, 141	3eqtr4d 2807	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
143	120, 142	eqtrd 2797	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
144	133	cjcld 15285	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘(𝑋‘(𝐿‘𝑚))) ∈ ℂ)
145	144, 135, 137	divcld 12018	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚) ∈ ℂ)
146	141, 145	eqeltrd 2862	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
147		eqid 2762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
148	2, 17, 18, 1, 3, 19, 9, 100, 147	dchrmusumlema 27727	. . . . . . . . . . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 27738	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5135	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3166	. . . . . . . . . . . . . . . . . . . . . . . . 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 1966	. . . . . . . . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 0)
162		seqex 14069	. . . . . . . . . . . . . . . . . . . . . . . 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 6887	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
165		id 23	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚)
166	164, 165	oveq12d 7434	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
167		ovex 7449	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V
168	166, 147, 167	fvmpt 6990	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ℕ → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
169	168	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
170	133, 135, 137	divcld 12018	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
171	169, 170	eqeltrd 2862	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) ∈ ℂ)
172	111, 112, 171	serf 14096	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))):ℕ⟶ℂ)
173	172	ffvelcdmda 7080	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘) ∈ ℂ)
174		fzfid 14039	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
175		simpl 488	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝜑)
176		elfznn 13610	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
177	175, 176, 170	syl2an 608	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
178	174, 177	fsumcj 15899	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∗‘Σ𝑚 ∈ (1...𝑘)((𝑋‘(𝐿‘𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...𝑘)(∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
179	175, 176, 169	syl2an 608	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
180		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
181	180, 111	eleqtrdi 2872	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ_≥‘1))
182	179, 181, 177	fsumser 15818	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((𝑋‘(𝐿‘𝑚)) / 𝑚) = (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘))
183	182	fveq2d 6886	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∗‘Σ𝑚 ∈ (1...𝑘)((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (∗‘(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘)))
184	175, 176, 120	syl2an 608	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
185	170	cjcld 15285	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ)
186	175, 176, 185	syl2an 608	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ)
187	184, 181, 186	fsumser 15818	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛))))‘𝑘))
188	178, 183, 187	3eqtr3rd 2806	. . . . . . . . . . . . . . . . . . . . . . 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 15694	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))) ⇝ (∗‘0))
190		cj0 15247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∗‘0) = 0
191	189, 190	breqtrdi 5150	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))) ⇝ 0)
192	111, 112, 143, 146, 191	isumclim 15845	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = 0)
193		fveq1 6881	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = (∗ ∘ 𝑋) → (𝑦‘(𝐿‘𝑚)) = ((∗ ∘ 𝑋)‘(𝐿‘𝑚)))
194	193	oveq1d 7431	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = (∗ ∘ 𝑋) → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
195	194	sumeq2sdv 15792	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (∗ ∘ 𝑋) → Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
196	195	eqeq1d 2764	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (∗ ∘ 𝑋) → (Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = 0))
197	196, 5	elrab2 3652	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∗ ∘ 𝑋) ∈ 𝑊 ↔ ((∗ ∘ 𝑋) ∈ (𝐷 ∖ { 1 }) ∧ Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = 0))
198	110, 192, 197	sylanbrc 595	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (∗ ∘ 𝑋) ∈ 𝑊)
199	198	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (∗ ∘ 𝑋) ∈ 𝑊)
200	7	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑋 ∈ 𝑊)
201		simplr 781	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (∗ ∘ 𝑋) ≠ 𝑋)
202	89, 199, 200, 201	nehash2 14541	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 2 ≤ (♯‘𝑊))
203		suble0 11755	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℝ ∧ (♯‘𝑊) ∈ ℝ) → ((2 − (♯‘𝑊)) ≤ 0 ↔ 2 ≤ (♯‘𝑊)))
204	77, 79, 203	sylancr 599	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((2 − (♯‘𝑊)) ≤ 0 ↔ 2 ≤ (♯‘𝑊)))
205	202, 204	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (2 − (♯‘𝑊)) ≤ 0)
206	80, 75, 72, 88, 205	lemul2ad 12182	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (2 − (♯‘𝑊))) ≤ ((log‘𝑥) · 0))
207		df-2 12330	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 = (1 + 1)
208	207	oveq1i 7426	. . . . . . . . . . . . . . . . . 18 ⊢ (2 − (♯‘𝑊)) = ((1 + 1) − (♯‘𝑊))
209		1cnd 11229	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ∈ ℂ)
210	79	recnd 11264	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (♯‘𝑊) ∈ ℂ)
211	209, 209, 210	addsubassd 11616	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((1 + 1) − (♯‘𝑊)) = (1 + (1 − (♯‘𝑊))))
212	208, 211	eqtrid 2809	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (2 − (♯‘𝑊)) = (1 + (1 − (♯‘𝑊))))
213	212	oveq2d 7432	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (2 − (♯‘𝑊))) = ((log‘𝑥) · (1 + (1 − (♯‘𝑊)))))
214	71	adantrr 730	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℂ)
215	64	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 − (♯‘𝑊)) ∈ ℝ)
216	215	recnd 11264	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 − (♯‘𝑊)) ∈ ℂ)
217	214, 209, 216	adddid 11260	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (1 + (1 − (♯‘𝑊)))) = (((log‘𝑥) · 1) + ((log‘𝑥) · (1 − (♯‘𝑊)))))
218	214	mulridd 11253	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · 1) = (log‘𝑥))
219	218	oveq1d 7431	. . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (2 − (♯‘𝑊))) = ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))))
221	214	mul01d 11436	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · 0) = 0)
222	206, 220, 221	3brtr3d 5140	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))) ≤ 0)
223	33	nnred 12275	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℝ)
224	223	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (ϕ‘𝑁) ∈ ℝ)
225	49	ad2ant2r 760	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛) ∈ ℝ)
226	34	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (ϕ‘𝑁) ∈ ℕ₀)
227	226	nn0ge0d 12595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ (ϕ‘𝑁))
228	44, 45	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → (Λ‘𝑛) ∈ ℝ)
229		vmage0 27355	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
230	44, 229	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 0 ≤ (Λ‘𝑛))
231	44	nnred 12275	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 𝑛 ∈ ℝ)
232	44	nngt0d 12312	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 0 < 𝑛)
233		divge0 12111	. . . . . . . . . . . . . . . . . 18 ⊢ ((((Λ‘𝑛) ∈ ℝ ∧ 0 ≤ (Λ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((Λ‘𝑛) / 𝑛))
234	228, 230, 231, 232, 233	syl22anc 852	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 0 ≤ ((Λ‘𝑛) / 𝑛))
235	40, 48, 234	fsumge0 15884	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛))
236	235	ad2ant2r 760	. . . . . . . . . . . . . . 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 11818	. . . . . . . . . . . . . 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 11394	. . . . . . . . . . . . 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 11717	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 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 235	. . . . . . . . . . . 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 15512	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(log‘𝑥)) = (log‘𝑥))
243	67	ad2ant2r 760	. . . . . . . . . . . . 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 11394	. . . . . . . . . . . . 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 15512	. . . . . . . . . . . 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 5141	. . . . . . . . . . 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 15742	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1))
248	247	ex 418	. . . . . . . . 9 ⊢ (𝜑 → ((∗ ∘ 𝑋) ≠ 𝑋 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1)))
249	248	necon1bd 2975	. . . . . . . 8 ⊢ (𝜑 → (¬ (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1) → (∗ ∘ 𝑋) = 𝑋))
250	15, 249	mpi 21	. . . . . . 7 ⊢ (𝜑 → (∗ ∘ 𝑋) = 𝑋)
251	250	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (∗ ∘ 𝑋) = 𝑋)
252	251	fveq1d 6884	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘𝑥) = (𝑋‘𝑥))
253	14, 252	eqtr3d 2799	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (∗‘(𝑋‘𝑥)) = (𝑋‘𝑥))
254	12, 253	cjrebd 15291	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℝ)
255	254	ralrimiva 3156	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(𝑋‘𝑥) ∈ ℝ)
256		ffnfv 7115	. 2 ⊢ (𝑋:(Base‘𝑍)⟶ℝ ↔ (𝑋 Fn (Base‘𝑍) ∧ ∀𝑥 ∈ (Base‘𝑍)(𝑋‘𝑥) ∈ ℝ))
257	11, 255, 256	sylanbrc 595	1 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2145 ≠ wne 2957 ∀wral 3078 ∃wrex 3088 {crab 3414 Vcvv 3453 ∖ cdif 3899 ∩ cin 3901 ⊆ wss 3902 {csn 4587 class class class wbr 5107 ↦ cmpt 5190 ^◡ccnv 5658 “ cima 5662 ∘ ccom 5663 Fn wfn 6532 ⟶wf 6533 –onto→wfo 6535 ‘cfv 6537 (class class class)co 7416 Fincfn 8955 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 +∞cpnf 11267 < clt 11270 ≤ cle 11271 − cmin 11468 / cdiv 11898 ℕcn 12260 2c2 12322 ℕ₀cn0 12531 ℤcz 12618 ℤ_≥cuz 12890 ℝ⁺crp 13044 [,)cico 13402 ...cfz 13563 ⌊cfl 13853 seqcseq 14067 ♯chash 14396 ∗ccj 15185 abscabs 15323 ⇝ cli 15573 𝑂(1)co1 15575 Σcsu 15775 ϕcphi 16859 Basecbs 17305 0_gc0g 17528 Grpcgrp 19058 inv_gcminusg 19059 Abelcabl 19909 1_rcur 20321 Ringcrg 20373 CRingccrg 20374 Unitcui 20497 ℤRHomczrh 21713 ℤ/nℤczn 21716 logclog 26789 Λcvma 27326 DChrcdchr 27466

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206 ax-mulf 11207

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-disj 5075 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-rpss 7727 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-oadd 8462 df-omul 8463 df-er 8699 df-ec 8701 df-qs 8705 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-dju 9909 df-card 9947 df-acn 9950 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-xnn0 12605 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13045 df-xneg 13165 df-xadd 13166 df-xmul 13167 df-ioo 13404 df-ioc 13405 df-ico 13406 df-icc 13407 df-fz 13564 df-fzo 13712 df-fl 13855 df-mod 13933 df-seq 14068 df-exp 14128 df-fac 14340 df-bc 14369 df-hash 14397 df-word 14581 df-concat 14638 df-s1 14665 df-shft 15142 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-limsup 15560 df-clim 15577 df-rlim 15578 df-o1 15579 df-lo1 15580 df-sum 15776 df-ef 16157 df-e 16158 df-sin 16159 df-cos 16160 df-tan 16161 df-pi 16162 df-dvds 16347 df-gcd 16589 df-prm 16766 df-phi 16861 df-pc 16933 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-starv 17361 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ds 17368 df-unif 17369 df-hom 17370 df-cco 17371 df-rest 17511 df-topn 17512 df-0g 17530 df-gsum 17531 df-topgen 17532 df-pt 17533 df-prds 17536 df-xrs 17592 df-qtop 17597 df-imas 17598 df-qus 17599 df-xps 17600 df-mre 17674 df-mrc 17675 df-acs 17677 df-mgm 18734 df-sgrp 18823 df-mnd 18839 df-mhm 18892 df-submnd 18893 df-grp 19061 df-minusg 19062 df-sbg 19063 df-mulg 19192 df-subg 19247 df-nsg 19248 df-eqg 19249 df-ghm 19342 df-gim 19387 df-ga 19418 df-cntz 19445 df-oppg 19474 df-od 19656 df-gex 19657 df-pgp 19658 df-lsm 19764 df-pj1 19765 df-cmn 19910 df-abl 19911 df-cyg 20006 df-dprd 20125 df-dpj 20126 df-mgp 20275 df-rng 20289 df-ur 20322 df-ring 20375 df-cring 20376 df-oppr 20479 df-dvdsr 20499 df-unit 20500 df-invr 20530 df-dvr 20543 df-rhm 20614 df-subrng 20709 df-subrg 20733 df-drng 20893 df-lmod 21047 df-lss 21117 df-lsp 21157 df-sra 21358 df-rgmod 21359 df-lidl 21396 df-rsp 21397 df-2idl 21453 df-psmet 21578 df-xmet 21579 df-met 21580 df-bl 21581 df-mopn 21582 df-fbas 21583 df-fg 21584 df-cnfld 21587 df-zring 21661 df-zrh 21717 df-zn 21720 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-cld 23245 df-ntr 23246 df-cls 23247 df-nei 23324 df-lp 23362 df-perf 23363 df-cn 23453 df-cnp 23454 df-haus 23541 df-cmp 23613 df-tx 23789 df-hmeo 23982 df-fil 24073 df-fm 24165 df-flim 24166 df-flf 24167 df-xms 24547 df-ms 24548 df-tms 24549 df-cncf 25107 df-0p 25899 df-limc 26095 df-dv 26096 df-ply 26415 df-idp 26416 df-coe 26417 df-dgr 26418 df-quot 26522 df-ulm 26610 df-log 26791 df-cxp 26792 df-atan 27102 df-em 27227 df-cht 27331 df-vma 27332 df-chp 27333 df-ppi 27334 df-mu 27335 df-dchr 27467

This theorem is used by: dchrisum0 27754

W3C validator