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Theorem dchrisum0re 26861

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 2736	. . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍)
5		rpvmasum2.w	. . . . . . 7 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
6	5	ssrab3 4040	. . . . . 6 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
7		dchrisum0.b	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
8	6, 7	sselid 3942	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
9	8	eldifad 3922	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
10	1, 2, 3, 4, 9	dchrf 26590	. . 3 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
11	10	ffnd 6669	. 2 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍))
12	10	ffvelcdmda 7035	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ)
13		fvco3 6940	. . . . . 6 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑥 ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘𝑥) = (∗‘(𝑋‘𝑥)))
14	10, 13	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘𝑥) = (∗‘(𝑋‘𝑥)))
15		logno1 25991	. . . . . . . 8 ⊢ ¬ (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1)
16		1red 11156	. . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
21	18	nnnn0d 12473	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
22	2	zncrng 20951	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
23	21, 22	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑍 ∈ CRing)
24		crngring 19976	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
25	23, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ Ring)
26		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
27	20, 26	1unit 20087	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ Ring → (1_r‘𝑍) ∈ (Unit‘𝑍))
28	25, 27	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1_r‘𝑍) ∈ (Unit‘𝑍))
29		eqid 2736	. . . . . . . . . . . . 13 ⊢ (^◡𝐿 “ {(1_r‘𝑍)}) = (^◡𝐿 “ {(1_r‘𝑍)})
30		eqidd 2737	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑊) → (1_r‘𝑍) = (1_r‘𝑍))
31	2, 17, 18, 1, 3, 19, 5, 20, 28, 29, 30	rpvmasum2 26860	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 16644	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
34	33	nnnn0d 12473	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ₀)
35	34	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (ϕ‘𝑁) ∈ ℕ₀)
36	35	nn0red 12474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (ϕ‘𝑁) ∈ ℝ)
37		fzfid 13878	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
38		inss1 4188	. . . . . . . . . . . . . . . . 17 ⊢ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) ⊆ (1...(⌊‘𝑥))
39		ssfi 9117	. . . . . . . . . . . . . . . . 17 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) ⊆ (1...(⌊‘𝑥))) → ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) ∈ Fin)
40	37, 38, 39	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) ∈ Fin)
41		elinel1 4155	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) → 𝑛 ∈ (1...(⌊‘𝑥)))
42		elfznn 13470	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
43	42	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
44	41, 43	sylan2 593	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 𝑛 ∈ ℕ)
45		vmacl 26467	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
46		nndivre 12194	. . . . . . . . . . . . . . . . . 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 15619	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛) ∈ ℝ)
50	36, 49	remulcld 11185	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ∈ ℝ)
51		relogcl 25931	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
52	51	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
53		1re 11155	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
54	1, 3	dchrfi 26603	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)
55	18, 54	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐷 ∈ Fin)
56		difss 4091	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∖ { 1 }) ⊆ 𝐷
57	6, 56	sstri 3953	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑊 ⊆ 𝐷
58		ssfi 9117	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ Fin ∧ 𝑊 ⊆ 𝐷) → 𝑊 ∈ Fin)
59	55, 57, 58	sylancl 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑊 ∈ Fin)
60		hashcl 14256	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ₀)
61	59, 60	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
62	61	nn0red 12474	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ)
63		resubcl 11465	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ (♯‘𝑊) ∈ ℝ) → (1 − (♯‘𝑊)) ∈ ℝ)
64	53, 62, 63	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 − (♯‘𝑊)) ∈ ℝ)
65	64	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 − (♯‘𝑊)) ∈ ℝ)
66	52, 65	remulcld 11185	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · (1 − (♯‘𝑊))) ∈ ℝ)
67	50, 66	resubcld 11583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈ ℝ)
68	67	recnd 11183	. . . . . . . . . . . 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 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
71	70	recnd 11183	. . . . . . . . . . 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 11184	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈ ℝ)
75		0red 11158	. . . . . . . . . . . . . 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 12227	. . . . . . . . . . . . . . . . . 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 11583	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (2 − (♯‘𝑊)) ∈ ℝ)
81		log1 25941	. . . . . . . . . . . . . . . . 17 ⊢ (log‘1) = 0
82		simprr 771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
83		1rp 12919	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ⁺
84		simprl 769	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
85		logleb 25958	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
86	83, 84, 85	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
87	82, 86	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘1) ≤ (log‘𝑥))
88	81, 87	eqbrtrrid 5141	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
91	1, 3, 9, 90	dchrinv 26609	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((inv_g‘𝐺)‘𝑋) = (∗ ∘ 𝑋))
92	1	dchrabl 26602	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
93	18, 92	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐺 ∈ Abel)
94		ablgrp 19567	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
95	93, 94	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐺 ∈ Grp)
96	3, 90	grpinvcl 18798	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐷) → ((inv_g‘𝐺)‘𝑋) ∈ 𝐷)
97	95, 9, 96	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((inv_g‘𝐺)‘𝑋) ∈ 𝐷)
98	91, 97	eqeltrrd 2839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (∗ ∘ 𝑋) ∈ 𝐷)
99		eldifsni 4750	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → 𝑋 ≠ 1 )
100	8, 99	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑋 ≠ 1 )
101	3, 19	grpidcl 18778	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐺 ∈ Grp → 1 ∈ 𝐷)
102	95, 101	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 1 ∈ 𝐷)
103	3, 90, 95, 9, 102	grpinv11 18816	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (((inv_g‘𝐺)‘𝑋) = ((inv_g‘𝐺)‘ 1 ) ↔ 𝑋 = 1 ))
104	103	necon3bid 2988	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((inv_g‘𝐺)‘𝑋) ≠ ((inv_g‘𝐺)‘ 1 ) ↔ 𝑋 ≠ 1 ))
105	100, 104	mpbird 256	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((inv_g‘𝐺)‘𝑋) ≠ ((inv_g‘𝐺)‘ 1 ))
106	19, 90	grpinvid 18808	. . . . . . . . . . . . . . . . . . . . . . 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 3020	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (∗ ∘ 𝑋) ≠ 1 )
109		eldifsn 4747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∗ ∘ 𝑋) ∈ (𝐷 ∖ { 1 }) ↔ ((∗ ∘ 𝑋) ∈ 𝐷 ∧ (∗ ∘ 𝑋) ≠ 1 ))
110	98, 108, 109	sylanbrc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (∗ ∘ 𝑋) ∈ (𝐷 ∖ { 1 }))
111		nnuz 12806	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ = (ℤ_≥‘1)
112		1zzd 12534	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 1 ∈ ℤ)
113		2fveq3 6847	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘𝑚)))
114		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → 𝑛 = 𝑚)
115	113, 114	oveq12d 7375	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑚 → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
116	115	fveq2d 6846	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
117		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛))) = (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))
118		fvex 6855	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ V
119	116, 117, 118	fvmpt 6948	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
120	119	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
121		nnre 12160	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
122	121	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
123	122	cjred 15111	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘𝑚) = 𝑚)
124	123	oveq2d 7373	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((∗‘(𝑋‘(𝐿‘𝑚))) / (∗‘𝑚)) = ((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚))
125	10	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋:(Base‘𝑍)⟶ℂ)
126	2, 4, 17	znzrhfo 20954	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
127	21, 126	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑍))
128		fof 6756	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
129	127, 128	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍))
130		nnz 12520	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
131		ffvelcdm 7032	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ (Base‘𝑍))
132	129, 130, 131	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘𝑚) ∈ (Base‘𝑍))
133	125, 132	ffvelcdmd 7036	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
134		nncn 12161	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
135	134	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
136		nnne0 12187	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
137	136	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
138	133, 135, 137	cjdivd 15108	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = ((∗‘(𝑋‘(𝐿‘𝑚))) / (∗‘𝑚)))
139		fvco3 6940	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ (𝐿‘𝑚) ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘(𝐿‘𝑚)) = (∗‘(𝑋‘(𝐿‘𝑚))))
140	125, 132, 139	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((∗ ∘ 𝑋)‘(𝐿‘𝑚)) = (∗‘(𝑋‘(𝐿‘𝑚))))
141	140	oveq1d 7372	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = ((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚))
142	124, 138, 141	3eqtr4d 2786	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
143	120, 142	eqtrd 2776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
144	133	cjcld 15081	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘(𝑋‘(𝐿‘𝑚))) ∈ ℂ)
145	144, 135, 137	divcld 11931	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚) ∈ ℂ)
146	141, 145	eqeltrd 2838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
147		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
148	2, 17, 18, 1, 3, 19, 9, 100, 147	dchrmusumlema 26841	. . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 26852	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5131	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3153	. . . . . . . . . . . . . . . . . . . . . . . . 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 1936	. . . . . . . . . . . . . . . . . . . . . . . 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 13908	. . . . . . . . . . . . . . . . . . . . . . . 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 6847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
165		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚)
166	164, 165	oveq12d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
167		ovex 7390	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V
168	166, 147, 167	fvmpt 6948	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ℕ → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
169	168	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
170	133, 135, 137	divcld 11931	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
171	169, 170	eqeltrd 2838	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) ∈ ℂ)
172	111, 112, 171	serf 13936	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))):ℕ⟶ℂ)
173	172	ffvelcdmda 7035	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘) ∈ ℂ)
174		fzfid 13878	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
175		simpl 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝜑)
176		elfznn 13470	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
177	175, 176, 170	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
178	174, 177	fsumcj 15695	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∗‘Σ𝑚 ∈ (1...𝑘)((𝑋‘(𝐿‘𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...𝑘)(∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
179	175, 176, 169	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
180		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
181	180, 111	eleqtrdi 2848	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ_≥‘1))
182	179, 181, 177	fsumser 15615	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((𝑋‘(𝐿‘𝑚)) / 𝑚) = (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘))
183	182	fveq2d 6846	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∗‘Σ𝑚 ∈ (1...𝑘)((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (∗‘(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘)))
184	175, 176, 120	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
185	170	cjcld 15081	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ)
186	175, 176, 185	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ)
187	184, 181, 186	fsumser 15615	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛))))‘𝑘))
188	178, 183, 187	3eqtr3rd 2785	. . . . . . . . . . . . . . . . . . . . . . 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 15487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))) ⇝ (∗‘0))
190		cj0 15043	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∗‘0) = 0
191	189, 190	breqtrdi 5146	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))) ⇝ 0)
192	111, 112, 143, 146, 191	isumclim 15642	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = 0)
193		fveq1 6841	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = (∗ ∘ 𝑋) → (𝑦‘(𝐿‘𝑚)) = ((∗ ∘ 𝑋)‘(𝐿‘𝑚)))
194	193	oveq1d 7372	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = (∗ ∘ 𝑋) → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
195	194	sumeq2sdv 15589	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (∗ ∘ 𝑋) → Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
196	195	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (∗ ∘ 𝑋) → (Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = 0))
197	196, 5	elrab2 3648	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∗ ∘ 𝑋) ∈ 𝑊 ↔ ((∗ ∘ 𝑋) ∈ (𝐷 ∖ { 1 }) ∧ Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = 0))
198	110, 192, 197	sylanbrc 583	. . . . . . . . . . . . . . . . . . 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 14373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 2 ≤ (♯‘𝑊))
203		suble0 11669	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℝ ∧ (♯‘𝑊) ∈ ℝ) → ((2 − (♯‘𝑊)) ≤ 0 ↔ 2 ≤ (♯‘𝑊)))
204	77, 79, 203	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((2 − (♯‘𝑊)) ≤ 0 ↔ 2 ≤ (♯‘𝑊)))
205	202, 204	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (2 − (♯‘𝑊)) ≤ 0)
206	80, 75, 72, 88, 205	lemul2ad 12095	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (2 − (♯‘𝑊))) ≤ ((log‘𝑥) · 0))
207		df-2 12216	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 = (1 + 1)
208	207	oveq1i 7367	. . . . . . . . . . . . . . . . . 18 ⊢ (2 − (♯‘𝑊)) = ((1 + 1) − (♯‘𝑊))
209		1cnd 11150	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ∈ ℂ)
210	79	recnd 11183	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (♯‘𝑊) ∈ ℂ)
211	209, 209, 210	addsubassd 11532	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((1 + 1) − (♯‘𝑊)) = (1 + (1 − (♯‘𝑊))))
212	208, 211	eqtrid 2788	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (2 − (♯‘𝑊)) = (1 + (1 − (♯‘𝑊))))
213	212	oveq2d 7373	. . . . . . . . . . . . . . . 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 11183	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 − (♯‘𝑊)) ∈ ℂ)
217	214, 209, 216	adddid 11179	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (1 + (1 − (♯‘𝑊)))) = (((log‘𝑥) · 1) + ((log‘𝑥) · (1 − (♯‘𝑊)))))
218	214	mulid1d 11172	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · 1) = (log‘𝑥))
219	218	oveq1d 7372	. . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (2 − (♯‘𝑊))) = ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))))
221	214	mul01d 11354	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · 0) = 0)
222	206, 220, 221	3brtr3d 5136	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))) ≤ 0)
223	33	nnred 12168	. . . . . . . . . . . . . . . 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 12476	. . . . . . . . . . . . . . 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 26470	. . . . . . . . . . . . . . . . . . 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 12168	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 𝑛 ∈ ℝ)
232	44	nngt0d 12202	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 0 < 𝑛)
233		divge0 12024	. . . . . . . . . . . . . . . . . 18 ⊢ ((((Λ‘𝑛) ∈ ℝ ∧ 0 ≤ (Λ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((Λ‘𝑛) / 𝑛))
234	228, 230, 231, 232, 233	syl22anc 837	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 0 ≤ ((Λ‘𝑛) / 𝑛))
235	40, 48, 234	fsumge0 15680	. . . . . . . . . . . . . . . 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 11732	. . . . . . . . . . . . . 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 11312	. . . . . . . . . . . . 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 11631	. . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 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 15307	. . . . . . . . . . . 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 11312	. . . . . . . . . . . . 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 15307	. . . . . . . . . . . 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 5137	. . . . . . . . . . 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 15537	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1))
248	247	ex 413	. . . . . . . . 9 ⊢ (𝜑 → ((∗ ∘ 𝑋) ≠ 𝑋 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1)))
249	248	necon1bd 2961	. . . . . . . 8 ⊢ (𝜑 → (¬ (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1) → (∗ ∘ 𝑋) = 𝑋))
250	15, 249	mpi 20	. . . . . . 7 ⊢ (𝜑 → (∗ ∘ 𝑋) = 𝑋)
251	250	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (∗ ∘ 𝑋) = 𝑋)
252	251	fveq1d 6844	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘𝑥) = (𝑋‘𝑥))
253	14, 252	eqtr3d 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (∗‘(𝑋‘𝑥)) = (𝑋‘𝑥))
254	12, 253	cjrebd 15087	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℝ)
255	254	ralrimiva 3143	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(𝑋‘𝑥) ∈ ℝ)
256		ffnfv 7066	. 2 ⊢ (𝑋:(Base‘𝑍)⟶ℝ ↔ (𝑋 Fn (Base‘𝑍) ∧ ∀𝑥 ∈ (Base‘𝑍)(𝑋‘𝑥) ∈ ℝ))
257	11, 255, 256	sylanbrc 583	1 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 {crab 3407 Vcvv 3445 ∖ cdif 3907 ∩ cin 3909 ⊆ wss 3910 {csn 4586 class class class wbr 5105 ↦ cmpt 5188 ^◡ccnv 5632 “ cima 5636 ∘ ccom 5637 Fn wfn 6491 ⟶wf 6492 –onto→wfo 6494 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 +∞cpnf 11186 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 ℝ⁺crp 12915 [,)cico 13266 ...cfz 13424 ⌊cfl 13695 seqcseq 13906 ♯chash 14230 ∗ccj 14981 abscabs 15119 ⇝ cli 15366 𝑂(1)co1 15368 Σcsu 15570 ϕcphi 16636 Basecbs 17083 0_gc0g 17321 Grpcgrp 18748 inv_gcminusg 18749 Abelcabl 19563 1_rcur 19913 Ringcrg 19964 CRingccrg 19965 Unitcui 20068 ℤRHomczrh 20900 ℤ/nℤczn 20903 logclog 25910 Λcvma 26441 DChrcdchr 26580

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-rpss 7660 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-ec 8650 df-qs 8654 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-word 14403 df-concat 14459 df-s1 14484 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-o1 15372 df-lo1 15373 df-sum 15571 df-ef 15950 df-e 15951 df-sin 15952 df-cos 15953 df-tan 15954 df-pi 15955 df-dvds 16137 df-gcd 16375 df-prm 16548 df-phi 16638 df-pc 16709 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-qus 17391 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-nsg 18926 df-eqg 18927 df-ghm 19006 df-gim 19049 df-ga 19070 df-cntz 19097 df-oppg 19124 df-od 19310 df-gex 19311 df-pgp 19312 df-lsm 19418 df-pj1 19419 df-cmn 19564 df-abl 19565 df-cyg 19655 df-dprd 19774 df-dpj 19775 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-rnghom 20146 df-drng 20187 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-sra 20633 df-rgmod 20634 df-lidl 20635 df-rsp 20636 df-2idl 20702 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-zring 20870 df-zrh 20904 df-zn 20907 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-0p 25034 df-limc 25230 df-dv 25231 df-ply 25549 df-idp 25550 df-coe 25551 df-dgr 25552 df-quot 25651 df-ulm 25736 df-log 25912 df-cxp 25913 df-atan 26217 df-em 26342 df-cht 26446 df-vma 26447 df-chp 26448 df-ppi 26449 df-mu 26450 df-dchr 26581

This theorem is referenced by: dchrisum0 26868

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