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Theorem dchrisum0re 27444

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 2730	. . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍)
5		rpvmasum2.w	. . . . . . 7 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
6	5	ssrab3 4030	. . . . . 6 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
7		dchrisum0.b	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
8	6, 7	sselid 3930	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
9	8	eldifad 3912	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
10	1, 2, 3, 4, 9	dchrf 27173	. . 3 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
11	10	ffnd 6648	. 2 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍))
12	10	ffvelcdmda 7012	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ)
13		fvco3 6916	. . . . . 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 26565	. . . . . . . 8 ⊢ ¬ (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1)
16		1red 11105	. . . . . . . . . . 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 2730	. . . . . . . . . . . . 13 ⊢ (Unit‘𝑍) = (Unit‘𝑍)
21	18	nnnn0d 12434	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
22	2	zncrng 21474	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
23	21, 22	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑍 ∈ CRing)
24		crngring 20156	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
25	23, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ Ring)
26		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
27	20, 26	1unit 20285	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ Ring → (1_r‘𝑍) ∈ (Unit‘𝑍))
28	25, 27	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1_r‘𝑍) ∈ (Unit‘𝑍))
29		eqid 2730	. . . . . . . . . . . . 13 ⊢ (^◡𝐿 “ {(1_r‘𝑍)}) = (^◡𝐿 “ {(1_r‘𝑍)})
30		eqidd 2731	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑊) → (1_r‘𝑍) = (1_r‘𝑍))
31	2, 17, 18, 1, 3, 19, 5, 20, 28, 29, 30	rpvmasum2 27443	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 16675	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
34	33	nnnn0d 12434	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ₀)
35	34	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (ϕ‘𝑁) ∈ ℕ₀)
36	35	nn0red 12435	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (ϕ‘𝑁) ∈ ℝ)
37		fzfid 13872	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
38		inss1 4185	. . . . . . . . . . . . . . . . 17 ⊢ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) ⊆ (1...(⌊‘𝑥))
39		ssfi 9077	. . . . . . . . . . . . . . . . 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 4149	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)})) → 𝑛 ∈ (1...(⌊‘𝑥)))
42		elfznn 13445	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
43	42	adantl 481	. . . . . . . . . . . . . . . . . 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 27048	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
46		nndivre 12158	. . . . . . . . . . . . . . . . . 18 ⊢ (((Λ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
47	45, 46	mpancom 688	. . . . . . . . . . . . . . . . 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 15633	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛) ∈ ℝ)
50	36, 49	remulcld 11134	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ∈ ℝ)
51		relogcl 26504	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
52	51	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
53		1re 11104	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
54	1, 3	dchrfi 27186	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)
55	18, 54	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐷 ∈ Fin)
56		difss 4084	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∖ { 1 }) ⊆ 𝐷
57	6, 56	sstri 3942	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑊 ⊆ 𝐷
58		ssfi 9077	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ Fin ∧ 𝑊 ⊆ 𝐷) → 𝑊 ∈ Fin)
59	55, 57, 58	sylancl 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑊 ∈ Fin)
60		hashcl 14255	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ₀)
61	59, 60	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
62	61	nn0red 12435	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ)
63		resubcl 11417	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ (♯‘𝑊) ∈ ℝ) → (1 − (♯‘𝑊)) ∈ ℝ)
64	53, 62, 63	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 − (♯‘𝑊)) ∈ ℝ)
65	64	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 − (♯‘𝑊)) ∈ ℝ)
66	52, 65	remulcld 11134	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · (1 − (♯‘𝑊))) ∈ ℝ)
67	50, 66	resubcld 11537	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈ ℝ)
68	67	recnd 11132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈ ℂ)
69	68	adantlr 715	. . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
71	70	recnd 11132	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
72	51	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
73	66	ad2ant2r 747	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (1 − (♯‘𝑊))) ∈ ℝ)
74	72, 73	readdcld 11133	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))) ∈ ℝ)
75		0red 11107	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ∈ ℝ)
76	50	ad2ant2r 747	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛)) ∈ ℝ)
77		2re 12191	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
78	77	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 2 ∈ ℝ)
79	62	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (♯‘𝑊) ∈ ℝ)
80	78, 79	resubcld 11537	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (2 − (♯‘𝑊)) ∈ ℝ)
81		log1 26514	. . . . . . . . . . . . . . . . 17 ⊢ (log‘1) = 0
82		simprr 772	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
83		1rp 12886	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ⁺
84		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
85		logleb 26532	. . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘1) ≤ (log‘𝑥))
88	81, 87	eqbrtrrid 5125	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ (log‘𝑥))
89	59	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑊 ∈ Fin)
90		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
91	1, 3, 9, 90	dchrinv 27192	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((inv_g‘𝐺)‘𝑋) = (∗ ∘ 𝑋))
92	1	dchrabl 27185	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
93	18, 92	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐺 ∈ Abel)
94		ablgrp 19690	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
95	93, 94	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐺 ∈ Grp)
96	3, 90	grpinvcl 18892	. . . . . . . . . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (∗ ∘ 𝑋) ∈ 𝐷)
99		eldifsni 4740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → 𝑋 ≠ 1 )
100	8, 99	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑋 ≠ 1 )
101	3, 19	grpidcl 18870	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐺 ∈ Grp → 1 ∈ 𝐷)
102	95, 101	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 1 ∈ 𝐷)
103	3, 90, 95, 9, 102	grpinv11 18912	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (((inv_g‘𝐺)‘𝑋) = ((inv_g‘𝐺)‘ 1 ) ↔ 𝑋 = 1 ))
104	103	necon3bid 2970	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((inv_g‘𝐺)‘𝑋) ≠ ((inv_g‘𝐺)‘ 1 ) ↔ 𝑋 ≠ 1 ))
105	100, 104	mpbird 257	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((inv_g‘𝐺)‘𝑋) ≠ ((inv_g‘𝐺)‘ 1 ))
106	19, 90	grpinvid 18904	. . . . . . . . . . . . . . . . . . . . . . 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 3002	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (∗ ∘ 𝑋) ≠ 1 )
109		eldifsn 4736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∗ ∘ 𝑋) ∈ (𝐷 ∖ { 1 }) ↔ ((∗ ∘ 𝑋) ∈ 𝐷 ∧ (∗ ∘ 𝑋) ≠ 1 ))
110	98, 108, 109	sylanbrc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (∗ ∘ 𝑋) ∈ (𝐷 ∖ { 1 }))
111		nnuz 12767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ = (ℤ_≥‘1)
112		1zzd 12495	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 1 ∈ ℤ)
113		2fveq3 6822	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘𝑚)))
114		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → 𝑛 = 𝑚)
115	113, 114	oveq12d 7359	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑚 → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
116	115	fveq2d 6821	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
117		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛))) = (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))
118		fvex 6830	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ V
119	116, 117, 118	fvmpt 6924	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
120	119	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)))
121		nnre 12124	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
122	121	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
123	122	cjred 15125	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘𝑚) = 𝑚)
124	123	oveq2d 7357	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((∗‘(𝑋‘(𝐿‘𝑚))) / (∗‘𝑚)) = ((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚))
125	10	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋:(Base‘𝑍)⟶ℂ)
126	2, 4, 17	znzrhfo 21477	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
127	21, 126	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑍))
128		fof 6731	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
129	127, 128	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍))
130		nnz 12481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
131		ffvelcdm 7009	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7013	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
134		nncn 12125	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
135	134	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
136		nnne0 12151	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
137	136	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
138	133, 135, 137	cjdivd 15122	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = ((∗‘(𝑋‘(𝐿‘𝑚))) / (∗‘𝑚)))
139		fvco3 6916	. . . . . . . . . . . . . . . . . . . . . . . . 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 7356	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = ((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚))
142	124, 138, 141	3eqtr4d 2775	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
143	120, 142	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))‘𝑚) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
144	133	cjcld 15095	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∗‘(𝑋‘(𝐿‘𝑚))) ∈ ℂ)
145	144, 135, 137	divcld 11889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((∗‘(𝑋‘(𝐿‘𝑚))) / 𝑚) ∈ ℂ)
146	141, 145	eqeltrd 2829	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
147		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
148	2, 17, 18, 1, 3, 19, 9, 100, 147	dchrmusumlema 27424	. . . . . . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 27435	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5115	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3132	. . . . . . . . . . . . . . . . . . . . . . . . 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 13902	. . . . . . . . . . . . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
165		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚)
166	164, 165	oveq12d 7359	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
167		ovex 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V
168	166, 147, 167	fvmpt 6924	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ℕ → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
169	168	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
170	133, 135, 137	divcld 11889	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
171	169, 170	eqeltrd 2829	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))‘𝑚) ∈ ℂ)
172	111, 112, 171	serf 13929	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))):ℕ⟶ℂ)
173	172	ffvelcdmda 7012	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘) ∈ ℂ)
174		fzfid 13872	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
175		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝜑)
176		elfznn 13445	. . . . . . . . . . . . . . . . . . . . . . . . . 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 15709	. . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
181	180, 111	eleqtrdi 2839	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ_≥‘1))
182	179, 181, 177	fsumser 15629	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((𝑋‘(𝐿‘𝑚)) / 𝑚) = (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘𝑘))
183	182	fveq2d 6821	. . . . . . . . . . . . . . . . . . . . . . . 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 15095	. . . . . . . . . . . . . . . . . . . . . . . . . 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 15629	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(∗‘((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛))))‘𝑘))
188	178, 183, 187	3eqtr3rd 2774	. . . . . . . . . . . . . . . . . . . . . . 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 15504	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))) ⇝ (∗‘0))
190		cj0 15057	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∗‘0) = 0
191	189, 190	breqtrdi 5130	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (∗‘((𝑋‘(𝐿‘𝑛)) / 𝑛)))) ⇝ 0)
192	111, 112, 143, 146, 191	isumclim 15656	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚) = 0)
193		fveq1 6816	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = (∗ ∘ 𝑋) → (𝑦‘(𝐿‘𝑚)) = ((∗ ∘ 𝑋)‘(𝐿‘𝑚)))
194	193	oveq1d 7356	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = (∗ ∘ 𝑋) → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
195	194	sumeq2sdv 15602	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (∗ ∘ 𝑋) → Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑚 ∈ ℕ (((∗ ∘ 𝑋)‘(𝐿‘𝑚)) / 𝑚))
196	195	eqeq1d 2732	. . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (∗ ∘ 𝑋) ∈ 𝑊)
200	7	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑋 ∈ 𝑊)
201		simplr 768	. . . . . . . . . . . . . . . . . 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 11623	. . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (2 − (♯‘𝑊)) ≤ 0)
206	80, 75, 72, 88, 205	lemul2ad 12054	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (2 − (♯‘𝑊))) ≤ ((log‘𝑥) · 0))
207		df-2 12180	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 = (1 + 1)
208	207	oveq1i 7351	. . . . . . . . . . . . . . . . . 18 ⊢ (2 − (♯‘𝑊)) = ((1 + 1) − (♯‘𝑊))
209		1cnd 11099	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ∈ ℂ)
210	79	recnd 11132	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (♯‘𝑊) ∈ ℂ)
211	209, 209, 210	addsubassd 11484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((1 + 1) − (♯‘𝑊)) = (1 + (1 − (♯‘𝑊))))
212	208, 211	eqtrid 2777	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (2 − (♯‘𝑊)) = (1 + (1 − (♯‘𝑊))))
213	212	oveq2d 7357	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (2 − (♯‘𝑊))) = ((log‘𝑥) · (1 + (1 − (♯‘𝑊)))))
214	71	adantrr 717	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℂ)
215	64	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 − (♯‘𝑊)) ∈ ℝ)
216	215	recnd 11132	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 − (♯‘𝑊)) ∈ ℂ)
217	214, 209, 216	adddid 11128	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (1 + (1 − (♯‘𝑊)))) = (((log‘𝑥) · 1) + ((log‘𝑥) · (1 − (♯‘𝑊)))))
218	214	mulridd 11121	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · 1) = (log‘𝑥))
219	218	oveq1d 7356	. . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · (2 − (♯‘𝑊))) = ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))))
221	214	mul01d 11304	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) · 0) = 0)
222	206, 220, 221	3brtr3d 5120	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) + ((log‘𝑥) · (1 − (♯‘𝑊)))) ≤ 0)
223	33	nnred 12132	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℝ)
224	223	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (ϕ‘𝑁) ∈ ℝ)
225	49	ad2ant2r 747	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛) ∈ ℝ)
226	34	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (ϕ‘𝑁) ∈ ℕ₀)
227	226	nn0ge0d 12437	. . . . . . . . . . . . . . 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 27051	. . . . . . . . . . . . . . . . . . 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 12132	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 𝑛 ∈ ℝ)
232	44	nngt0d 12166	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 0 < 𝑛)
233		divge0 11983	. . . . . . . . . . . . . . . . . 18 ⊢ ((((Λ‘𝑛) ∈ ℝ ∧ 0 ≤ (Λ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((Λ‘𝑛) / 𝑛))
234	228, 230, 231, 232, 233	syl22anc 838	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))) → 0 ≤ ((Λ‘𝑛) / 𝑛))
235	40, 48, 234	fsumge0 15694	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ (^◡𝐿 “ {(1_r‘𝑍)}))((Λ‘𝑛) / 𝑛))
236	235	ad2ant2r 747	. . . . . . . . . . . . . . 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 11686	. . . . . . . . . . . . . 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 11262	. . . . . . . . . . . . 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 11585	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 15322	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(log‘𝑥)) = (log‘𝑥))
243	67	ad2ant2r 747	. . . . . . . . . . . . 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 11262	. . . . . . . . . . . . 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 15322	. . . . . . . . . . . 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 5121	. . . . . . . . . . 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 15552	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (∗ ∘ 𝑋) ≠ 𝑋) → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1))
248	247	ex 412	. . . . . . . . 9 ⊢ (𝜑 → ((∗ ∘ 𝑋) ≠ 𝑋 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1)))
249	248	necon1bd 2944	. . . . . . . 8 ⊢ (𝜑 → (¬ (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1) → (∗ ∘ 𝑋) = 𝑋))
250	15, 249	mpi 20	. . . . . . 7 ⊢ (𝜑 → (∗ ∘ 𝑋) = 𝑋)
251	250	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (∗ ∘ 𝑋) = 𝑋)
252	251	fveq1d 6819	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → ((∗ ∘ 𝑋)‘𝑥) = (𝑋‘𝑥))
253	14, 252	eqtr3d 2767	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (∗‘(𝑋‘𝑥)) = (𝑋‘𝑥))
254	12, 253	cjrebd 15101	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℝ)
255	254	ralrimiva 3122	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑍)(𝑋‘𝑥) ∈ ℝ)
256		ffnfv 7047	. 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 206 ∧ wa 395 = wceq 1541 ∃wex 1780 ∈ wcel 2110 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 {crab 3393 Vcvv 3434 ∖ cdif 3897 ∩ cin 3899 ⊆ wss 3900 {csn 4574 class class class wbr 5089 ↦ cmpt 5170 ^◡ccnv 5613 “ cima 5617 ∘ ccom 5618 Fn wfn 6472 ⟶wf 6473 –onto→wfo 6475 ‘cfv 6477 (class class class)co 7341 Fincfn 8864 ℂcc 10996 ℝcr 10997 0cc0 10998 1c1 10999 + caddc 11001 · cmul 11003 +∞cpnf 11135 < clt 11138 ≤ cle 11139 − cmin 11336 / cdiv 11766 ℕcn 12117 2c2 12172 ℕ₀cn0 12373 ℤcz 12460 ℤ_≥cuz 12724 ℝ⁺crp 12882 [,)cico 13239 ...cfz 13399 ⌊cfl 13686 seqcseq 13900 ♯chash 14229 ∗ccj 14995 abscabs 15133 ⇝ cli 15383 𝑂(1)co1 15385 Σcsu 15585 ϕcphi 16667 Basecbs 17112 0_gc0g 17335 Grpcgrp 18838 inv_gcminusg 18839 Abelcabl 19686 1_rcur 20092 Ringcrg 20144 CRingccrg 20145 Unitcui 20266 ℤRHomczrh 21429 ℤ/nℤczn 21432 logclog 26483 Λcvma 27022 DChrcdchr 27163

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 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-inf2 9526 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076 ax-addf 11077 ax-mulf 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-rpss 7651 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-omul 8385 df-er 8617 df-ec 8619 df-qs 8623 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-dju 9786 df-card 9824 df-acn 9827 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-xnn0 12447 df-z 12461 df-dec 12581 df-uz 12725 df-q 12839 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-ioo 13241 df-ioc 13242 df-ico 13243 df-icc 13244 df-fz 13400 df-fzo 13547 df-fl 13688 df-mod 13766 df-seq 13901 df-exp 13961 df-fac 14173 df-bc 14202 df-hash 14230 df-word 14413 df-concat 14470 df-s1 14496 df-shft 14966 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-limsup 15370 df-clim 15387 df-rlim 15388 df-o1 15389 df-lo1 15390 df-sum 15586 df-ef 15966 df-e 15967 df-sin 15968 df-cos 15969 df-tan 15970 df-pi 15971 df-dvds 16156 df-gcd 16398 df-prm 16575 df-phi 16669 df-pc 16741 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-starv 17168 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-unif 17176 df-hom 17177 df-cco 17178 df-rest 17318 df-topn 17319 df-0g 17337 df-gsum 17338 df-topgen 17339 df-pt 17340 df-prds 17343 df-xrs 17398 df-qtop 17403 df-imas 17404 df-qus 17405 df-xps 17406 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-mhm 18683 df-submnd 18684 df-grp 18841 df-minusg 18842 df-sbg 18843 df-mulg 18973 df-subg 19028 df-nsg 19029 df-eqg 19030 df-ghm 19118 df-gim 19164 df-ga 19195 df-cntz 19222 df-oppg 19251 df-od 19433 df-gex 19434 df-pgp 19435 df-lsm 19541 df-pj1 19542 df-cmn 19687 df-abl 19688 df-cyg 19783 df-dprd 19902 df-dpj 19903 df-mgp 20052 df-rng 20064 df-ur 20093 df-ring 20146 df-cring 20147 df-oppr 20248 df-dvdsr 20268 df-unit 20269 df-invr 20299 df-dvr 20312 df-rhm 20383 df-subrng 20454 df-subrg 20478 df-drng 20639 df-lmod 20788 df-lss 20858 df-lsp 20898 df-sra 21100 df-rgmod 21101 df-lidl 21138 df-rsp 21139 df-2idl 21180 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-zring 21377 df-zrh 21433 df-zn 21436 df-top 22802 df-topon 22819 df-topsp 22841 df-bases 22854 df-cld 22927 df-ntr 22928 df-cls 22929 df-nei 23006 df-lp 23044 df-perf 23045 df-cn 23135 df-cnp 23136 df-haus 23223 df-cmp 23295 df-tx 23470 df-hmeo 23663 df-fil 23754 df-fm 23846 df-flim 23847 df-flf 23848 df-xms 24228 df-ms 24229 df-tms 24230 df-cncf 24791 df-0p 25591 df-limc 25787 df-dv 25788 df-ply 26113 df-idp 26114 df-coe 26115 df-dgr 26116 df-quot 26219 df-ulm 26306 df-log 26485 df-cxp 26486 df-atan 26797 df-em 26923 df-cht 27027 df-vma 27028 df-chp 27029 df-ppi 27030 df-mu 27031 df-dchr 27164

This theorem is referenced by: dchrisum0 27451

W3C validator