cvgratnnlemrate - Intuitionistic Logic Explorer

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Theorem cvgratnnlemrate 10987

Description: Lemma for cvgratnn 10988. (Contributed by Jim Kingdon, 21-Nov-2022.)

**Hypotheses**
Ref	Expression
cvgratnn.3	⊢ (𝜑 → 𝐴 ∈ ℝ)
cvgratnn.4	⊢ (𝜑 → 𝐴 < 1)
cvgratnn.gt0	⊢ (𝜑 → 0 < 𝐴)
cvgratnn.6	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
cvgratnn.7	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))
cvgratnnlemrate.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
cvgratnnlemrate.n	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))

**Assertion**
Ref	Expression
cvgratnnlemrate	⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) < (((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) · (𝐴 / (1 − 𝐴))) / 𝑀))

Distinct variable groups: 𝐴,𝑘 𝑘,𝐹 𝑘,𝑁 𝜑,𝑘 𝑘,𝑀

Proof of Theorem cvgratnnlemrate Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 9117	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 8840	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
3		cvgratnn.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
4	1, 2, 3	serf 9963	. . . . . 6 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
5		cvgratnnlemrate.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6		cvgratnnlemrate.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
7		eluznn 9150	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑁 ∈ ℕ)
8	5, 6, 7	syl2anc 404	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9	4, 8	ffvelrnd 5451	. . . . 5 ⊢ (𝜑 → (seq1( + , 𝐹)‘𝑁) ∈ ℂ)
10	4, 5	ffvelrnd 5451	. . . . 5 ⊢ (𝜑 → (seq1( + , 𝐹)‘𝑀) ∈ ℂ)
11	9, 10	subcld 7856	. . . 4 ⊢ (𝜑 → ((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀)) ∈ ℂ)
12	11	abscld 10677	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) ∈ ℝ)
13		fveq2 5320	. . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
14	13	eleq1d 2157	. . . . . 6 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ))
15	3	ralrimiva 2447	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
16	14, 15, 5	rspcdva 2730	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ)
17	16	abscld 10677	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℝ)
18	5	nnzd 8930	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
19	18	peano2zd 8934	. . . . . 6 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
20		eluzelz 9091	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
21	6, 20	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
22	19, 21	fzfigd 9901	. . . . 5 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ Fin)
23		cvgratnn.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
24	23	adantr 271	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝐴 ∈ ℝ)
25	5	nnred 8498	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
26	25	adantr 271	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ∈ ℝ)
27		peano2re 7681	. . . . . . . . 9 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
28	26, 27	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑀 + 1) ∈ ℝ)
29		elfzelz 9503	. . . . . . . . . 10 ⊢ (𝑖 ∈ ((𝑀 + 1)...𝑁) → 𝑖 ∈ ℤ)
30	29	adantl 272	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑖 ∈ ℤ)
31	30	zred 8931	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑖 ∈ ℝ)
32	26	lep1d 8455	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ≤ (𝑀 + 1))
33		elfzle1 9504	. . . . . . . . 9 ⊢ (𝑖 ∈ ((𝑀 + 1)...𝑁) → (𝑀 + 1) ≤ 𝑖)
34	33	adantl 272	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑀 + 1) ≤ 𝑖)
35	26, 28, 31, 32, 34	letrd 7670	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ≤ 𝑖)
36		znn0sub 8878	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑀 ≤ 𝑖 ↔ (𝑖 − 𝑀) ∈ ℕ₀))
37	18, 29, 36	syl2an 284	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑀 ≤ 𝑖 ↔ (𝑖 − 𝑀) ∈ ℕ₀))
38	35, 37	mpbid 146	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑖 − 𝑀) ∈ ℕ₀)
39	24, 38	reexpcld 10166	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝐴↑(𝑖 − 𝑀)) ∈ ℝ)
40	22, 39	fsumrecl 10858	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) ∈ ℝ)
41	17, 40	remulcld 7581	. . 3 ⊢ (𝜑 → ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀))) ∈ ℝ)
42		cvgratnn.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 < 1)
43		cvgratnn.gt0	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 𝐴)
44	23, 43	elrpd 9234	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
45	44	reclt1d 9250	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
46	42, 45	mpbid 146	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (1 / 𝐴))
47		1re 7550	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
48	44	rprecred 9248	. . . . . . . . . . 11 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
49		difrp 9233	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ (1 / 𝐴) ∈ ℝ) → (1 < (1 / 𝐴) ↔ ((1 / 𝐴) − 1) ∈ ℝ⁺))
50	47, 48, 49	sylancr 406	. . . . . . . . . 10 ⊢ (𝜑 → (1 < (1 / 𝐴) ↔ ((1 / 𝐴) − 1) ∈ ℝ⁺))
51	46, 50	mpbid 146	. . . . . . . . 9 ⊢ (𝜑 → ((1 / 𝐴) − 1) ∈ ℝ⁺)
52	51	rpreccld 9247	. . . . . . . 8 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ⁺)
53	52, 44	rpdivcld 9254	. . . . . . 7 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ⁺)
54		fveq2 5320	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
55	54	eleq1d 2157	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘1) ∈ ℂ))
56		1nn 8496	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
57	56	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℕ)
58	55, 15, 57	rspcdva 2730	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
59	58	abscld 10677	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℝ)
60	58	absge0d 10680	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘1)))
61	59, 60	ge0p1rpd 9267	. . . . . . 7 ⊢ (𝜑 → ((abs‘(𝐹‘1)) + 1) ∈ ℝ⁺)
62	53, 61	rpmulcld 9253	. . . . . 6 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ⁺)
63	62	rpred 9236	. . . . 5 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ)
64	63, 5	nndivred 8535	. . . 4 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀) ∈ ℝ)
65		1red 7566	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
66	65, 23	resubcld 7922	. . . . . . 7 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ)
67	23, 65	posdifd 8072	. . . . . . . 8 ⊢ (𝜑 → (𝐴 < 1 ↔ 0 < (1 − 𝐴)))
68	42, 67	mpbid 146	. . . . . . 7 ⊢ (𝜑 → 0 < (1 − 𝐴))
69	66, 68	elrpd 9234	. . . . . 6 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ⁺)
70	44, 69	rpdivcld 9254	. . . . 5 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℝ⁺)
71	70	rpred 9236	. . . 4 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℝ)
72	64, 71	remulcld 7581	. . 3 ⊢ (𝜑 → (((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀) · (𝐴 / (1 − 𝐴))) ∈ ℝ)
73		cvgratnn.7	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))
74	23, 42, 43, 3, 73, 5, 6	cvgratnnlemseq 10983	. . . . 5 ⊢ (𝜑 → ((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀)) = Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖))
75	74	fveq2d 5324	. . . 4 ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) = (abs‘Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖)))
76	23, 42, 43, 3, 73, 5, 6	cvgratnnlemabsle 10984	. . . 4 ⊢ (𝜑 → (abs‘Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖)) ≤ ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀))))
77	75, 76	eqbrtrd 3873	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) ≤ ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀))))
78	16	absge0d 10680	. . . 4 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘𝑀)))
79	23, 42, 43, 3, 73, 5	cvgratnnlemfm 10986	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) < ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀))
80	44	adantr 271	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝐴 ∈ ℝ⁺)
81	38	nn0zd 8929	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑖 − 𝑀) ∈ ℤ)
82	80, 81	rpexpcld 10173	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝐴↑(𝑖 − 𝑀)) ∈ ℝ⁺)
83	82	rpge0d 9240	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 0 ≤ (𝐴↑(𝑖 − 𝑀)))
84	22, 39, 83	fsumge0 10916	. . . 4 ⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)))
85	23, 42, 43, 3, 73, 5, 6	cvgratnnlemsumlt 10985	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) < (𝐴 / (1 − 𝐴)))
86	17, 64, 40, 71, 78, 79, 84, 85	ltmul12ad 8465	. . 3 ⊢ (𝜑 → ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀))) < (((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀) · (𝐴 / (1 − 𝐴))))
87	12, 41, 72, 77, 86	lelttrd 7671	. 2 ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) < (((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀) · (𝐴 / (1 − 𝐴))))
88	63	recnd 7579	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℂ)
89	71	recnd 7579	. . 3 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℂ)
90	5	nncnd 8499	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℂ)
91	5	nnap0d 8531	. . 3 ⊢ (𝜑 → 𝑀 # 0)
92	88, 89, 90, 91	div23apd 8358	. 2 ⊢ (𝜑 → (((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) · (𝐴 / (1 − 𝐴))) / 𝑀) = (((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀) · (𝐴 / (1 − 𝐴))))
93	87, 92	breqtrrd 3879	1 ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) < (((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) · (𝐴 / (1 − 𝐴))) / 𝑀))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1290 ∈ wcel 1439 class class class wbr 3853 ‘cfv 5030 (class class class)co 5668 ℂcc 7411 ℝcr 7412 0cc0 7413 1c1 7414 + caddc 7416 · cmul 7418 < clt 7585 ≤ cle 7586 − cmin 7716 / cdiv 8202 ℕcn 8485 ℕ₀cn0 8736 ℤcz 8813 ℤ_≥cuz 9082 ℝ⁺crp 9197 ...cfz 9487 seqcseq 9915 ↑cexp 10017 abscabs 10493 Σcsu 10805

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3962 ax-sep 3965 ax-nul 3973 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-iinf 4418 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-mulrcl 7507 ax-addcom 7508 ax-mulcom 7509 ax-addass 7510 ax-mulass 7511 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-1rid 7515 ax-0id 7516 ax-rnegex 7517 ax-precex 7518 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-apti 7523 ax-pre-ltadd 7524 ax-pre-mulgt0 7525 ax-pre-mulext 7526 ax-arch 7527 ax-caucvg 7528

This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-if 3400 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-iun 3740 df-br 3854 df-opab 3908 df-mpt 3909 df-tr 3945 df-id 4131 df-po 4134 df-iso 4135 df-iord 4204 df-on 4206 df-ilim 4207 df-suc 4209 df-iom 4421 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 df-fv 5038 df-isom 5039 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-1st 5927 df-2nd 5928 df-recs 6086 df-irdg 6151 df-frec 6172 df-1o 6197 df-oadd 6201 df-er 6308 df-en 6514 df-dom 6515 df-fin 6516 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-sub 7718 df-neg 7719 df-reap 8115 df-ap 8122 df-div 8203 df-inn 8486 df-2 8544 df-3 8545 df-4 8546 df-n0 8737 df-z 8814 df-uz 9083 df-q 9168 df-rp 9198 df-ico 9375 df-fz 9488 df-fzo 9617 df-iseq 9916 df-seq3 9917 df-exp 10018 df-ihash 10247 df-cj 10339 df-re 10340 df-im 10341 df-rsqrt 10494 df-abs 10495 df-clim 10730 df-isum 10806

This theorem is referenced by: cvgratnn 10988

W3C validator