cvgratnnlemrate - Intuitionistic Logic Explorer

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Theorem cvgratnnlemrate 11695

Description: Lemma for cvgratnn 11696. (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 9637	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 9353	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
3		cvgratnn.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
4	1, 2, 3	serf 10575	. . . . . 6 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
5		cvgratnnlemrate.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6		cvgratnnlemrate.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
7		eluznn 9674	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑁 ∈ ℕ)
8	5, 6, 7	syl2anc 411	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9	4, 8	ffvelcdmd 5698	. . . . 5 ⊢ (𝜑 → (seq1( + , 𝐹)‘𝑁) ∈ ℂ)
10	4, 5	ffvelcdmd 5698	. . . . 5 ⊢ (𝜑 → (seq1( + , 𝐹)‘𝑀) ∈ ℂ)
11	9, 10	subcld 8337	. . . 4 ⊢ (𝜑 → ((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀)) ∈ ℂ)
12	11	abscld 11346	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) ∈ ℝ)
13		fveq2 5558	. . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
14	13	eleq1d 2265	. . . . . 6 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ))
15	3	ralrimiva 2570	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
16	14, 15, 5	rspcdva 2873	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ)
17	16	abscld 11346	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℝ)
18	5	nnzd 9447	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
19	18	peano2zd 9451	. . . . . 6 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
20		eluzelz 9610	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
21	6, 20	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
22	19, 21	fzfigd 10523	. . . . 5 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ Fin)
23		cvgratnn.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
24	23	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝐴 ∈ ℝ)
25	5	nnred 9003	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
26	25	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ∈ ℝ)
27		peano2re 8162	. . . . . . . . 9 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
28	26, 27	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑀 + 1) ∈ ℝ)
29		elfzelz 10100	. . . . . . . . . 10 ⊢ (𝑖 ∈ ((𝑀 + 1)...𝑁) → 𝑖 ∈ ℤ)
30	29	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑖 ∈ ℤ)
31	30	zred 9448	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑖 ∈ ℝ)
32	26	lep1d 8958	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ≤ (𝑀 + 1))
33		elfzle1 10102	. . . . . . . . 9 ⊢ (𝑖 ∈ ((𝑀 + 1)...𝑁) → (𝑀 + 1) ≤ 𝑖)
34	33	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑀 + 1) ≤ 𝑖)
35	26, 28, 31, 32, 34	letrd 8150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ≤ 𝑖)
36		znn0sub 9391	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑀 ≤ 𝑖 ↔ (𝑖 − 𝑀) ∈ ℕ₀))
37	18, 29, 36	syl2an 289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑀 ≤ 𝑖 ↔ (𝑖 − 𝑀) ∈ ℕ₀))
38	35, 37	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑖 − 𝑀) ∈ ℕ₀)
39	24, 38	reexpcld 10782	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝐴↑(𝑖 − 𝑀)) ∈ ℝ)
40	22, 39	fsumrecl 11566	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) ∈ ℝ)
41	17, 40	remulcld 8057	. . 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 9768	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
45	44	reclt1d 9785	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
46	42, 45	mpbid 147	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (1 / 𝐴))
47		1re 8025	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
48	44	rprecred 9783	. . . . . . . . . . 11 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
49		difrp 9767	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ (1 / 𝐴) ∈ ℝ) → (1 < (1 / 𝐴) ↔ ((1 / 𝐴) − 1) ∈ ℝ⁺))
50	47, 48, 49	sylancr 414	. . . . . . . . . 10 ⊢ (𝜑 → (1 < (1 / 𝐴) ↔ ((1 / 𝐴) − 1) ∈ ℝ⁺))
51	46, 50	mpbid 147	. . . . . . . . 9 ⊢ (𝜑 → ((1 / 𝐴) − 1) ∈ ℝ⁺)
52	51	rpreccld 9782	. . . . . . . 8 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ⁺)
53	52, 44	rpdivcld 9789	. . . . . . 7 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ⁺)
54		fveq2 5558	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
55	54	eleq1d 2265	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘1) ∈ ℂ))
56		1nn 9001	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
57	56	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℕ)
58	55, 15, 57	rspcdva 2873	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
59	58	abscld 11346	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℝ)
60	58	absge0d 11349	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘1)))
61	59, 60	ge0p1rpd 9802	. . . . . . 7 ⊢ (𝜑 → ((abs‘(𝐹‘1)) + 1) ∈ ℝ⁺)
62	53, 61	rpmulcld 9788	. . . . . 6 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ⁺)
63	62	rpred 9771	. . . . 5 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ)
64	63, 5	nndivred 9040	. . . 4 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀) ∈ ℝ)
65		1red 8041	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
66	65, 23	resubcld 8407	. . . . . . 7 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ)
67	23, 65	posdifd 8559	. . . . . . . 8 ⊢ (𝜑 → (𝐴 < 1 ↔ 0 < (1 − 𝐴)))
68	42, 67	mpbid 147	. . . . . . 7 ⊢ (𝜑 → 0 < (1 − 𝐴))
69	66, 68	elrpd 9768	. . . . . 6 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ⁺)
70	44, 69	rpdivcld 9789	. . . . 5 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℝ⁺)
71	70	rpred 9771	. . . 4 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℝ)
72	64, 71	remulcld 8057	. . 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 11691	. . . . 5 ⊢ (𝜑 → ((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀)) = Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖))
75	74	fveq2d 5562	. . . 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 11692	. . . 4 ⊢ (𝜑 → (abs‘Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖)) ≤ ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀))))
77	75, 76	eqbrtrd 4055	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) ≤ ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀))))
78	16	absge0d 11349	. . . 4 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘𝑀)))
79	23, 42, 43, 3, 73, 5	cvgratnnlemfm 11694	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) < ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀))
80	44	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝐴 ∈ ℝ⁺)
81	38	nn0zd 9446	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑖 − 𝑀) ∈ ℤ)
82	80, 81	rpexpcld 10789	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝐴↑(𝑖 − 𝑀)) ∈ ℝ⁺)
83	82	rpge0d 9775	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 0 ≤ (𝐴↑(𝑖 − 𝑀)))
84	22, 39, 83	fsumge0 11624	. . . 4 ⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)))
85	23, 42, 43, 3, 73, 5, 6	cvgratnnlemsumlt 11693	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) < (𝐴 / (1 − 𝐴)))
86	17, 64, 40, 71, 78, 79, 84, 85	ltmul12ad 8968	. . 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 8151	. 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 8055	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℂ)
89	71	recnd 8055	. . 3 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℂ)
90	5	nncnd 9004	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℂ)
91	5	nnap0d 9036	. . 3 ⊢ (𝜑 → 𝑀 # 0)
92	88, 89, 90, 91	div23apd 8855	. 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 4061	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 104 ↔ wb 105 = wceq 1364 ∈ wcel 2167 class class class wbr 4033 ‘cfv 5258 (class class class)co 5922 ℂcc 7877 ℝcr 7878 0cc0 7879 1c1 7880 + caddc 7882 · cmul 7884 < clt 8061 ≤ cle 8062 − cmin 8197 / cdiv 8699 ℕcn 8990 ℕ₀cn0 9249 ℤcz 9326 ℤ_≥cuz 9601 ℝ⁺crp 9728 ...cfz 10083 seqcseq 10539 ↑cexp 10630 abscabs 11162 Σcsu 11518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-frec 6449 df-1o 6474 df-oadd 6478 df-er 6592 df-en 6800 df-dom 6801 df-fin 6802 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-ico 9969 df-fz 10084 df-fzo 10218 df-seqfrec 10540 df-exp 10631 df-ihash 10868 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-clim 11444 df-sumdc 11519

This theorem is referenced by: cvgratnn 11696

W3C validator