cvgratnnlemrate - Intuitionistic Logic Explorer

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Theorem cvgratnnlemrate 11540

Description: Lemma for cvgratnn 11541. (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 9565	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 9282	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
3		cvgratnn.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
4	1, 2, 3	serf 10476	. . . . . 6 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
5		cvgratnnlemrate.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6		cvgratnnlemrate.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
7		eluznn 9602	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑁 ∈ ℕ)
8	5, 6, 7	syl2anc 411	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9	4, 8	ffvelcdmd 5654	. . . . 5 ⊢ (𝜑 → (seq1( + , 𝐹)‘𝑁) ∈ ℂ)
10	4, 5	ffvelcdmd 5654	. . . . 5 ⊢ (𝜑 → (seq1( + , 𝐹)‘𝑀) ∈ ℂ)
11	9, 10	subcld 8270	. . . 4 ⊢ (𝜑 → ((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀)) ∈ ℂ)
12	11	abscld 11192	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) ∈ ℝ)
13		fveq2 5517	. . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
14	13	eleq1d 2246	. . . . . 6 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ))
15	3	ralrimiva 2550	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
16	14, 15, 5	rspcdva 2848	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ)
17	16	abscld 11192	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℝ)
18	5	nnzd 9376	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
19	18	peano2zd 9380	. . . . . 6 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
20		eluzelz 9539	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
21	6, 20	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
22	19, 21	fzfigd 10433	. . . . 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 8934	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
26	25	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ∈ ℝ)
27		peano2re 8095	. . . . . . . . 9 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
28	26, 27	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑀 + 1) ∈ ℝ)
29		elfzelz 10027	. . . . . . . . . 10 ⊢ (𝑖 ∈ ((𝑀 + 1)...𝑁) → 𝑖 ∈ ℤ)
30	29	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑖 ∈ ℤ)
31	30	zred 9377	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑖 ∈ ℝ)
32	26	lep1d 8890	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ≤ (𝑀 + 1))
33		elfzle1 10029	. . . . . . . . 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 8083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ≤ 𝑖)
36		znn0sub 9320	. . . . . . . 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 10673	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝐴↑(𝑖 − 𝑀)) ∈ ℝ)
40	22, 39	fsumrecl 11411	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) ∈ ℝ)
41	17, 40	remulcld 7990	. . 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 9695	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
45	44	reclt1d 9712	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
46	42, 45	mpbid 147	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (1 / 𝐴))
47		1re 7958	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
48	44	rprecred 9710	. . . . . . . . . . 11 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
49		difrp 9694	. . . . . . . . . . 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 9709	. . . . . . . 8 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ⁺)
53	52, 44	rpdivcld 9716	. . . . . . 7 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ⁺)
54		fveq2 5517	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
55	54	eleq1d 2246	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘1) ∈ ℂ))
56		1nn 8932	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
57	56	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℕ)
58	55, 15, 57	rspcdva 2848	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
59	58	abscld 11192	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℝ)
60	58	absge0d 11195	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘1)))
61	59, 60	ge0p1rpd 9729	. . . . . . 7 ⊢ (𝜑 → ((abs‘(𝐹‘1)) + 1) ∈ ℝ⁺)
62	53, 61	rpmulcld 9715	. . . . . 6 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ⁺)
63	62	rpred 9698	. . . . 5 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ)
64	63, 5	nndivred 8971	. . . 4 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀) ∈ ℝ)
65		1red 7974	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
66	65, 23	resubcld 8340	. . . . . . 7 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ)
67	23, 65	posdifd 8491	. . . . . . . 8 ⊢ (𝜑 → (𝐴 < 1 ↔ 0 < (1 − 𝐴)))
68	42, 67	mpbid 147	. . . . . . 7 ⊢ (𝜑 → 0 < (1 − 𝐴))
69	66, 68	elrpd 9695	. . . . . 6 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ⁺)
70	44, 69	rpdivcld 9716	. . . . 5 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℝ⁺)
71	70	rpred 9698	. . . 4 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℝ)
72	64, 71	remulcld 7990	. . 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 11536	. . . . 5 ⊢ (𝜑 → ((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀)) = Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖))
75	74	fveq2d 5521	. . . 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 11537	. . . 4 ⊢ (𝜑 → (abs‘Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖)) ≤ ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀))))
77	75, 76	eqbrtrd 4027	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) ≤ ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀))))
78	16	absge0d 11195	. . . 4 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘𝑀)))
79	23, 42, 43, 3, 73, 5	cvgratnnlemfm 11539	. . . 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 9375	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑖 − 𝑀) ∈ ℤ)
82	80, 81	rpexpcld 10680	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝐴↑(𝑖 − 𝑀)) ∈ ℝ⁺)
83	82	rpge0d 9702	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 0 ≤ (𝐴↑(𝑖 − 𝑀)))
84	22, 39, 83	fsumge0 11469	. . . 4 ⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)))
85	23, 42, 43, 3, 73, 5, 6	cvgratnnlemsumlt 11538	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) < (𝐴 / (1 − 𝐴)))
86	17, 64, 40, 71, 78, 79, 84, 85	ltmul12ad 8900	. . 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 8084	. 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 7988	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℂ)
89	71	recnd 7988	. . 3 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℂ)
90	5	nncnd 8935	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℂ)
91	5	nnap0d 8967	. . 3 ⊢ (𝜑 → 𝑀 # 0)
92	88, 89, 90, 91	div23apd 8787	. 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 4033	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 1353 ∈ wcel 2148 class class class wbr 4005 ‘cfv 5218 (class class class)co 5877 ℂcc 7811 ℝcr 7812 0cc0 7813 1c1 7814 + caddc 7816 · cmul 7818 < clt 7994 ≤ cle 7995 − cmin 8130 / cdiv 8631 ℕcn 8921 ℕ₀cn0 9178 ℤcz 9255 ℤ_≥cuz 9530 ℝ⁺crp 9655 ...cfz 10010 seqcseq 10447 ↑cexp 10521 abscabs 11008 Σcsu 11363

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-frec 6394 df-1o 6419 df-oadd 6423 df-er 6537 df-en 6743 df-dom 6744 df-fin 6745 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-ico 9896 df-fz 10011 df-fzo 10145 df-seqfrec 10448 df-exp 10522 df-ihash 10758 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-clim 11289 df-sumdc 11364

This theorem is referenced by: cvgratnn 11541

W3C validator