cvgratnnlemrate - Intuitionistic Logic Explorer

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Theorem cvgratnnlemrate 12057

Description: Lemma for cvgratnn 12058. (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 9770	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 9484	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
3		cvgratnn.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
4	1, 2, 3	serf 10717	. . . . . 6 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
5		cvgratnnlemrate.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6		cvgratnnlemrate.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
7		eluznn 9807	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑁 ∈ ℕ)
8	5, 6, 7	syl2anc 411	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9	4, 8	ffvelcdmd 5773	. . . . 5 ⊢ (𝜑 → (seq1( + , 𝐹)‘𝑁) ∈ ℂ)
10	4, 5	ffvelcdmd 5773	. . . . 5 ⊢ (𝜑 → (seq1( + , 𝐹)‘𝑀) ∈ ℂ)
11	9, 10	subcld 8468	. . . 4 ⊢ (𝜑 → ((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀)) ∈ ℂ)
12	11	abscld 11708	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) ∈ ℝ)
13		fveq2 5629	. . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
14	13	eleq1d 2298	. . . . . 6 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ))
15	3	ralrimiva 2603	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
16	14, 15, 5	rspcdva 2912	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ)
17	16	abscld 11708	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℝ)
18	5	nnzd 9579	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
19	18	peano2zd 9583	. . . . . 6 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
20		eluzelz 9743	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
21	6, 20	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
22	19, 21	fzfigd 10665	. . . . 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 9134	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
26	25	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ∈ ℝ)
27		peano2re 8293	. . . . . . . . 9 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
28	26, 27	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑀 + 1) ∈ ℝ)
29		elfzelz 10233	. . . . . . . . . 10 ⊢ (𝑖 ∈ ((𝑀 + 1)...𝑁) → 𝑖 ∈ ℤ)
30	29	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑖 ∈ ℤ)
31	30	zred 9580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑖 ∈ ℝ)
32	26	lep1d 9089	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ≤ (𝑀 + 1))
33		elfzle1 10235	. . . . . . . . 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 8281	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ≤ 𝑖)
36		znn0sub 9523	. . . . . . . 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 10924	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝐴↑(𝑖 − 𝑀)) ∈ ℝ)
40	22, 39	fsumrecl 11928	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) ∈ ℝ)
41	17, 40	remulcld 8188	. . 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 9901	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
45	44	reclt1d 9918	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
46	42, 45	mpbid 147	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (1 / 𝐴))
47		1re 8156	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
48	44	rprecred 9916	. . . . . . . . . . 11 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
49		difrp 9900	. . . . . . . . . . 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 9915	. . . . . . . 8 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ⁺)
53	52, 44	rpdivcld 9922	. . . . . . 7 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ⁺)
54		fveq2 5629	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
55	54	eleq1d 2298	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘1) ∈ ℂ))
56		1nn 9132	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
57	56	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℕ)
58	55, 15, 57	rspcdva 2912	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
59	58	abscld 11708	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℝ)
60	58	absge0d 11711	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘1)))
61	59, 60	ge0p1rpd 9935	. . . . . . 7 ⊢ (𝜑 → ((abs‘(𝐹‘1)) + 1) ∈ ℝ⁺)
62	53, 61	rpmulcld 9921	. . . . . 6 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ⁺)
63	62	rpred 9904	. . . . 5 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ)
64	63, 5	nndivred 9171	. . . 4 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀) ∈ ℝ)
65		1red 8172	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
66	65, 23	resubcld 8538	. . . . . . 7 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ)
67	23, 65	posdifd 8690	. . . . . . . 8 ⊢ (𝜑 → (𝐴 < 1 ↔ 0 < (1 − 𝐴)))
68	42, 67	mpbid 147	. . . . . . 7 ⊢ (𝜑 → 0 < (1 − 𝐴))
69	66, 68	elrpd 9901	. . . . . 6 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ⁺)
70	44, 69	rpdivcld 9922	. . . . 5 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℝ⁺)
71	70	rpred 9904	. . . 4 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℝ)
72	64, 71	remulcld 8188	. . 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 12053	. . . . 5 ⊢ (𝜑 → ((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀)) = Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖))
75	74	fveq2d 5633	. . . 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 12054	. . . 4 ⊢ (𝜑 → (abs‘Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖)) ≤ ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀))))
77	75, 76	eqbrtrd 4105	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) ≤ ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀))))
78	16	absge0d 11711	. . . 4 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘𝑀)))
79	23, 42, 43, 3, 73, 5	cvgratnnlemfm 12056	. . . 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 9578	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑖 − 𝑀) ∈ ℤ)
82	80, 81	rpexpcld 10931	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝐴↑(𝑖 − 𝑀)) ∈ ℝ⁺)
83	82	rpge0d 9908	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 0 ≤ (𝐴↑(𝑖 − 𝑀)))
84	22, 39, 83	fsumge0 11986	. . . 4 ⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)))
85	23, 42, 43, 3, 73, 5, 6	cvgratnnlemsumlt 12055	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) < (𝐴 / (1 − 𝐴)))
86	17, 64, 40, 71, 78, 79, 84, 85	ltmul12ad 9099	. . 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 8282	. 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 8186	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℂ)
89	71	recnd 8186	. . 3 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℂ)
90	5	nncnd 9135	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℂ)
91	5	nnap0d 9167	. . 3 ⊢ (𝜑 → 𝑀 # 0)
92	88, 89, 90, 91	div23apd 8986	. 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 4111	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 1395 ∈ wcel 2200 class class class wbr 4083 ‘cfv 5318 (class class class)co 6007 ℂcc 8008 ℝcr 8009 0cc0 8010 1c1 8011 + caddc 8013 · cmul 8015 < clt 8192 ≤ cle 8193 − cmin 8328 / cdiv 8830 ℕcn 9121 ℕ₀cn0 9380 ℤcz 9457 ℤ_≥cuz 9733 ℝ⁺crp 9861 ...cfz 10216 seqcseq 10681 ↑cexp 10772 abscabs 11524 Σcsu 11880

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128 ax-arch 8129 ax-caucvg 8130

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-frec 6543 df-1o 6568 df-oadd 6572 df-er 6688 df-en 6896 df-dom 6897 df-fin 6898 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-inn 9122 df-2 9180 df-3 9181 df-4 9182 df-n0 9381 df-z 9458 df-uz 9734 df-q 9827 df-rp 9862 df-ico 10102 df-fz 10217 df-fzo 10351 df-seqfrec 10682 df-exp 10773 df-ihash 11010 df-cj 11369 df-re 11370 df-im 11371 df-rsqrt 11525 df-abs 11526 df-clim 11806 df-sumdc 11881

This theorem is referenced by: cvgratnn 12058

W3C validator