cvgratnnlemrate - Intuitionistic Logic Explorer

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Theorem cvgratnnlemrate 11956

Description: Lemma for cvgratnn 11957. (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 9719	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 9434	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
3		cvgratnn.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
4	1, 2, 3	serf 10665	. . . . . 6 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
5		cvgratnnlemrate.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6		cvgratnnlemrate.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
7		eluznn 9756	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑁 ∈ ℕ)
8	5, 6, 7	syl2anc 411	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9	4, 8	ffvelcdmd 5739	. . . . 5 ⊢ (𝜑 → (seq1( + , 𝐹)‘𝑁) ∈ ℂ)
10	4, 5	ffvelcdmd 5739	. . . . 5 ⊢ (𝜑 → (seq1( + , 𝐹)‘𝑀) ∈ ℂ)
11	9, 10	subcld 8418	. . . 4 ⊢ (𝜑 → ((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀)) ∈ ℂ)
12	11	abscld 11607	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) ∈ ℝ)
13		fveq2 5599	. . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
14	13	eleq1d 2276	. . . . . 6 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ))
15	3	ralrimiva 2581	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
16	14, 15, 5	rspcdva 2889	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ)
17	16	abscld 11607	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℝ)
18	5	nnzd 9529	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
19	18	peano2zd 9533	. . . . . 6 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
20		eluzelz 9692	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
21	6, 20	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
22	19, 21	fzfigd 10613	. . . . 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 9084	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
26	25	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ∈ ℝ)
27		peano2re 8243	. . . . . . . . 9 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
28	26, 27	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑀 + 1) ∈ ℝ)
29		elfzelz 10182	. . . . . . . . . 10 ⊢ (𝑖 ∈ ((𝑀 + 1)...𝑁) → 𝑖 ∈ ℤ)
30	29	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑖 ∈ ℤ)
31	30	zred 9530	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑖 ∈ ℝ)
32	26	lep1d 9039	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ≤ (𝑀 + 1))
33		elfzle1 10184	. . . . . . . . 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 8231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ≤ 𝑖)
36		znn0sub 9473	. . . . . . . 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 10872	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝐴↑(𝑖 − 𝑀)) ∈ ℝ)
40	22, 39	fsumrecl 11827	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) ∈ ℝ)
41	17, 40	remulcld 8138	. . 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 9850	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
45	44	reclt1d 9867	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
46	42, 45	mpbid 147	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (1 / 𝐴))
47		1re 8106	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
48	44	rprecred 9865	. . . . . . . . . . 11 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
49		difrp 9849	. . . . . . . . . . 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 9864	. . . . . . . 8 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ⁺)
53	52, 44	rpdivcld 9871	. . . . . . 7 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ⁺)
54		fveq2 5599	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
55	54	eleq1d 2276	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘1) ∈ ℂ))
56		1nn 9082	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
57	56	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℕ)
58	55, 15, 57	rspcdva 2889	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
59	58	abscld 11607	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℝ)
60	58	absge0d 11610	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘1)))
61	59, 60	ge0p1rpd 9884	. . . . . . 7 ⊢ (𝜑 → ((abs‘(𝐹‘1)) + 1) ∈ ℝ⁺)
62	53, 61	rpmulcld 9870	. . . . . 6 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ⁺)
63	62	rpred 9853	. . . . 5 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ)
64	63, 5	nndivred 9121	. . . 4 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀) ∈ ℝ)
65		1red 8122	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
66	65, 23	resubcld 8488	. . . . . . 7 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ)
67	23, 65	posdifd 8640	. . . . . . . 8 ⊢ (𝜑 → (𝐴 < 1 ↔ 0 < (1 − 𝐴)))
68	42, 67	mpbid 147	. . . . . . 7 ⊢ (𝜑 → 0 < (1 − 𝐴))
69	66, 68	elrpd 9850	. . . . . 6 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ⁺)
70	44, 69	rpdivcld 9871	. . . . 5 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℝ⁺)
71	70	rpred 9853	. . . 4 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℝ)
72	64, 71	remulcld 8138	. . 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 11952	. . . . 5 ⊢ (𝜑 → ((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀)) = Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖))
75	74	fveq2d 5603	. . . 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 11953	. . . 4 ⊢ (𝜑 → (abs‘Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖)) ≤ ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀))))
77	75, 76	eqbrtrd 4081	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) ≤ ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀))))
78	16	absge0d 11610	. . . 4 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘𝑀)))
79	23, 42, 43, 3, 73, 5	cvgratnnlemfm 11955	. . . 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 9528	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝑖 − 𝑀) ∈ ℤ)
82	80, 81	rpexpcld 10879	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → (𝐴↑(𝑖 − 𝑀)) ∈ ℝ⁺)
83	82	rpge0d 9857	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑀 + 1)...𝑁)) → 0 ≤ (𝐴↑(𝑖 − 𝑀)))
84	22, 39, 83	fsumge0 11885	. . . 4 ⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)))
85	23, 42, 43, 3, 73, 5, 6	cvgratnnlemsumlt 11954	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) < (𝐴 / (1 − 𝐴)))
86	17, 64, 40, 71, 78, 79, 84, 85	ltmul12ad 9049	. . 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 8232	. 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 8136	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℂ)
89	71	recnd 8136	. . 3 ⊢ (𝜑 → (𝐴 / (1 − 𝐴)) ∈ ℂ)
90	5	nncnd 9085	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℂ)
91	5	nnap0d 9117	. . 3 ⊢ (𝜑 → 𝑀 # 0)
92	88, 89, 90, 91	div23apd 8936	. 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 4087	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 1373 ∈ wcel 2178 class class class wbr 4059 ‘cfv 5290 (class class class)co 5967 ℂcc 7958 ℝcr 7959 0cc0 7960 1c1 7961 + caddc 7963 · cmul 7965 < clt 8142 ≤ cle 8143 − cmin 8278 / cdiv 8780 ℕcn 9071 ℕ₀cn0 9330 ℤcz 9407 ℤ_≥cuz 9683 ℝ⁺crp 9810 ...cfz 10165 seqcseq 10629 ↑cexp 10720 abscabs 11423 Σcsu 11779

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077 ax-pre-mulext 8078 ax-arch 8079 ax-caucvg 8080

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rmo 2494 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-if 3580 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-id 4358 df-po 4361 df-iso 4362 df-iord 4431 df-on 4433 df-ilim 4434 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-isom 5299 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-1st 6249 df-2nd 6250 df-recs 6414 df-irdg 6479 df-frec 6500 df-1o 6525 df-oadd 6529 df-er 6643 df-en 6851 df-dom 6852 df-fin 6853 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-reap 8683 df-ap 8690 df-div 8781 df-inn 9072 df-2 9130 df-3 9131 df-4 9132 df-n0 9331 df-z 9408 df-uz 9684 df-q 9776 df-rp 9811 df-ico 10051 df-fz 10166 df-fzo 10300 df-seqfrec 10630 df-exp 10721 df-ihash 10958 df-cj 11268 df-re 11269 df-im 11270 df-rsqrt 11424 df-abs 11425 df-clim 11705 df-sumdc 11780

This theorem is referenced by: cvgratnn 11957

W3C validator