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Theorem cvgratnnlemfm 11492

Description: Lemma for cvgratnn 11494. (Contributed by Jim Kingdon, 23-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))))
cvgratnnlemfm.m	⊢ (𝜑 → 𝑀 ∈ ℕ)

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

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

**Proof of Theorem cvgratnnlemfm**
Step	Hyp	Ref	Expression
1		fveq2 5496	. . . . 5 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
2	1	eleq1d 2239	. . . 4 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ))
3		cvgratnn.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
4	3	ralrimiva 2543	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
5		cvgratnnlemfm.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6	2, 4, 5	rspcdva 2839	. . 3 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ)
7	6	abscld 11145	. 2 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℝ)
8		cvgratnn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9		cvgratnn.gt0	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝐴)
10	8, 9	gt0ap0d 8548	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 # 0)
11	8, 10	rerecclapd 8751	. . . . . . . . 9 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
12		1red 7935	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ)
13	11, 12	resubcld 8300	. . . . . . . 8 ⊢ (𝜑 → ((1 / 𝐴) − 1) ∈ ℝ)
14		cvgratnn.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < 1)
15	8, 9	elrpd 9650	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
16	15	reclt1d 9667	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
17	14, 16	mpbid 146	. . . . . . . . 9 ⊢ (𝜑 → 1 < (1 / 𝐴))
18	12, 11	posdifd 8451	. . . . . . . . 9 ⊢ (𝜑 → (1 < (1 / 𝐴) ↔ 0 < ((1 / 𝐴) − 1)))
19	17, 18	mpbid 146	. . . . . . . 8 ⊢ (𝜑 → 0 < ((1 / 𝐴) − 1))
20	13, 19	elrpd 9650	. . . . . . 7 ⊢ (𝜑 → ((1 / 𝐴) − 1) ∈ ℝ⁺)
21	20	rpreccld 9664	. . . . . 6 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ⁺)
22	21, 15	rpdivcld 9671	. . . . 5 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ⁺)
23	22	rpred 9653	. . . 4 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ)
24		fveq2 5496	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
25	24	eleq1d 2239	. . . . . 6 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘1) ∈ ℂ))
26		1nn 8889	. . . . . . 7 ⊢ 1 ∈ ℕ
27	26	a1i 9	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ)
28	25, 4, 27	rspcdva 2839	. . . . 5 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
29	28	abscld 11145	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℝ)
30	23, 29	remulcld 7950	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) ∈ ℝ)
31	30, 5	nndivred 8928	. 2 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀) ∈ ℝ)
32		peano2re 8055	. . . . 5 ⊢ ((abs‘(𝐹‘1)) ∈ ℝ → ((abs‘(𝐹‘1)) + 1) ∈ ℝ)
33	29, 32	syl 14	. . . 4 ⊢ (𝜑 → ((abs‘(𝐹‘1)) + 1) ∈ ℝ)
34	23, 33	remulcld 7950	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ)
35	34, 5	nndivred 8928	. 2 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀) ∈ ℝ)
36		nnm1nn0 9176	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ₀)
37	5, 36	syl 14	. . . . 5 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ₀)
38	8, 37	reexpcld 10626	. . . 4 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) ∈ ℝ)
39	29, 38	remulcld 7950	. . 3 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))) ∈ ℝ)
40		cvgratnn.7	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))
41	8, 14, 9, 3, 40, 5	cvgratnnlemnexp 11487	. . 3 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))))
42	23, 5	nndivred 8928	. . . . 5 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀) ∈ ℝ)
43	28	absge0d 11148	. . . . 5 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘1)))
44	8	recnd 7948	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
45	5	nnzd 9333	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
46	44, 10, 45	expm1apd 10619	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) = ((𝐴↑𝑀) / 𝐴))
47	5	nnnn0d 9188	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
48	8, 47	reexpcld 10626	. . . . . . . . 9 ⊢ (𝜑 → (𝐴↑𝑀) ∈ ℝ)
49	21	rpred 9653	. . . . . . . . . 10 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ)
50	49, 5	nndivred 8928	. . . . . . . . 9 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝑀) ∈ ℝ)
51	8, 14, 9, 5	cvgratnnlembern 11486	. . . . . . . . 9 ⊢ (𝜑 → (𝐴↑𝑀) < ((1 / ((1 / 𝐴) − 1)) / 𝑀))
52	48, 50, 15, 51	ltdiv1dd 9711	. . . . . . . 8 ⊢ (𝜑 → ((𝐴↑𝑀) / 𝐴) < (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴))
53	46, 52	eqbrtrd 4011	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) < (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴))
54	49	recnd 7948	. . . . . . . 8 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℂ)
55	5	nncnd 8892	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ)
56	5	nnap0d 8924	. . . . . . . 8 ⊢ (𝜑 → 𝑀 # 0)
57	54, 55, 44, 56, 10	divdiv32apd 8733	. . . . . . 7 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴) = (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
58	53, 57	breqtrd 4015	. . . . . 6 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) < (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
59	38, 42, 58	ltled 8038	. . . . 5 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) ≤ (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
60	38, 42, 29, 43, 59	lemul2ad 8856	. . . 4 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))) ≤ ((abs‘(𝐹‘1)) · (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀)))
61	29	recnd 7948	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℂ)
62	23	recnd 7948	. . . . . . 7 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℂ)
63	61, 62	mulcomd 7941	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · ((1 / ((1 / 𝐴) − 1)) / 𝐴)) = (((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))))
64	63	oveq1d 5868	. . . . 5 ⊢ (𝜑 → (((abs‘(𝐹‘1)) · ((1 / ((1 / 𝐴) − 1)) / 𝐴)) / 𝑀) = ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀))
65	61, 62, 55, 56	divassapd 8743	. . . . 5 ⊢ (𝜑 → (((abs‘(𝐹‘1)) · ((1 / ((1 / 𝐴) − 1)) / 𝐴)) / 𝑀) = ((abs‘(𝐹‘1)) · (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀)))
66	64, 65	eqtr3d 2205	. . . 4 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀) = ((abs‘(𝐹‘1)) · (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀)))
67	60, 66	breqtrrd 4017	. . 3 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))) ≤ ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀))
68	7, 39, 31, 41, 67	letrd 8043	. 2 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀))
69	5	nnrpd 9651	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
70	29	ltp1d 8846	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘1)) < ((abs‘(𝐹‘1)) + 1))
71	29, 33, 22, 70	ltmul2dd 9710	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) < (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)))
72	30, 34, 69, 71	ltdiv1dd 9711	. 2 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀) < ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀))
73	7, 31, 35, 68, 72	lelttrd 8044	1 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) < ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 ℂcc 7772 ℝcr 7773 0cc0 7774 1c1 7775 + caddc 7777 · cmul 7779 < clt 7954 ≤ cle 7955 − cmin 8090 / cdiv 8589 ℕcn 8878 ℕ₀cn0 9135 ↑cexp 10475 abscabs 10961

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-rp 9611 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963

This theorem is referenced by: cvgratnnlemrate 11493

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