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Theorem cvgratnnlemfm 11521

Description: Lemma for cvgratnn 11523. (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 5511	. . . . 5 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
2	1	eleq1d 2246	. . . 4 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ))
3		cvgratnn.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
4	3	ralrimiva 2550	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
5		cvgratnnlemfm.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6	2, 4, 5	rspcdva 2846	. . 3 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ)
7	6	abscld 11174	. 2 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℝ)
8		cvgratnn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9		cvgratnn.gt0	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝐴)
10	8, 9	gt0ap0d 8576	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 # 0)
11	8, 10	rerecclapd 8780	. . . . . . . . 9 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
12		1red 7963	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ)
13	11, 12	resubcld 8328	. . . . . . . 8 ⊢ (𝜑 → ((1 / 𝐴) − 1) ∈ ℝ)
14		cvgratnn.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < 1)
15	8, 9	elrpd 9680	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
16	15	reclt1d 9697	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
17	14, 16	mpbid 147	. . . . . . . . 9 ⊢ (𝜑 → 1 < (1 / 𝐴))
18	12, 11	posdifd 8479	. . . . . . . . 9 ⊢ (𝜑 → (1 < (1 / 𝐴) ↔ 0 < ((1 / 𝐴) − 1)))
19	17, 18	mpbid 147	. . . . . . . 8 ⊢ (𝜑 → 0 < ((1 / 𝐴) − 1))
20	13, 19	elrpd 9680	. . . . . . 7 ⊢ (𝜑 → ((1 / 𝐴) − 1) ∈ ℝ⁺)
21	20	rpreccld 9694	. . . . . 6 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ⁺)
22	21, 15	rpdivcld 9701	. . . . 5 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ⁺)
23	22	rpred 9683	. . . 4 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ)
24		fveq2 5511	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
25	24	eleq1d 2246	. . . . . 6 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘1) ∈ ℂ))
26		1nn 8919	. . . . . . 7 ⊢ 1 ∈ ℕ
27	26	a1i 9	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ)
28	25, 4, 27	rspcdva 2846	. . . . 5 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
29	28	abscld 11174	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℝ)
30	23, 29	remulcld 7978	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) ∈ ℝ)
31	30, 5	nndivred 8958	. 2 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀) ∈ ℝ)
32		peano2re 8083	. . . . 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 7978	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ)
35	34, 5	nndivred 8958	. 2 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀) ∈ ℝ)
36		nnm1nn0 9206	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ₀)
37	5, 36	syl 14	. . . . 5 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ₀)
38	8, 37	reexpcld 10656	. . . 4 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) ∈ ℝ)
39	29, 38	remulcld 7978	. . 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 11516	. . 3 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))))
42	23, 5	nndivred 8958	. . . . 5 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀) ∈ ℝ)
43	28	absge0d 11177	. . . . 5 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘1)))
44	8	recnd 7976	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
45	5	nnzd 9363	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
46	44, 10, 45	expm1apd 10649	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) = ((𝐴↑𝑀) / 𝐴))
47	5	nnnn0d 9218	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
48	8, 47	reexpcld 10656	. . . . . . . . 9 ⊢ (𝜑 → (𝐴↑𝑀) ∈ ℝ)
49	21	rpred 9683	. . . . . . . . . 10 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ)
50	49, 5	nndivred 8958	. . . . . . . . 9 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝑀) ∈ ℝ)
51	8, 14, 9, 5	cvgratnnlembern 11515	. . . . . . . . 9 ⊢ (𝜑 → (𝐴↑𝑀) < ((1 / ((1 / 𝐴) − 1)) / 𝑀))
52	48, 50, 15, 51	ltdiv1dd 9741	. . . . . . . 8 ⊢ (𝜑 → ((𝐴↑𝑀) / 𝐴) < (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴))
53	46, 52	eqbrtrd 4022	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) < (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴))
54	49	recnd 7976	. . . . . . . 8 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℂ)
55	5	nncnd 8922	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ)
56	5	nnap0d 8954	. . . . . . . 8 ⊢ (𝜑 → 𝑀 # 0)
57	54, 55, 44, 56, 10	divdiv32apd 8762	. . . . . . 7 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴) = (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
58	53, 57	breqtrd 4026	. . . . . 6 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) < (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
59	38, 42, 58	ltled 8066	. . . . 5 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) ≤ (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
60	38, 42, 29, 43, 59	lemul2ad 8886	. . . 4 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))) ≤ ((abs‘(𝐹‘1)) · (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀)))
61	29	recnd 7976	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℂ)
62	23	recnd 7976	. . . . . . 7 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℂ)
63	61, 62	mulcomd 7969	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · ((1 / ((1 / 𝐴) − 1)) / 𝐴)) = (((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))))
64	63	oveq1d 5884	. . . . 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 8772	. . . . 5 ⊢ (𝜑 → (((abs‘(𝐹‘1)) · ((1 / ((1 / 𝐴) − 1)) / 𝐴)) / 𝑀) = ((abs‘(𝐹‘1)) · (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀)))
66	64, 65	eqtr3d 2212	. . . 4 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀) = ((abs‘(𝐹‘1)) · (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀)))
67	60, 66	breqtrrd 4028	. . 3 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))) ≤ ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀))
68	7, 39, 31, 41, 67	letrd 8071	. 2 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀))
69	5	nnrpd 9681	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
70	29	ltp1d 8876	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘1)) < ((abs‘(𝐹‘1)) + 1))
71	29, 33, 22, 70	ltmul2dd 9740	. . 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 9741	. 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 8072	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 104 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 ‘cfv 5212 (class class class)co 5869 ℂcc 7800 ℝcr 7801 0cc0 7802 1c1 7803 + caddc 7805 · cmul 7807 < clt 7982 ≤ cle 7983 − cmin 8118 / cdiv 8618 ℕcn 8908 ℕ₀cn0 9165 ↑cexp 10505 abscabs 10990

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 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922

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 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-rp 9641 df-seqfrec 10432 df-exp 10506 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992

This theorem is referenced by: cvgratnnlemrate 11522

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