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Theorem cvgratnnlemfm 12215

Description: Lemma for cvgratnn 12217. (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 5670	. . . . 5 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
2	1	eleq1d 2301	. . . 4 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ))
3		cvgratnn.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
4	3	ralrimiva 2615	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
5		cvgratnnlemfm.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6	2, 4, 5	rspcdva 2926	. . 3 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ)
7	6	abscld 11866	. 2 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℝ)
8		cvgratnn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9		cvgratnn.gt0	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝐴)
10	8, 9	gt0ap0d 8903	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 # 0)
11	8, 10	rerecclapd 9108	. . . . . . . . 9 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
12		1red 8289	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ)
13	11, 12	resubcld 8654	. . . . . . . 8 ⊢ (𝜑 → ((1 / 𝐴) − 1) ∈ ℝ)
14		cvgratnn.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < 1)
15	8, 9	elrpd 10026	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
16	15	reclt1d 10043	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
17	14, 16	mpbid 147	. . . . . . . . 9 ⊢ (𝜑 → 1 < (1 / 𝐴))
18	12, 11	posdifd 8806	. . . . . . . . 9 ⊢ (𝜑 → (1 < (1 / 𝐴) ↔ 0 < ((1 / 𝐴) − 1)))
19	17, 18	mpbid 147	. . . . . . . 8 ⊢ (𝜑 → 0 < ((1 / 𝐴) − 1))
20	13, 19	elrpd 10026	. . . . . . 7 ⊢ (𝜑 → ((1 / 𝐴) − 1) ∈ ℝ⁺)
21	20	rpreccld 10040	. . . . . 6 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ⁺)
22	21, 15	rpdivcld 10047	. . . . 5 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ⁺)
23	22	rpred 10029	. . . 4 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ)
24		fveq2 5670	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
25	24	eleq1d 2301	. . . . . 6 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘1) ∈ ℂ))
26		1nn 9248	. . . . . . 7 ⊢ 1 ∈ ℕ
27	26	a1i 9	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ)
28	25, 4, 27	rspcdva 2926	. . . . 5 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
29	28	abscld 11866	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℝ)
30	23, 29	remulcld 8304	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) ∈ ℝ)
31	30, 5	nndivred 9287	. 2 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀) ∈ ℝ)
32		peano2re 8409	. . . . 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 8304	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ)
35	34, 5	nndivred 9287	. 2 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀) ∈ ℝ)
36		nnm1nn0 9537	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ₀)
37	5, 36	syl 14	. . . . 5 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ₀)
38	8, 37	reexpcld 11052	. . . 4 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) ∈ ℝ)
39	29, 38	remulcld 8304	. . 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 12210	. . 3 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))))
42	23, 5	nndivred 9287	. . . . 5 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀) ∈ ℝ)
43	28	absge0d 11869	. . . . 5 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘1)))
44	8	recnd 8302	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
45	5	nnzd 9699	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
46	44, 10, 45	expm1apd 11045	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) = ((𝐴↑𝑀) / 𝐴))
47	5	nnnn0d 9553	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
48	8, 47	reexpcld 11052	. . . . . . . . 9 ⊢ (𝜑 → (𝐴↑𝑀) ∈ ℝ)
49	21	rpred 10029	. . . . . . . . . 10 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ)
50	49, 5	nndivred 9287	. . . . . . . . 9 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝑀) ∈ ℝ)
51	8, 14, 9, 5	cvgratnnlembern 12209	. . . . . . . . 9 ⊢ (𝜑 → (𝐴↑𝑀) < ((1 / ((1 / 𝐴) − 1)) / 𝑀))
52	48, 50, 15, 51	ltdiv1dd 10087	. . . . . . . 8 ⊢ (𝜑 → ((𝐴↑𝑀) / 𝐴) < (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴))
53	46, 52	eqbrtrd 4131	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) < (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴))
54	49	recnd 8302	. . . . . . . 8 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℂ)
55	5	nncnd 9251	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ)
56	5	nnap0d 9283	. . . . . . . 8 ⊢ (𝜑 → 𝑀 # 0)
57	54, 55, 44, 56, 10	divdiv32apd 9090	. . . . . . 7 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴) = (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
58	53, 57	breqtrd 4135	. . . . . 6 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) < (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
59	38, 42, 58	ltled 8392	. . . . 5 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) ≤ (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
60	38, 42, 29, 43, 59	lemul2ad 9214	. . . 4 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))) ≤ ((abs‘(𝐹‘1)) · (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀)))
61	29	recnd 8302	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℂ)
62	23	recnd 8302	. . . . . . 7 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℂ)
63	61, 62	mulcomd 8295	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · ((1 / ((1 / 𝐴) − 1)) / 𝐴)) = (((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))))
64	63	oveq1d 6065	. . . . 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 9100	. . . . 5 ⊢ (𝜑 → (((abs‘(𝐹‘1)) · ((1 / ((1 / 𝐴) − 1)) / 𝐴)) / 𝑀) = ((abs‘(𝐹‘1)) · (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀)))
66	64, 65	eqtr3d 2267	. . . 4 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀) = ((abs‘(𝐹‘1)) · (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀)))
67	60, 66	breqtrrd 4137	. . 3 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))) ≤ ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀))
68	7, 39, 31, 41, 67	letrd 8397	. 2 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀))
69	5	nnrpd 10027	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
70	29	ltp1d 9204	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘1)) < ((abs‘(𝐹‘1)) + 1))
71	29, 33, 22, 70	ltmul2dd 10086	. . 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 10087	. 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 8398	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 1398 ∈ wcel 2203 class class class wbr 4109 ‘cfv 5352 (class class class)co 6050 ℂcc 8125 ℝcr 8126 0cc0 8127 1c1 8128 + caddc 8130 · cmul 8132 < clt 8308 ≤ cle 8309 − cmin 8444 / cdiv 8946 ℕcn 9237 ℕ₀cn0 9496 ↑cexp 10900 abscabs 11682

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-frec 6622 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-n0 9497 df-z 9578 df-uz 9854 df-rp 9987 df-seqfrec 10810 df-exp 10901 df-cj 11527 df-re 11528 df-im 11529 df-rsqrt 11683 df-abs 11684

This theorem is referenced by: cvgratnnlemrate 12216

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