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Theorem cvgratnnlemfm 11537

Description: Lemma for cvgratnn 11539. (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 5516	. . . . 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 2847	. . 3 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ)
7	6	abscld 11190	. 2 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℝ)
8		cvgratnn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9		cvgratnn.gt0	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝐴)
10	8, 9	gt0ap0d 8586	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 # 0)
11	8, 10	rerecclapd 8791	. . . . . . . . 9 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
12		1red 7972	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ)
13	11, 12	resubcld 8338	. . . . . . . 8 ⊢ (𝜑 → ((1 / 𝐴) − 1) ∈ ℝ)
14		cvgratnn.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < 1)
15	8, 9	elrpd 9693	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
16	15	reclt1d 9710	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
17	14, 16	mpbid 147	. . . . . . . . 9 ⊢ (𝜑 → 1 < (1 / 𝐴))
18	12, 11	posdifd 8489	. . . . . . . . 9 ⊢ (𝜑 → (1 < (1 / 𝐴) ↔ 0 < ((1 / 𝐴) − 1)))
19	17, 18	mpbid 147	. . . . . . . 8 ⊢ (𝜑 → 0 < ((1 / 𝐴) − 1))
20	13, 19	elrpd 9693	. . . . . . 7 ⊢ (𝜑 → ((1 / 𝐴) − 1) ∈ ℝ⁺)
21	20	rpreccld 9707	. . . . . 6 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ⁺)
22	21, 15	rpdivcld 9714	. . . . 5 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ⁺)
23	22	rpred 9696	. . . 4 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ)
24		fveq2 5516	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
25	24	eleq1d 2246	. . . . . 6 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘1) ∈ ℂ))
26		1nn 8930	. . . . . . 7 ⊢ 1 ∈ ℕ
27	26	a1i 9	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ)
28	25, 4, 27	rspcdva 2847	. . . . 5 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
29	28	abscld 11190	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℝ)
30	23, 29	remulcld 7988	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) ∈ ℝ)
31	30, 5	nndivred 8969	. 2 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀) ∈ ℝ)
32		peano2re 8093	. . . . 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 7988	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ)
35	34, 5	nndivred 8969	. 2 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀) ∈ ℝ)
36		nnm1nn0 9217	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ₀)
37	5, 36	syl 14	. . . . 5 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ₀)
38	8, 37	reexpcld 10671	. . . 4 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) ∈ ℝ)
39	29, 38	remulcld 7988	. . 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 11532	. . 3 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))))
42	23, 5	nndivred 8969	. . . . 5 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀) ∈ ℝ)
43	28	absge0d 11193	. . . . 5 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘1)))
44	8	recnd 7986	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
45	5	nnzd 9374	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
46	44, 10, 45	expm1apd 10664	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) = ((𝐴↑𝑀) / 𝐴))
47	5	nnnn0d 9229	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
48	8, 47	reexpcld 10671	. . . . . . . . 9 ⊢ (𝜑 → (𝐴↑𝑀) ∈ ℝ)
49	21	rpred 9696	. . . . . . . . . 10 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ)
50	49, 5	nndivred 8969	. . . . . . . . 9 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝑀) ∈ ℝ)
51	8, 14, 9, 5	cvgratnnlembern 11531	. . . . . . . . 9 ⊢ (𝜑 → (𝐴↑𝑀) < ((1 / ((1 / 𝐴) − 1)) / 𝑀))
52	48, 50, 15, 51	ltdiv1dd 9754	. . . . . . . 8 ⊢ (𝜑 → ((𝐴↑𝑀) / 𝐴) < (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴))
53	46, 52	eqbrtrd 4026	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) < (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴))
54	49	recnd 7986	. . . . . . . 8 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℂ)
55	5	nncnd 8933	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ)
56	5	nnap0d 8965	. . . . . . . 8 ⊢ (𝜑 → 𝑀 # 0)
57	54, 55, 44, 56, 10	divdiv32apd 8773	. . . . . . 7 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴) = (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
58	53, 57	breqtrd 4030	. . . . . 6 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) < (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
59	38, 42, 58	ltled 8076	. . . . 5 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) ≤ (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
60	38, 42, 29, 43, 59	lemul2ad 8897	. . . 4 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))) ≤ ((abs‘(𝐹‘1)) · (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀)))
61	29	recnd 7986	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℂ)
62	23	recnd 7986	. . . . . . 7 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℂ)
63	61, 62	mulcomd 7979	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · ((1 / ((1 / 𝐴) − 1)) / 𝐴)) = (((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))))
64	63	oveq1d 5890	. . . . 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 8783	. . . . 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 4032	. . 3 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))) ≤ ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀))
68	7, 39, 31, 41, 67	letrd 8081	. 2 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀))
69	5	nnrpd 9694	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
70	29	ltp1d 8887	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘1)) < ((abs‘(𝐹‘1)) + 1))
71	29, 33, 22, 70	ltmul2dd 9753	. . 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 9754	. 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 8082	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 4004 ‘cfv 5217 (class class class)co 5875 ℂcc 7809 ℝcr 7810 0cc0 7811 1c1 7812 + caddc 7814 · cmul 7816 < clt 7992 ≤ cle 7993 − cmin 8128 / cdiv 8629 ℕcn 8919 ℕ₀cn0 9176 ↑cexp 10519 abscabs 11006

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 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 ax-arch 7930 ax-caucvg 7931

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 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-frec 6392 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-n0 9177 df-z 9254 df-uz 9529 df-rp 9654 df-seqfrec 10446 df-exp 10520 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008

This theorem is referenced by: cvgratnnlemrate 11538

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