cvgratnnlemfm - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > cvgratnnlemfm		GIF version

Theorem cvgratnnlemfm 12321

Description: Lemma for cvgratnn 12323. (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 5698	. . . . 5 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
2	1	eleq1d 2307	. . . 4 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ))
3		cvgratnn.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
4	3	ralrimiva 2623	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
5		cvgratnnlemfm.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6	2, 4, 5	rspcdva 2934	. . 3 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ)
7	6	abscld 11970	. 2 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℝ)
8		cvgratnn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9		cvgratnn.gt0	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝐴)
10	8, 9	gt0ap0d 8965	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 # 0)
11	8, 10	rerecclapd 9172	. . . . . . . . 9 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
12		1red 8347	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ)
13	11, 12	resubcld 8715	. . . . . . . 8 ⊢ (𝜑 → ((1 / 𝐴) − 1) ∈ ℝ)
14		cvgratnn.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < 1)
15	8, 9	elrpd 10111	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
16	15	reclt1d 10128	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
17	14, 16	mpbid 147	. . . . . . . . 9 ⊢ (𝜑 → 1 < (1 / 𝐴))
18	12, 11	posdifd 8867	. . . . . . . . 9 ⊢ (𝜑 → (1 < (1 / 𝐴) ↔ 0 < ((1 / 𝐴) − 1)))
19	17, 18	mpbid 147	. . . . . . . 8 ⊢ (𝜑 → 0 < ((1 / 𝐴) − 1))
20	13, 19	elrpd 10111	. . . . . . 7 ⊢ (𝜑 → ((1 / 𝐴) − 1) ∈ ℝ⁺)
21	20	rpreccld 10125	. . . . . 6 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ⁺)
22	21, 15	rpdivcld 10132	. . . . 5 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ⁺)
23	22	rpred 10114	. . . 4 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℝ)
24		fveq2 5698	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
25	24	eleq1d 2307	. . . . . 6 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘1) ∈ ℂ))
26		1nn 9324	. . . . . . 7 ⊢ 1 ∈ ℕ
27	26	a1i 9	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ)
28	25, 4, 27	rspcdva 2934	. . . . 5 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
29	28	abscld 11970	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℝ)
30	23, 29	remulcld 8362	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) ∈ ℝ)
31	30, 5	nndivred 9363	. 2 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀) ∈ ℝ)
32		peano2re 8469	. . . . 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 8362	. . 3 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) ∈ ℝ)
35	34, 5	nndivred 9363	. 2 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀) ∈ ℝ)
36		nnm1nn0 9615	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ₀)
37	5, 36	syl 14	. . . . 5 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ₀)
38	8, 37	reexpcld 11149	. . . 4 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) ∈ ℝ)
39	29, 38	remulcld 8362	. . 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 12316	. . 3 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))))
42	23, 5	nndivred 9363	. . . . 5 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀) ∈ ℝ)
43	28	absge0d 11973	. . . . 5 ⊢ (𝜑 → 0 ≤ (abs‘(𝐹‘1)))
44	8	recnd 8360	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
45	5	nnzd 9778	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
46	44, 10, 45	expm1apd 11142	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) = ((𝐴↑𝑀) / 𝐴))
47	5	nnnn0d 9631	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
48	8, 47	reexpcld 11149	. . . . . . . . 9 ⊢ (𝜑 → (𝐴↑𝑀) ∈ ℝ)
49	21	rpred 10114	. . . . . . . . . 10 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℝ)
50	49, 5	nndivred 9363	. . . . . . . . 9 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝑀) ∈ ℝ)
51	8, 14, 9, 5	cvgratnnlembern 12315	. . . . . . . . 9 ⊢ (𝜑 → (𝐴↑𝑀) < ((1 / ((1 / 𝐴) − 1)) / 𝑀))
52	48, 50, 15, 51	ltdiv1dd 10172	. . . . . . . 8 ⊢ (𝜑 → ((𝐴↑𝑀) / 𝐴) < (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴))
53	46, 52	eqbrtrd 4153	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) < (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴))
54	49	recnd 8360	. . . . . . . 8 ⊢ (𝜑 → (1 / ((1 / 𝐴) − 1)) ∈ ℂ)
55	5	nncnd 9327	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ)
56	5	nnap0d 9359	. . . . . . . 8 ⊢ (𝜑 → 𝑀 # 0)
57	54, 55, 44, 56, 10	divdiv32apd 9154	. . . . . . 7 ⊢ (𝜑 → (((1 / ((1 / 𝐴) − 1)) / 𝑀) / 𝐴) = (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
58	53, 57	breqtrd 4157	. . . . . 6 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) < (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
59	38, 42, 58	ltled 8452	. . . . 5 ⊢ (𝜑 → (𝐴↑(𝑀 − 1)) ≤ (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀))
60	38, 42, 29, 43, 59	lemul2ad 9278	. . . 4 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))) ≤ ((abs‘(𝐹‘1)) · (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀)))
61	29	recnd 8360	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℂ)
62	23	recnd 8360	. . . . . . 7 ⊢ (𝜑 → ((1 / ((1 / 𝐴) − 1)) / 𝐴) ∈ ℂ)
63	61, 62	mulcomd 8353	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · ((1 / ((1 / 𝐴) − 1)) / 𝐴)) = (((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))))
64	63	oveq1d 6103	. . . . 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 9164	. . . . 5 ⊢ (𝜑 → (((abs‘(𝐹‘1)) · ((1 / ((1 / 𝐴) − 1)) / 𝐴)) / 𝑀) = ((abs‘(𝐹‘1)) · (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀)))
66	64, 65	eqtr3d 2273	. . . 4 ⊢ (𝜑 → ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀) = ((abs‘(𝐹‘1)) · (((1 / ((1 / 𝐴) − 1)) / 𝐴) / 𝑀)))
67	60, 66	breqtrrd 4159	. . 3 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑀 − 1))) ≤ ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀))
68	7, 39, 31, 41, 67	letrd 8457	. 2 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · (abs‘(𝐹‘1))) / 𝑀))
69	5	nnrpd 10112	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
70	29	ltp1d 9268	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘1)) < ((abs‘(𝐹‘1)) + 1))
71	29, 33, 22, 70	ltmul2dd 10171	. . 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 10172	. 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 8458	1 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) < ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 class class class wbr 4131 ‘cfv 5380 (class class class)co 6088 ℂcc 8183 ℝcr 8184 0cc0 8185 1c1 8186 + caddc 8188 · cmul 8190 < clt 8366 ≤ cle 8367 − cmin 8504 / cdiv 9010 ℕcn 9313 ℕ₀cn0 9574 ↑cexp 10996 abscabs 11785

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-n0 9575 df-z 9656 df-uz 9938 df-rp 10072 df-seqfrec 10906 df-exp 10997 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787

This theorem is used by: cvgratnnlemrate 12322

W3C validator