cvgratnnlemsumlt - Intuitionistic Logic Explorer

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Theorem cvgratnnlemsumlt 11536

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))))
cvgratnn.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
cvgratnn.n	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))

**Assertion**
Ref	Expression
cvgratnnlemsumlt	⊢ (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) < (𝐴 / (1 − 𝐴)))

Distinct variable groups: 𝐴,𝑘 𝑘,𝐹 𝑘,𝑁 𝜑,𝑘 𝐴,𝑖,𝑘 𝑖,𝑀,𝑘 𝑖,𝑁 𝜑,𝑖 Allowed substitution hint: 𝐹(𝑖)

**Proof of Theorem cvgratnnlemsumlt**
Step	Hyp	Ref	Expression
1		cvgratnn.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ)
2	1	nnzd 9374	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		1zzd 9280	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
4		cvgratnn.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		eluzelz 9537	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
6	4, 5	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7	6, 2	zsubcld 9380	. . . 4 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℤ)
8		cvgratnn.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9	8	recnd 7986	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
10	9	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 𝑀))) → 𝐴 ∈ ℂ)
11		elfznn 10054	. . . . . . 7 ⊢ (𝑘 ∈ (1...(𝑁 − 𝑀)) → 𝑘 ∈ ℕ)
12	11	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 𝑀))) → 𝑘 ∈ ℕ)
13	12	nnnn0d 9229	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 𝑀))) → 𝑘 ∈ ℕ₀)
14	10, 13	expcld 10654	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 𝑀))) → (𝐴↑𝑘) ∈ ℂ)
15		oveq2 5883	. . . 4 ⊢ (𝑘 = (𝑖 − 𝑀) → (𝐴↑𝑘) = (𝐴↑(𝑖 − 𝑀)))
16	2, 3, 7, 14, 15	fsumshft 11452	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (1...(𝑁 − 𝑀))(𝐴↑𝑘) = Σ𝑖 ∈ ((1 + 𝑀)...((𝑁 − 𝑀) + 𝑀))(𝐴↑(𝑖 − 𝑀)))
17		1cnd 7973	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
18	1	nncnd 8933	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ)
19	17, 18	addcomd 8108	. . . . 5 ⊢ (𝜑 → (1 + 𝑀) = (𝑀 + 1))
20	6	zcnd 9376	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
21	20, 18	npcand 8272	. . . . 5 ⊢ (𝜑 → ((𝑁 − 𝑀) + 𝑀) = 𝑁)
22	19, 21	oveq12d 5893	. . . 4 ⊢ (𝜑 → ((1 + 𝑀)...((𝑁 − 𝑀) + 𝑀)) = ((𝑀 + 1)...𝑁))
23	22	sumeq1d 11374	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ ((1 + 𝑀)...((𝑁 − 𝑀) + 𝑀))(𝐴↑(𝑖 − 𝑀)) = Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)))
24	16, 23	eqtrd 2210	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (1...(𝑁 − 𝑀))(𝐴↑𝑘) = Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)))
25		fzval3 10204	. . . . 5 ⊢ ((𝑁 − 𝑀) ∈ ℤ → (1...(𝑁 − 𝑀)) = (1..^((𝑁 − 𝑀) + 1)))
26	25	sumeq1d 11374	. . . 4 ⊢ ((𝑁 − 𝑀) ∈ ℤ → Σ𝑘 ∈ (1...(𝑁 − 𝑀))(𝐴↑𝑘) = Σ𝑘 ∈ (1..^((𝑁 − 𝑀) + 1))(𝐴↑𝑘))
27	7, 26	syl 14	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (1...(𝑁 − 𝑀))(𝐴↑𝑘) = Σ𝑘 ∈ (1..^((𝑁 − 𝑀) + 1))(𝐴↑𝑘))
28		1red 7972	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
29		cvgratnn.4	. . . . . 6 ⊢ (𝜑 → 𝐴 < 1)
30	8, 28, 29	ltapd 8595	. . . . 5 ⊢ (𝜑 → 𝐴 # 1)
31		1nn0 9192	. . . . . 6 ⊢ 1 ∈ ℕ₀
32	31	a1i 9	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
33	7	peano2zd 9378	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 𝑀) + 1) ∈ ℤ)
34		eluzle 9540	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
35	4, 34	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
36	6	zred 9375	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
37	1	nnred 8932	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
38	36, 37	subge0d 8492	. . . . . . . 8 ⊢ (𝜑 → (0 ≤ (𝑁 − 𝑀) ↔ 𝑀 ≤ 𝑁))
39	35, 38	mpbird 167	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑁 − 𝑀))
40	7	zred 9375	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℝ)
41	28, 40	addge02d 8491	. . . . . . 7 ⊢ (𝜑 → (0 ≤ (𝑁 − 𝑀) ↔ 1 ≤ ((𝑁 − 𝑀) + 1)))
42	39, 41	mpbid 147	. . . . . 6 ⊢ (𝜑 → 1 ≤ ((𝑁 − 𝑀) + 1))
43		eluz2 9534	. . . . . 6 ⊢ (((𝑁 − 𝑀) + 1) ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ ((𝑁 − 𝑀) + 1) ∈ ℤ ∧ 1 ≤ ((𝑁 − 𝑀) + 1)))
44	3, 33, 42, 43	syl3anbrc 1181	. . . . 5 ⊢ (𝜑 → ((𝑁 − 𝑀) + 1) ∈ (ℤ_≥‘1))
45	9, 30, 32, 44	geosergap 11514	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (1..^((𝑁 − 𝑀) + 1))(𝐴↑𝑘) = (((𝐴↑1) − (𝐴↑((𝑁 − 𝑀) + 1))) / (1 − 𝐴)))
46	9	exp1d 10649	. . . . . . 7 ⊢ (𝜑 → (𝐴↑1) = 𝐴)
47	46, 8	eqeltrd 2254	. . . . . 6 ⊢ (𝜑 → (𝐴↑1) ∈ ℝ)
48		cvgratnn.gt0	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝐴)
49	8, 48	elrpd 9693	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
50	49, 33	rpexpcld 10678	. . . . . . 7 ⊢ (𝜑 → (𝐴↑((𝑁 − 𝑀) + 1)) ∈ ℝ⁺)
51	50	rpred 9696	. . . . . 6 ⊢ (𝜑 → (𝐴↑((𝑁 − 𝑀) + 1)) ∈ ℝ)
52	47, 51	resubcld 8338	. . . . 5 ⊢ (𝜑 → ((𝐴↑1) − (𝐴↑((𝑁 − 𝑀) + 1))) ∈ ℝ)
53	28, 8	resubcld 8338	. . . . . 6 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ)
54	8, 28	posdifd 8489	. . . . . . 7 ⊢ (𝜑 → (𝐴 < 1 ↔ 0 < (1 − 𝐴)))
55	29, 54	mpbid 147	. . . . . 6 ⊢ (𝜑 → 0 < (1 − 𝐴))
56	53, 55	elrpd 9693	. . . . 5 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ⁺)
57	46	oveq1d 5890	. . . . . 6 ⊢ (𝜑 → ((𝐴↑1) − (𝐴↑((𝑁 − 𝑀) + 1))) = (𝐴 − (𝐴↑((𝑁 − 𝑀) + 1))))
58	8, 50	ltsubrpd 9729	. . . . . 6 ⊢ (𝜑 → (𝐴 − (𝐴↑((𝑁 − 𝑀) + 1))) < 𝐴)
59	57, 58	eqbrtrd 4026	. . . . 5 ⊢ (𝜑 → ((𝐴↑1) − (𝐴↑((𝑁 − 𝑀) + 1))) < 𝐴)
60	52, 8, 56, 59	ltdiv1dd 9754	. . . 4 ⊢ (𝜑 → (((𝐴↑1) − (𝐴↑((𝑁 − 𝑀) + 1))) / (1 − 𝐴)) < (𝐴 / (1 − 𝐴)))
61	45, 60	eqbrtrd 4026	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (1..^((𝑁 − 𝑀) + 1))(𝐴↑𝑘) < (𝐴 / (1 − 𝐴)))
62	27, 61	eqbrtrd 4026	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (1...(𝑁 − 𝑀))(𝐴↑𝑘) < (𝐴 / (1 − 𝐴)))
63	24, 62	eqbrtrrd 4028	1 ⊢ (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) < (𝐴 / (1 − 𝐴)))

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 ℤcz 9253 ℤ_≥cuz 9528 ...cfz 10008 ..^cfzo 10142 ↑cexp 10519 abscabs 11006 Σcsu 11361

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-isom 5226 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-irdg 6371 df-frec 6392 df-1o 6417 df-oadd 6421 df-er 6535 df-en 6741 df-dom 6742 df-fin 6743 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-q 9620 df-rp 9654 df-fz 10009 df-fzo 10143 df-seqfrec 10446 df-exp 10520 df-ihash 10756 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008 df-clim 11287 df-sumdc 11362

This theorem is referenced by: cvgratnnlemrate 11538

W3C validator