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Mirrors > Home > ILE Home > Th. List > pwm1geoserap1 | GIF version |
Description: The n-th power of a number decreased by 1 expressed by the finite geometric series 1 + 𝐴↑1 + 𝐴↑2 +... + 𝐴↑(𝑁 − 1). (Contributed by AV, 14-Aug-2021.) (Revised by Jim Kingdon, 24-Oct-2022.) |
Ref | Expression |
---|---|
pwm1geoser.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
pwm1geoser.3 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
pwm1geoserap1.ap | ⊢ (𝜑 → 𝐴 # 1) |
Ref | Expression |
---|---|
pwm1geoserap1 | ⊢ (𝜑 → ((𝐴↑𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwm1geoser.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | pwm1geoserap1.ap | . . 3 ⊢ (𝜑 → 𝐴 # 1) | |
3 | pwm1geoser.3 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
4 | 1, 2, 3 | geoserap 11279 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴))) |
5 | eqcom 2141 | . . 3 ⊢ (Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴)) ↔ ((1 − (𝐴↑𝑁)) / (1 − 𝐴)) = Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘)) | |
6 | 1cnd 7785 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
7 | 1, 3 | expcld 10427 | . . . . . 6 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
8 | apsym 8371 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 # 1 ↔ 1 # 𝐴)) | |
9 | 1, 6, 8 | syl2anc 408 | . . . . . . 7 ⊢ (𝜑 → (𝐴 # 1 ↔ 1 # 𝐴)) |
10 | 2, 9 | mpbid 146 | . . . . . 6 ⊢ (𝜑 → 1 # 𝐴) |
11 | 6, 7, 6, 1, 10 | div2subapd 8600 | . . . . 5 ⊢ (𝜑 → ((1 − (𝐴↑𝑁)) / (1 − 𝐴)) = (((𝐴↑𝑁) − 1) / (𝐴 − 1))) |
12 | 11 | eqeq1d 2148 | . . . 4 ⊢ (𝜑 → (((1 − (𝐴↑𝑁)) / (1 − 𝐴)) = Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) ↔ (((𝐴↑𝑁) − 1) / (𝐴 − 1)) = Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘))) |
13 | peano2cnm 8031 | . . . . . 6 ⊢ ((𝐴↑𝑁) ∈ ℂ → ((𝐴↑𝑁) − 1) ∈ ℂ) | |
14 | 7, 13 | syl 14 | . . . . 5 ⊢ (𝜑 → ((𝐴↑𝑁) − 1) ∈ ℂ) |
15 | 0zd 9069 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ) | |
16 | 3 | nn0zd 9174 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
17 | peano2zm 9095 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
18 | 16, 17 | syl 14 | . . . . . . 7 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
19 | 15, 18 | fzfigd 10207 | . . . . . 6 ⊢ (𝜑 → (0...(𝑁 − 1)) ∈ Fin) |
20 | 1 | adantr 274 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝐴 ∈ ℂ) |
21 | elfznn0 9897 | . . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ0) | |
22 | 21 | adantl 275 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ0) |
23 | 20, 22 | expcld 10427 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴↑𝑘) ∈ ℂ) |
24 | 19, 23 | fsumcl 11172 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) ∈ ℂ) |
25 | peano2cnm 8031 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 − 1) ∈ ℂ) | |
26 | 1, 25 | syl 14 | . . . . 5 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ) |
27 | 1, 6, 2 | subap0d 8409 | . . . . 5 ⊢ (𝜑 → (𝐴 − 1) # 0) |
28 | 14, 24, 26, 27 | divmulap2d 8587 | . . . 4 ⊢ (𝜑 → ((((𝐴↑𝑁) − 1) / (𝐴 − 1)) = Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) ↔ ((𝐴↑𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘)))) |
29 | 12, 28 | bitrd 187 | . . 3 ⊢ (𝜑 → (((1 − (𝐴↑𝑁)) / (1 − 𝐴)) = Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) ↔ ((𝐴↑𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘)))) |
30 | 5, 29 | syl5bb 191 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴)) ↔ ((𝐴↑𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘)))) |
31 | 4, 30 | mpbid 146 | 1 ⊢ (𝜑 → ((𝐴↑𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 (class class class)co 5774 ℂcc 7621 0cc0 7623 1c1 7624 · cmul 7628 − cmin 7936 # cap 8346 / cdiv 8435 ℕ0cn0 8980 ℤcz 9057 ...cfz 9793 ↑cexp 10295 Σcsu 11125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 ax-arch 7742 ax-caucvg 7743 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-div 8436 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-n0 8981 df-z 9058 df-uz 9330 df-q 9415 df-rp 9445 df-fz 9794 df-fzo 9923 df-seqfrec 10222 df-exp 10296 df-ihash 10525 df-cj 10617 df-re 10618 df-im 10619 df-rsqrt 10773 df-abs 10774 df-clim 11051 df-sumdc 11126 |
This theorem is referenced by: (None) |
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