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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 11404 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴))) |
5 | eqcom 2159 | . . 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 7894 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
7 | 1, 3 | expcld 10551 | . . . . . 6 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
8 | apsym 8481 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 # 1 ↔ 1 # 𝐴)) | |
9 | 1, 6, 8 | syl2anc 409 | . . . . . . 7 ⊢ (𝜑 → (𝐴 # 1 ↔ 1 # 𝐴)) |
10 | 2, 9 | mpbid 146 | . . . . . 6 ⊢ (𝜑 → 1 # 𝐴) |
11 | 6, 7, 6, 1, 10 | div2subapd 8710 | . . . . 5 ⊢ (𝜑 → ((1 − (𝐴↑𝑁)) / (1 − 𝐴)) = (((𝐴↑𝑁) − 1) / (𝐴 − 1))) |
12 | 11 | eqeq1d 2166 | . . . 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 8141 | . . . . . 6 ⊢ ((𝐴↑𝑁) ∈ ℂ → ((𝐴↑𝑁) − 1) ∈ ℂ) | |
14 | 7, 13 | syl 14 | . . . . 5 ⊢ (𝜑 → ((𝐴↑𝑁) − 1) ∈ ℂ) |
15 | 0zd 9179 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ) | |
16 | 3 | nn0zd 9284 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
17 | peano2zm 9205 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
18 | 16, 17 | syl 14 | . . . . . . 7 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
19 | 15, 18 | fzfigd 10330 | . . . . . 6 ⊢ (𝜑 → (0...(𝑁 − 1)) ∈ Fin) |
20 | 1 | adantr 274 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝐴 ∈ ℂ) |
21 | elfznn0 10016 | . . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ0) | |
22 | 21 | adantl 275 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ0) |
23 | 20, 22 | expcld 10551 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴↑𝑘) ∈ ℂ) |
24 | 19, 23 | fsumcl 11297 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) ∈ ℂ) |
25 | peano2cnm 8141 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 − 1) ∈ ℂ) | |
26 | 1, 25 | syl 14 | . . . . 5 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ) |
27 | 1, 6, 2 | subap0d 8519 | . . . . 5 ⊢ (𝜑 → (𝐴 − 1) # 0) |
28 | 14, 24, 26, 27 | divmulap2d 8697 | . . . 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 1335 ∈ wcel 2128 class class class wbr 3965 (class class class)co 5824 ℂcc 7730 0cc0 7732 1c1 7733 · cmul 7737 − cmin 8046 # cap 8456 / cdiv 8545 ℕ0cn0 9090 ℤcz 9167 ...cfz 9912 ↑cexp 10418 Σcsu 11250 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 ax-caucvg 7852 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-isom 5179 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-frec 6338 df-1o 6363 df-oadd 6367 df-er 6480 df-en 6686 df-dom 6687 df-fin 6688 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-4 8894 df-n0 9091 df-z 9168 df-uz 9440 df-q 9529 df-rp 9561 df-fz 9913 df-fzo 10042 df-seqfrec 10345 df-exp 10419 df-ihash 10650 df-cj 10742 df-re 10743 df-im 10744 df-rsqrt 10898 df-abs 10899 df-clim 11176 df-sumdc 11251 |
This theorem is referenced by: (None) |
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