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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 10955 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴))) |
5 | eqcom 2091 | . . 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 7558 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
7 | 1, 3 | expcld 10140 | . . . . . 6 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
8 | apsym 8137 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 # 1 ↔ 1 # 𝐴)) | |
9 | 1, 6, 8 | syl2anc 404 | . . . . . . 7 ⊢ (𝜑 → (𝐴 # 1 ↔ 1 # 𝐴)) |
10 | 2, 9 | mpbid 146 | . . . . . 6 ⊢ (𝜑 → 1 # 𝐴) |
11 | 6, 7, 6, 1, 10 | div2subapd 8357 | . . . . 5 ⊢ (𝜑 → ((1 − (𝐴↑𝑁)) / (1 − 𝐴)) = (((𝐴↑𝑁) − 1) / (𝐴 − 1))) |
12 | 11 | eqeq1d 2097 | . . . 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 7802 | . . . . . 6 ⊢ ((𝐴↑𝑁) ∈ ℂ → ((𝐴↑𝑁) − 1) ∈ ℂ) | |
14 | 7, 13 | syl 14 | . . . . 5 ⊢ (𝜑 → ((𝐴↑𝑁) − 1) ∈ ℂ) |
15 | 0zd 8816 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ) | |
16 | 3 | nn0zd 8920 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
17 | peano2zm 8842 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
18 | 16, 17 | syl 14 | . . . . . . 7 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
19 | 15, 18 | fzfigd 9892 | . . . . . 6 ⊢ (𝜑 → (0...(𝑁 − 1)) ∈ Fin) |
20 | 1 | adantr 271 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝐴 ∈ ℂ) |
21 | elfznn0 9582 | . . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ0) | |
22 | 21 | adantl 272 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ0) |
23 | 20, 22 | expcld 10140 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴↑𝑘) ∈ ℂ) |
24 | 19, 23 | fsumcl 10848 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) ∈ ℂ) |
25 | peano2cnm 7802 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 − 1) ∈ ℂ) | |
26 | 1, 25 | syl 14 | . . . . 5 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ) |
27 | 1, 6, 2 | subap0d 8173 | . . . . 5 ⊢ (𝜑 → (𝐴 − 1) # 0) |
28 | 14, 24, 26, 27 | divmulap2d 8345 | . . . 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 1290 ∈ wcel 1439 class class class wbr 3851 (class class class)co 5666 ℂcc 7402 0cc0 7404 1c1 7405 · cmul 7409 − cmin 7707 # cap 8112 / cdiv 8193 ℕ0cn0 8727 ℤcz 8804 ...cfz 9478 ↑cexp 10008 Σcsu 10796 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-mulrcl 7498 ax-addcom 7499 ax-mulcom 7500 ax-addass 7501 ax-mulass 7502 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-1rid 7506 ax-0id 7507 ax-rnegex 7508 ax-precex 7509 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-apti 7514 ax-pre-ltadd 7515 ax-pre-mulgt0 7516 ax-pre-mulext 7517 ax-arch 7518 ax-caucvg 7519 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-if 3398 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-isom 5037 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-frec 6170 df-1o 6195 df-oadd 6199 df-er 6306 df-en 6512 df-dom 6513 df-fin 6514 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-reap 8106 df-ap 8113 df-div 8194 df-inn 8477 df-2 8535 df-3 8536 df-4 8537 df-n0 8728 df-z 8805 df-uz 9074 df-q 9159 df-rp 9189 df-fz 9479 df-fzo 9608 df-iseq 9907 df-seq3 9908 df-exp 10009 df-ihash 10238 df-cj 10330 df-re 10331 df-im 10332 df-rsqrt 10485 df-abs 10486 df-clim 10721 df-isum 10797 |
This theorem is referenced by: (None) |
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