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Mirrors > Home > MPE Home > Th. List > pwm1geoser | Structured version Visualization version 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.) (Proof shortened by AV, 19-Aug-2021.) |
Ref | Expression |
---|---|
pwm1geoser.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
pwm1geoser.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
pwm1geoser | ⊢ (𝜑 → ((𝐴↑𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwm1geoser.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
2 | 1 | nn0zd 11898 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
3 | 1exp 13273 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝜑 → (1↑𝑁) = 1) |
5 | 4 | eqcomd 2784 | . . 3 ⊢ (𝜑 → 1 = (1↑𝑁)) |
6 | 5 | oveq2d 6992 | . 2 ⊢ (𝜑 → ((𝐴↑𝑁) − 1) = ((𝐴↑𝑁) − (1↑𝑁))) |
7 | pwm1geoser.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
8 | 1cnd 10434 | . . 3 ⊢ (𝜑 → 1 ∈ ℂ) | |
9 | pwdif 15083 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴↑𝑁) − (1↑𝑁)) = ((𝐴 − 1) · Σ𝑘 ∈ (0..^𝑁)((𝐴↑𝑘) · (1↑((𝑁 − 𝑘) − 1))))) | |
10 | 1, 7, 8, 9 | syl3anc 1351 | . 2 ⊢ (𝜑 → ((𝐴↑𝑁) − (1↑𝑁)) = ((𝐴 − 1) · Σ𝑘 ∈ (0..^𝑁)((𝐴↑𝑘) · (1↑((𝑁 − 𝑘) − 1))))) |
11 | fzoval 12855 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (0..^𝑁) = (0...(𝑁 − 1))) | |
12 | 2, 11 | syl 17 | . . . 4 ⊢ (𝜑 → (0..^𝑁) = (0...(𝑁 − 1))) |
13 | 2 | adantr 473 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ) |
14 | elfzoelz 12854 | . . . . . . . . 9 ⊢ (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ ℤ) | |
15 | 14 | adantl 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℤ) |
16 | 13, 15 | zsubcld 11905 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑁 − 𝑘) ∈ ℤ) |
17 | peano2zm 11838 | . . . . . . 7 ⊢ ((𝑁 − 𝑘) ∈ ℤ → ((𝑁 − 𝑘) − 1) ∈ ℤ) | |
18 | 1exp 13273 | . . . . . . 7 ⊢ (((𝑁 − 𝑘) − 1) ∈ ℤ → (1↑((𝑁 − 𝑘) − 1)) = 1) | |
19 | 16, 17, 18 | 3syl 18 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (1↑((𝑁 − 𝑘) − 1)) = 1) |
20 | 19 | oveq2d 6992 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐴↑𝑘) · (1↑((𝑁 − 𝑘) − 1))) = ((𝐴↑𝑘) · 1)) |
21 | 7 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐴 ∈ ℂ) |
22 | elfzonn0 12897 | . . . . . . . 8 ⊢ (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ ℕ0) | |
23 | 22 | adantl 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℕ0) |
24 | 21, 23 | expcld 13325 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐴↑𝑘) ∈ ℂ) |
25 | 24 | mulid1d 10457 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐴↑𝑘) · 1) = (𝐴↑𝑘)) |
26 | 20, 25 | eqtrd 2814 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐴↑𝑘) · (1↑((𝑁 − 𝑘) − 1))) = (𝐴↑𝑘)) |
27 | 12, 26 | sumeq12dv 14923 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0..^𝑁)((𝐴↑𝑘) · (1↑((𝑁 − 𝑘) − 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘)) |
28 | 27 | oveq2d 6992 | . 2 ⊢ (𝜑 → ((𝐴 − 1) · Σ𝑘 ∈ (0..^𝑁)((𝐴↑𝑘) · (1↑((𝑁 − 𝑘) − 1)))) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘))) |
29 | 6, 10, 28 | 3eqtrd 2818 | 1 ⊢ (𝜑 → ((𝐴↑𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 (class class class)co 6976 ℂcc 10333 0cc0 10335 1c1 10336 · cmul 10340 − cmin 10670 ℕ0cn0 11707 ℤcz 11793 ...cfz 12708 ..^cfzo 12849 ↑cexp 13244 Σcsu 14903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-oi 8769 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-n0 11708 df-z 11794 df-uz 12059 df-rp 12205 df-fz 12709 df-fzo 12850 df-seq 13185 df-exp 13245 df-hash 13506 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-clim 14706 df-sum 14904 |
This theorem is referenced by: lighneallem3 43146 |
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