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Mirrors > Home > MPE Home > Th. List > oddpwp1fsum | Structured version Visualization version GIF version |
Description: An odd power of a number increased by 1 expressed by a product with a finite sum. (Contributed by AV, 15-Aug-2021.) |
Ref | Expression |
---|---|
pwp1fsum.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
pwp1fsum.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
oddpwp1fsum.n | ⊢ (𝜑 → ¬ 2 ∥ 𝑁) |
Ref | Expression |
---|---|
oddpwp1fsum | ⊢ (𝜑 → ((𝐴↑𝑁) + 1) = ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑁 − 1))((-1↑𝑘) · (𝐴↑𝑘)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddpwp1fsum.n | . . . . . 6 ⊢ (𝜑 → ¬ 2 ∥ 𝑁) | |
2 | pwp1fsum.n | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
3 | 2 | nnzd 12080 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
4 | oddm1even 15686 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1))) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1))) |
6 | 1, 5 | mpbid 234 | . . . . 5 ⊢ (𝜑 → 2 ∥ (𝑁 − 1)) |
7 | m1expe 15719 | . . . . 5 ⊢ (2 ∥ (𝑁 − 1) → (-1↑(𝑁 − 1)) = 1) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (-1↑(𝑁 − 1)) = 1) |
9 | 8 | oveq1d 7165 | . . 3 ⊢ (𝜑 → ((-1↑(𝑁 − 1)) · (𝐴↑𝑁)) = (1 · (𝐴↑𝑁))) |
10 | 9 | oveq1d 7165 | . 2 ⊢ (𝜑 → (((-1↑(𝑁 − 1)) · (𝐴↑𝑁)) + 1) = ((1 · (𝐴↑𝑁)) + 1)) |
11 | pwp1fsum.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
12 | 11, 2 | pwp1fsum 15736 | . 2 ⊢ (𝜑 → (((-1↑(𝑁 − 1)) · (𝐴↑𝑁)) + 1) = ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑁 − 1))((-1↑𝑘) · (𝐴↑𝑘)))) |
13 | 2 | nnnn0d 11949 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
14 | 11, 13 | expcld 13504 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
15 | 14 | mulid2d 10653 | . . 3 ⊢ (𝜑 → (1 · (𝐴↑𝑁)) = (𝐴↑𝑁)) |
16 | 15 | oveq1d 7165 | . 2 ⊢ (𝜑 → ((1 · (𝐴↑𝑁)) + 1) = ((𝐴↑𝑁) + 1)) |
17 | 10, 12, 16 | 3eqtr3rd 2865 | 1 ⊢ (𝜑 → ((𝐴↑𝑁) + 1) = ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑁 − 1))((-1↑𝑘) · (𝐴↑𝑘)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 (class class class)co 7150 ℂcc 10529 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 − cmin 10864 -cneg 10865 ℕcn 11632 2c2 11686 ℤcz 11975 ...cfz 12886 ↑cexp 13423 Σcsu 15036 ∥ cdvds 15601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-dvds 15602 |
This theorem is referenced by: lighneallem4b 43767 lighneallem4 43768 |
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