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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 12160 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
4 | oddm1even 15781 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1))) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1))) |
6 | 1, 5 | mpbid 235 | . . . . 5 ⊢ (𝜑 → 2 ∥ (𝑁 − 1)) |
7 | m1expe 15812 | . . . . 5 ⊢ (2 ∥ (𝑁 − 1) → (-1↑(𝑁 − 1)) = 1) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (-1↑(𝑁 − 1)) = 1) |
9 | 8 | oveq1d 7179 | . . 3 ⊢ (𝜑 → ((-1↑(𝑁 − 1)) · (𝐴↑𝑁)) = (1 · (𝐴↑𝑁))) |
10 | 9 | oveq1d 7179 | . 2 ⊢ (𝜑 → (((-1↑(𝑁 − 1)) · (𝐴↑𝑁)) + 1) = ((1 · (𝐴↑𝑁)) + 1)) |
11 | pwp1fsum.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
12 | 11, 2 | pwp1fsum 15829 | . 2 ⊢ (𝜑 → (((-1↑(𝑁 − 1)) · (𝐴↑𝑁)) + 1) = ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑁 − 1))((-1↑𝑘) · (𝐴↑𝑘)))) |
13 | 2 | nnnn0d 12029 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
14 | 11, 13 | expcld 13595 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
15 | 14 | mulid2d 10730 | . . 3 ⊢ (𝜑 → (1 · (𝐴↑𝑁)) = (𝐴↑𝑁)) |
16 | 15 | oveq1d 7179 | . 2 ⊢ (𝜑 → ((1 · (𝐴↑𝑁)) + 1) = ((𝐴↑𝑁) + 1)) |
17 | 10, 12, 16 | 3eqtr3rd 2782 | 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 209 = wceq 1542 ∈ wcel 2113 class class class wbr 5027 (class class class)co 7164 ℂcc 10606 0cc0 10608 1c1 10609 + caddc 10611 · cmul 10613 − cmin 10941 -cneg 10942 ℕcn 11709 2c2 11764 ℤcz 12055 ...cfz 12974 ↑cexp 13514 Σcsu 15128 ∥ cdvds 15692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-inf2 9170 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-sup 8972 df-oi 9040 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-n0 11970 df-z 12056 df-uz 12318 df-rp 12466 df-fz 12975 df-fzo 13118 df-seq 13454 df-exp 13515 df-hash 13776 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-clim 14928 df-sum 15129 df-dvds 15693 |
This theorem is referenced by: lighneallem4b 44579 lighneallem4 44580 |
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