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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 12592 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
4 | oddm1even 16293 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1))) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1))) |
6 | 1, 5 | mpbid 231 | . . . . 5 ⊢ (𝜑 → 2 ∥ (𝑁 − 1)) |
7 | m1expe 16324 | . . . . 5 ⊢ (2 ∥ (𝑁 − 1) → (-1↑(𝑁 − 1)) = 1) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (-1↑(𝑁 − 1)) = 1) |
9 | 8 | oveq1d 7427 | . . 3 ⊢ (𝜑 → ((-1↑(𝑁 − 1)) · (𝐴↑𝑁)) = (1 · (𝐴↑𝑁))) |
10 | 9 | oveq1d 7427 | . 2 ⊢ (𝜑 → (((-1↑(𝑁 − 1)) · (𝐴↑𝑁)) + 1) = ((1 · (𝐴↑𝑁)) + 1)) |
11 | pwp1fsum.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
12 | 11, 2 | pwp1fsum 16341 | . 2 ⊢ (𝜑 → (((-1↑(𝑁 − 1)) · (𝐴↑𝑁)) + 1) = ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑁 − 1))((-1↑𝑘) · (𝐴↑𝑘)))) |
13 | 2 | nnnn0d 12539 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
14 | 11, 13 | expcld 14118 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
15 | 14 | mullidd 11239 | . . 3 ⊢ (𝜑 → (1 · (𝐴↑𝑁)) = (𝐴↑𝑁)) |
16 | 15 | oveq1d 7427 | . 2 ⊢ (𝜑 → ((1 · (𝐴↑𝑁)) + 1) = ((𝐴↑𝑁) + 1)) |
17 | 10, 12, 16 | 3eqtr3rd 2780 | 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 205 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 (class class class)co 7412 ℂcc 11114 0cc0 11116 1c1 11117 + caddc 11119 · cmul 11121 − cmin 11451 -cneg 11452 ℕcn 12219 2c2 12274 ℤcz 12565 ...cfz 13491 ↑cexp 14034 Σcsu 15639 ∥ cdvds 16204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-fz 13492 df-fzo 13635 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-sum 15640 df-dvds 16205 |
This theorem is referenced by: lighneallem4b 46736 lighneallem4 46737 |
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