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Mirrors > Home > MPE Home > Th. List > Mathboxes > m1expoddALTV | Structured version Visualization version GIF version |
Description: Exponentiation of -1 by an odd power. (Contributed by AV, 6-Jul-2020.) |
Ref | Expression |
---|---|
m1expoddALTV | ⊢ (𝑁 ∈ Odd → (-1↑𝑁) = -1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddz 42367 | . . . . 5 ⊢ (𝑁 ∈ Odd → 𝑁 ∈ ℤ) | |
2 | 1 | zcnd 11811 | . . . 4 ⊢ (𝑁 ∈ Odd → 𝑁 ∈ ℂ) |
3 | npcan1 10779 | . . . . 5 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) | |
4 | 3 | eqcomd 2831 | . . . 4 ⊢ (𝑁 ∈ ℂ → 𝑁 = ((𝑁 − 1) + 1)) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑁 ∈ Odd → 𝑁 = ((𝑁 − 1) + 1)) |
6 | 5 | oveq2d 6921 | . 2 ⊢ (𝑁 ∈ Odd → (-1↑𝑁) = (-1↑((𝑁 − 1) + 1))) |
7 | neg1cn 11472 | . . . 4 ⊢ -1 ∈ ℂ | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝑁 ∈ Odd → -1 ∈ ℂ) |
9 | neg1ne0 11474 | . . . 4 ⊢ -1 ≠ 0 | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑁 ∈ Odd → -1 ≠ 0) |
11 | peano2zm 11748 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
12 | 1, 11 | syl 17 | . . 3 ⊢ (𝑁 ∈ Odd → (𝑁 − 1) ∈ ℤ) |
13 | 8, 10, 12 | expp1zd 13311 | . 2 ⊢ (𝑁 ∈ Odd → (-1↑((𝑁 − 1) + 1)) = ((-1↑(𝑁 − 1)) · -1)) |
14 | oddm1eveni 42378 | . . . . 5 ⊢ (𝑁 ∈ Odd → (𝑁 − 1) ∈ Even ) | |
15 | m1expevenALTV 42383 | . . . . 5 ⊢ ((𝑁 − 1) ∈ Even → (-1↑(𝑁 − 1)) = 1) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ (𝑁 ∈ Odd → (-1↑(𝑁 − 1)) = 1) |
17 | 16 | oveq1d 6920 | . . 3 ⊢ (𝑁 ∈ Odd → ((-1↑(𝑁 − 1)) · -1) = (1 · -1)) |
18 | 8 | mulid2d 10375 | . . 3 ⊢ (𝑁 ∈ Odd → (1 · -1) = -1) |
19 | 17, 18 | eqtrd 2861 | . 2 ⊢ (𝑁 ∈ Odd → ((-1↑(𝑁 − 1)) · -1) = -1) |
20 | 6, 13, 19 | 3eqtrd 2865 | 1 ⊢ (𝑁 ∈ Odd → (-1↑𝑁) = -1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 (class class class)co 6905 ℂcc 10250 0cc0 10252 1c1 10253 + caddc 10255 · cmul 10257 − cmin 10585 -cneg 10586 ℤcz 11704 ↑cexp 13154 Even ceven 42360 Odd codd 42361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-n0 11619 df-z 11705 df-uz 11969 df-seq 13096 df-exp 13155 df-even 42362 df-odd 42363 |
This theorem is referenced by: (None) |
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