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Mirrors > Home > ILE Home > Th. List > m1expo | GIF version |
Description: Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.) |
Ref | Expression |
---|---|
m1expo | ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (-1↑𝑁) = -1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odd2np1 11570 | . . 3 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁)) | |
2 | oveq2 5782 | . . . . . . 7 ⊢ (𝑁 = ((2 · 𝑛) + 1) → (-1↑𝑁) = (-1↑((2 · 𝑛) + 1))) | |
3 | 2 | eqcoms 2142 | . . . . . 6 ⊢ (((2 · 𝑛) + 1) = 𝑁 → (-1↑𝑁) = (-1↑((2 · 𝑛) + 1))) |
4 | neg1cn 8825 | . . . . . . . . . 10 ⊢ -1 ∈ ℂ | |
5 | 4 | a1i 9 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → -1 ∈ ℂ) |
6 | neg1ap0 8829 | . . . . . . . . . 10 ⊢ -1 # 0 | |
7 | 6 | a1i 9 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → -1 # 0) |
8 | 2z 9082 | . . . . . . . . . . 11 ⊢ 2 ∈ ℤ | |
9 | 8 | a1i 9 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 2 ∈ ℤ) |
10 | id 19 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℤ) | |
11 | 9, 10 | zmulcld 9179 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → (2 · 𝑛) ∈ ℤ) |
12 | 5, 7, 11 | expp1zapd 10433 | . . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (-1↑((2 · 𝑛) + 1)) = ((-1↑(2 · 𝑛)) · -1)) |
13 | m1expeven 10340 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → (-1↑(2 · 𝑛)) = 1) | |
14 | 13 | oveq1d 5789 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → ((-1↑(2 · 𝑛)) · -1) = (1 · -1)) |
15 | 4 | mulid2i 7769 | . . . . . . . . 9 ⊢ (1 · -1) = -1 |
16 | 14, 15 | syl6eq 2188 | . . . . . . . 8 ⊢ (𝑛 ∈ ℤ → ((-1↑(2 · 𝑛)) · -1) = -1) |
17 | 12, 16 | eqtrd 2172 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → (-1↑((2 · 𝑛) + 1)) = -1) |
18 | 17 | adantl 275 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (-1↑((2 · 𝑛) + 1)) = -1) |
19 | 3, 18 | sylan9eqr 2194 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ((2 · 𝑛) + 1) = 𝑁) → (-1↑𝑁) = -1) |
20 | 19 | ex 114 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((2 · 𝑛) + 1) = 𝑁 → (-1↑𝑁) = -1)) |
21 | 20 | rexlimdva 2549 | . . 3 ⊢ (𝑁 ∈ ℤ → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁 → (-1↑𝑁) = -1)) |
22 | 1, 21 | sylbid 149 | . 2 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 → (-1↑𝑁) = -1)) |
23 | 22 | imp 123 | 1 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (-1↑𝑁) = -1) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 class class class wbr 3929 (class class class)co 5774 ℂcc 7618 0cc0 7620 1c1 7621 + caddc 7623 · cmul 7625 -cneg 7934 # cap 8343 2c2 8771 ℤcz 9054 ↑cexp 10292 ∥ cdvds 11493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-xor 1354 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-n0 8978 df-z 9055 df-uz 9327 df-seqfrec 10219 df-exp 10293 df-dvds 11494 |
This theorem is referenced by: (None) |
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