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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 11497 | . . 3 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁)) | |
2 | oveq2 5750 | . . . . . . 7 ⊢ (𝑁 = ((2 · 𝑛) + 1) → (-1↑𝑁) = (-1↑((2 · 𝑛) + 1))) | |
3 | 2 | eqcoms 2120 | . . . . . 6 ⊢ (((2 · 𝑛) + 1) = 𝑁 → (-1↑𝑁) = (-1↑((2 · 𝑛) + 1))) |
4 | neg1cn 8793 | . . . . . . . . . 10 ⊢ -1 ∈ ℂ | |
5 | 4 | a1i 9 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → -1 ∈ ℂ) |
6 | neg1ap0 8797 | . . . . . . . . . 10 ⊢ -1 # 0 | |
7 | 6 | a1i 9 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → -1 # 0) |
8 | 2z 9050 | . . . . . . . . . . 11 ⊢ 2 ∈ ℤ | |
9 | 8 | a1i 9 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 2 ∈ ℤ) |
10 | id 19 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℤ) | |
11 | 9, 10 | zmulcld 9147 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → (2 · 𝑛) ∈ ℤ) |
12 | 5, 7, 11 | expp1zapd 10401 | . . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (-1↑((2 · 𝑛) + 1)) = ((-1↑(2 · 𝑛)) · -1)) |
13 | m1expeven 10308 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → (-1↑(2 · 𝑛)) = 1) | |
14 | 13 | oveq1d 5757 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → ((-1↑(2 · 𝑛)) · -1) = (1 · -1)) |
15 | 4 | mulid2i 7737 | . . . . . . . . 9 ⊢ (1 · -1) = -1 |
16 | 14, 15 | syl6eq 2166 | . . . . . . . 8 ⊢ (𝑛 ∈ ℤ → ((-1↑(2 · 𝑛)) · -1) = -1) |
17 | 12, 16 | eqtrd 2150 | . . . . . . 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 2172 | . . . . 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 2526 | . . 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 1316 ∈ wcel 1465 ∃wrex 2394 class class class wbr 3899 (class class class)co 5742 ℂcc 7586 0cc0 7588 1c1 7589 + caddc 7591 · cmul 7593 -cneg 7902 # cap 8311 2c2 8739 ℤcz 9022 ↑cexp 10260 ∥ cdvds 11420 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-xor 1339 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-n0 8946 df-z 9023 df-uz 9295 df-seqfrec 10187 df-exp 10261 df-dvds 11421 |
This theorem is referenced by: (None) |
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