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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 11821 | . . 3 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁)) | |
2 | oveq2 5859 | . . . . . . 7 ⊢ (𝑁 = ((2 · 𝑛) + 1) → (-1↑𝑁) = (-1↑((2 · 𝑛) + 1))) | |
3 | 2 | eqcoms 2173 | . . . . . 6 ⊢ (((2 · 𝑛) + 1) = 𝑁 → (-1↑𝑁) = (-1↑((2 · 𝑛) + 1))) |
4 | neg1cn 8972 | . . . . . . . . . 10 ⊢ -1 ∈ ℂ | |
5 | 4 | a1i 9 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → -1 ∈ ℂ) |
6 | neg1ap0 8976 | . . . . . . . . . 10 ⊢ -1 # 0 | |
7 | 6 | a1i 9 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → -1 # 0) |
8 | 2z 9229 | . . . . . . . . . . 11 ⊢ 2 ∈ ℤ | |
9 | 8 | a1i 9 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 2 ∈ ℤ) |
10 | id 19 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℤ) | |
11 | 9, 10 | zmulcld 9329 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → (2 · 𝑛) ∈ ℤ) |
12 | 5, 7, 11 | expp1zapd 10607 | . . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (-1↑((2 · 𝑛) + 1)) = ((-1↑(2 · 𝑛)) · -1)) |
13 | m1expeven 10512 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → (-1↑(2 · 𝑛)) = 1) | |
14 | 13 | oveq1d 5866 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → ((-1↑(2 · 𝑛)) · -1) = (1 · -1)) |
15 | 4 | mulid2i 7912 | . . . . . . . . 9 ⊢ (1 · -1) = -1 |
16 | 14, 15 | eqtrdi 2219 | . . . . . . . 8 ⊢ (𝑛 ∈ ℤ → ((-1↑(2 · 𝑛)) · -1) = -1) |
17 | 12, 16 | eqtrd 2203 | . . . . . . 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 2225 | . . . . 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 2587 | . . 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 1348 ∈ wcel 2141 ∃wrex 2449 class class class wbr 3987 (class class class)co 5851 ℂcc 7761 0cc0 7763 1c1 7764 + caddc 7766 · cmul 7768 -cneg 8080 # cap 8489 2c2 8918 ℤcz 9201 ↑cexp 10464 ∥ cdvds 11738 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-mulrcl 7862 ax-addcom 7863 ax-mulcom 7864 ax-addass 7865 ax-mulass 7866 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-1rid 7870 ax-0id 7871 ax-rnegex 7872 ax-precex 7873 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-apti 7878 ax-pre-ltadd 7879 ax-pre-mulgt0 7880 ax-pre-mulext 7881 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-xor 1371 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 df-reap 8483 df-ap 8490 df-div 8579 df-inn 8868 df-2 8926 df-n0 9125 df-z 9202 df-uz 9477 df-seqfrec 10391 df-exp 10465 df-dvds 11739 |
This theorem is referenced by: (None) |
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