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Mirrors > Home > MPE Home > Th. List > m1expo | Structured version Visualization version 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 15799 | . . 3 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁)) | |
2 | oveq2 7191 | . . . . . 6 ⊢ (𝑁 = ((2 · 𝑛) + 1) → (-1↑𝑁) = (-1↑((2 · 𝑛) + 1))) | |
3 | 2 | eqcoms 2747 | . . . . 5 ⊢ (((2 · 𝑛) + 1) = 𝑁 → (-1↑𝑁) = (-1↑((2 · 𝑛) + 1))) |
4 | neg1cn 11843 | . . . . . . . . 9 ⊢ -1 ∈ ℂ | |
5 | 4 | a1i 11 | . . . . . . . 8 ⊢ (𝑛 ∈ ℤ → -1 ∈ ℂ) |
6 | neg1ne0 11845 | . . . . . . . . 9 ⊢ -1 ≠ 0 | |
7 | 6 | a1i 11 | . . . . . . . 8 ⊢ (𝑛 ∈ ℤ → -1 ≠ 0) |
8 | 2z 12108 | . . . . . . . . . 10 ⊢ 2 ∈ ℤ | |
9 | 8 | a1i 11 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → 2 ∈ ℤ) |
10 | id 22 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℤ) | |
11 | 9, 10 | zmulcld 12187 | . . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (2 · 𝑛) ∈ ℤ) |
12 | 5, 7, 11 | expp1zd 13624 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → (-1↑((2 · 𝑛) + 1)) = ((-1↑(2 · 𝑛)) · -1)) |
13 | m1expeven 13581 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → (-1↑(2 · 𝑛)) = 1) | |
14 | 13 | oveq1d 7198 | . . . . . . . 8 ⊢ (𝑛 ∈ ℤ → ((-1↑(2 · 𝑛)) · -1) = (1 · -1)) |
15 | 4 | mulid2i 10737 | . . . . . . . 8 ⊢ (1 · -1) = -1 |
16 | 14, 15 | eqtrdi 2790 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → ((-1↑(2 · 𝑛)) · -1) = -1) |
17 | 12, 16 | eqtrd 2774 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → (-1↑((2 · 𝑛) + 1)) = -1) |
18 | 17 | adantl 485 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (-1↑((2 · 𝑛) + 1)) = -1) |
19 | 3, 18 | sylan9eqr 2796 | . . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ((2 · 𝑛) + 1) = 𝑁) → (-1↑𝑁) = -1) |
20 | 19 | rexlimdva2 3198 | . . 3 ⊢ (𝑁 ∈ ℤ → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁 → (-1↑𝑁) = -1)) |
21 | 1, 20 | sylbid 243 | . 2 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 → (-1↑𝑁) = -1)) |
22 | 21 | imp 410 | 1 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (-1↑𝑁) = -1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ∃wrex 3055 class class class wbr 5040 (class class class)co 7183 ℂcc 10626 0cc0 10628 1c1 10629 + caddc 10631 · cmul 10633 -cneg 10962 2c2 11784 ℤcz 12075 ↑cexp 13534 ∥ cdvds 15712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 df-nn 11730 df-2 11792 df-n0 11990 df-z 12076 df-uz 12338 df-seq 13474 df-exp 13535 df-dvds 15713 |
This theorem is referenced by: 2lgsoddprm 26165 negexpidd 40117 |
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