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Mirrors > Home > MPE Home > Th. List > m1expeven | Structured version Visualization version GIF version |
Description: Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.) |
Ref | Expression |
---|---|
m1expeven | ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 12060 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1 | 2timesd 11952 | . . 3 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) = (𝑁 + 𝑁)) |
3 | 2 | oveq2d 7180 | . 2 ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = (-1↑(𝑁 + 𝑁))) |
4 | neg1cn 11823 | . . . 4 ⊢ -1 ∈ ℂ | |
5 | neg1ne0 11825 | . . . 4 ⊢ -1 ≠ 0 | |
6 | expaddz 13558 | . . . 4 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) | |
7 | 4, 5, 6 | mpanl12 702 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) |
8 | 7 | anidms 570 | . 2 ⊢ (𝑁 ∈ ℤ → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) |
9 | m1expcl2 13536 | . . 3 ⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1}) | |
10 | ovex 7197 | . . . . 5 ⊢ (-1↑𝑁) ∈ V | |
11 | 10 | elpr 4536 | . . . 4 ⊢ ((-1↑𝑁) ∈ {-1, 1} ↔ ((-1↑𝑁) = -1 ∨ (-1↑𝑁) = 1)) |
12 | oveq12 7173 | . . . . . . 7 ⊢ (((-1↑𝑁) = -1 ∧ (-1↑𝑁) = -1) → ((-1↑𝑁) · (-1↑𝑁)) = (-1 · -1)) | |
13 | 12 | anidms 570 | . . . . . 6 ⊢ ((-1↑𝑁) = -1 → ((-1↑𝑁) · (-1↑𝑁)) = (-1 · -1)) |
14 | neg1mulneg1e1 11922 | . . . . . 6 ⊢ (-1 · -1) = 1 | |
15 | 13, 14 | eqtrdi 2789 | . . . . 5 ⊢ ((-1↑𝑁) = -1 → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
16 | oveq12 7173 | . . . . . . 7 ⊢ (((-1↑𝑁) = 1 ∧ (-1↑𝑁) = 1) → ((-1↑𝑁) · (-1↑𝑁)) = (1 · 1)) | |
17 | 16 | anidms 570 | . . . . . 6 ⊢ ((-1↑𝑁) = 1 → ((-1↑𝑁) · (-1↑𝑁)) = (1 · 1)) |
18 | 1t1e1 11871 | . . . . . 6 ⊢ (1 · 1) = 1 | |
19 | 17, 18 | eqtrdi 2789 | . . . . 5 ⊢ ((-1↑𝑁) = 1 → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
20 | 15, 19 | jaoi 856 | . . . 4 ⊢ (((-1↑𝑁) = -1 ∨ (-1↑𝑁) = 1) → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
21 | 11, 20 | sylbi 220 | . . 3 ⊢ ((-1↑𝑁) ∈ {-1, 1} → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
22 | 9, 21 | syl 17 | . 2 ⊢ (𝑁 ∈ ℤ → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
23 | 3, 8, 22 | 3eqtrd 2777 | 1 ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 {cpr 4515 (class class class)co 7164 ℂcc 10606 0cc0 10608 1c1 10609 + caddc 10611 · cmul 10613 -cneg 10942 2c2 11764 ℤcz 12055 ↑cexp 13514 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
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 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-n0 11970 df-z 12056 df-uz 12318 df-seq 13454 df-exp 13515 |
This theorem is referenced by: fallrisefac 15464 m1expe 15812 m1expo 15813 m1exp1 15814 gausslemma2d 26102 stirlinglem5 43145 |
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