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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 11708 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1 | 2timesd 11600 | . . 3 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) = (𝑁 + 𝑁)) |
3 | 2 | oveq2d 6920 | . 2 ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = (-1↑(𝑁 + 𝑁))) |
4 | neg1cn 11471 | . . . 4 ⊢ -1 ∈ ℂ | |
5 | neg1ne0 11473 | . . . 4 ⊢ -1 ≠ 0 | |
6 | expaddz 13197 | . . . 4 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) | |
7 | 4, 5, 6 | mpanl12 695 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) |
8 | 7 | anidms 564 | . 2 ⊢ (𝑁 ∈ ℤ → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) |
9 | m1expcl2 13175 | . . 3 ⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1}) | |
10 | ovex 6936 | . . . . 5 ⊢ (-1↑𝑁) ∈ V | |
11 | 10 | elpr 4419 | . . . 4 ⊢ ((-1↑𝑁) ∈ {-1, 1} ↔ ((-1↑𝑁) = -1 ∨ (-1↑𝑁) = 1)) |
12 | oveq12 6913 | . . . . . . 7 ⊢ (((-1↑𝑁) = -1 ∧ (-1↑𝑁) = -1) → ((-1↑𝑁) · (-1↑𝑁)) = (-1 · -1)) | |
13 | 12 | anidms 564 | . . . . . 6 ⊢ ((-1↑𝑁) = -1 → ((-1↑𝑁) · (-1↑𝑁)) = (-1 · -1)) |
14 | neg1mulneg1e1 11570 | . . . . . 6 ⊢ (-1 · -1) = 1 | |
15 | 13, 14 | syl6eq 2876 | . . . . 5 ⊢ ((-1↑𝑁) = -1 → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
16 | oveq12 6913 | . . . . . . 7 ⊢ (((-1↑𝑁) = 1 ∧ (-1↑𝑁) = 1) → ((-1↑𝑁) · (-1↑𝑁)) = (1 · 1)) | |
17 | 16 | anidms 564 | . . . . . 6 ⊢ ((-1↑𝑁) = 1 → ((-1↑𝑁) · (-1↑𝑁)) = (1 · 1)) |
18 | 1t1e1 11519 | . . . . . 6 ⊢ (1 · 1) = 1 | |
19 | 17, 18 | syl6eq 2876 | . . . . 5 ⊢ ((-1↑𝑁) = 1 → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
20 | 15, 19 | jaoi 890 | . . . 4 ⊢ (((-1↑𝑁) = -1 ∨ (-1↑𝑁) = 1) → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
21 | 11, 20 | sylbi 209 | . . 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 2864 | 1 ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 880 = wceq 1658 ∈ wcel 2166 ≠ wne 2998 {cpr 4398 (class class class)co 6904 ℂcc 10249 0cc0 10251 1c1 10252 + caddc 10254 · cmul 10256 -cneg 10585 2c2 11405 ℤcz 11703 ↑cexp 13153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-n0 11618 df-z 11704 df-uz 11968 df-seq 13095 df-exp 13154 |
This theorem is referenced by: fallrisefac 15127 m1expe 15464 m1expo 15465 m1exp1 15466 gausslemma2d 25511 stirlinglem5 41088 |
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