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Mirrors > Home > MPE Home > Th. List > i3 | Structured version Visualization version GIF version |
Description: i cubed. (Contributed by NM, 31-Jan-2007.) |
Ref | Expression |
---|---|
i3 | ⊢ (i↑3) = -i |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3 11439 | . . 3 ⊢ 3 = (2 + 1) | |
2 | 1 | oveq2i 6933 | . 2 ⊢ (i↑3) = (i↑(2 + 1)) |
3 | ax-icn 10331 | . . . 4 ⊢ i ∈ ℂ | |
4 | 2nn0 11661 | . . . 4 ⊢ 2 ∈ ℕ0 | |
5 | expp1 13185 | . . . 4 ⊢ ((i ∈ ℂ ∧ 2 ∈ ℕ0) → (i↑(2 + 1)) = ((i↑2) · i)) | |
6 | 3, 4, 5 | mp2an 682 | . . 3 ⊢ (i↑(2 + 1)) = ((i↑2) · i) |
7 | i2 13284 | . . . . 5 ⊢ (i↑2) = -1 | |
8 | 7 | oveq1i 6932 | . . . 4 ⊢ ((i↑2) · i) = (-1 · i) |
9 | 3 | mulm1i 10820 | . . . 4 ⊢ (-1 · i) = -i |
10 | 8, 9 | eqtri 2802 | . . 3 ⊢ ((i↑2) · i) = -i |
11 | 6, 10 | eqtri 2802 | . 2 ⊢ (i↑(2 + 1)) = -i |
12 | 2, 11 | eqtri 2802 | 1 ⊢ (i↑3) = -i |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 (class class class)co 6922 ℂcc 10270 1c1 10273 ici 10274 + caddc 10275 · cmul 10277 -cneg 10607 2c2 11430 3c3 11431 ℕ0cn0 11642 ↑cexp 13178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-seq 13120 df-exp 13179 |
This theorem is referenced by: efi4p 15269 cphipval 23449 iblcnlem1 23991 itgcnlem 23993 ipval2 28134 |
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