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Mirrors > Home > MPE Home > Th. List > cu2 | Structured version Visualization version GIF version |
Description: The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.) |
Ref | Expression |
---|---|
cu2 | ⊢ (2↑3) = 8 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3 12272 | . . 3 ⊢ 3 = (2 + 1) | |
2 | 1 | oveq2i 7415 | . 2 ⊢ (2↑3) = (2↑(2 + 1)) |
3 | 2cn 12283 | . . . 4 ⊢ 2 ∈ ℂ | |
4 | 2nn0 12485 | . . . 4 ⊢ 2 ∈ ℕ0 | |
5 | expp1 14030 | . . . 4 ⊢ ((2 ∈ ℂ ∧ 2 ∈ ℕ0) → (2↑(2 + 1)) = ((2↑2) · 2)) | |
6 | 3, 4, 5 | mp2an 691 | . . 3 ⊢ (2↑(2 + 1)) = ((2↑2) · 2) |
7 | sq2 14157 | . . . . 5 ⊢ (2↑2) = 4 | |
8 | 7 | oveq1i 7414 | . . . 4 ⊢ ((2↑2) · 2) = (4 · 2) |
9 | 4t2e8 12376 | . . . 4 ⊢ (4 · 2) = 8 | |
10 | 8, 9 | eqtri 2761 | . . 3 ⊢ ((2↑2) · 2) = 8 |
11 | 6, 10 | eqtri 2761 | . 2 ⊢ (2↑(2 + 1)) = 8 |
12 | 2, 11 | eqtri 2761 | 1 ⊢ (2↑3) = 8 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7404 ℂcc 11104 1c1 11107 + caddc 11109 · cmul 11111 2c2 12263 3c3 12264 4c4 12265 8c8 12269 ℕ0cn0 12468 ↑cexp 14023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-n0 12469 df-z 12555 df-uz 12819 df-seq 13963 df-exp 14024 |
This theorem is referenced by: ef01bndlem 16123 2exp5 17015 2exp6 17016 2exp11 17019 2503lem2 17067 quartlem1 26342 chtub 26695 bposlem8 26774 3lexlogpow2ineq1 40861 aks4d1p1 40879 2ap1caineq 40899 lhe4.4ex1a 43021 fmtno3 46154 fmtnoprmfac2lem1 46169 fmtno4sqrt 46174 m3prm 46195 5tcu2e40 46218 41prothprm 46222 ackval3012 47280 |
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