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Mirrors > Home > MPE Home > Th. List > sq2 | Structured version Visualization version GIF version |
Description: The square of 2 is 4. (Contributed by NM, 22-Aug-1999.) |
Ref | Expression |
---|---|
sq2 | ⊢ (2↑2) = 4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 11303 | . . 3 ⊢ 2 ∈ ℂ | |
2 | 1 | sqvali 13157 | . 2 ⊢ (2↑2) = (2 · 2) |
3 | 2t2e4 11389 | . 2 ⊢ (2 · 2) = 4 | |
4 | 2, 3 | eqtri 2782 | 1 ⊢ (2↑2) = 4 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 (class class class)co 6814 · cmul 10153 2c2 11282 4c4 11284 ↑cexp 13074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-n0 11505 df-z 11590 df-uz 11900 df-seq 13016 df-exp 13075 |
This theorem is referenced by: sq4e2t8 13176 cu2 13177 sqoddm1div8 13242 faclbnd2 13292 sqrt4 14232 amgm2 14328 ef01bndlem 15133 cos2bnd 15137 pythagtriplem1 15743 4sqlem12 15882 2exp4 16016 lt6abl 18516 csbren 23402 minveclem2 23417 sincos6thpi 24487 heron 24785 quad2 24786 dcubic2 24791 mcubic 24794 dquartlem2 24799 dquart 24800 quart1 24803 quartlem1 24804 chtublem 25156 chtub 25157 bclbnd 25225 bposlem6 25234 bposlem8 25236 chebbnd1lem3 25380 chebbnd1 25381 ipidsq 27895 minvecolem2 28061 normpar2i 28343 sqsscirc1 30284 wallispi2lem1 40809 stirlinglem3 40814 stirlinglem10 40821 fmtno1 41981 fmtno2 41990 fmtnofac1 42010 m2prm 42033 2exp5 42035 lighneallem2 42051 lighneallem4a 42053 exple2lt6 42673 |
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