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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 11700 | . . 3 ⊢ 2 ∈ ℂ | |
2 | 1 | sqvali 13539 | . 2 ⊢ (2↑2) = (2 · 2) |
3 | 2t2e4 11789 | . 2 ⊢ (2 · 2) = 4 | |
4 | 2, 3 | eqtri 2821 | 1 ⊢ (2↑2) = 4 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 (class class class)co 7135 · cmul 10531 2c2 11680 4c4 11682 ↑cexp 13425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 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 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13365 df-exp 13426 |
This theorem is referenced by: sq4e2t8 13558 cu2 13559 sqoddm1div8 13600 faclbnd2 13647 sqrt4 14624 amgm2 14721 ef01bndlem 15529 cos2bnd 15533 pythagtriplem1 16143 4sqlem12 16282 2exp4 16411 2exp5 16412 efmnd2hash 18051 lt6abl 19008 csbren 24003 minveclem2 24030 sincos6thpi 25108 heron 25424 quad2 25425 dcubic2 25430 mcubic 25433 dquartlem2 25438 dquart 25439 quart1 25442 quartlem1 25443 chtublem 25795 chtub 25796 bclbnd 25864 bposlem6 25873 bposlem8 25875 addsqnreup 26027 addsq2nreurex 26028 chebbnd1lem3 26055 chebbnd1 26056 ipidsq 28493 minvecolem2 28658 normpar2i 28939 sqsscirc1 31261 lcmineqlem21 39337 wallispi2lem1 42713 stirlinglem3 42718 stirlinglem10 42725 fmtno1 44058 fmtno2 44067 fmtnofac1 44087 m2prm 44108 lighneallem2 44124 lighneallem4a 44126 exple2lt6 44766 ackval3 45097 ackval42 45110 ackval42a 45111 itsclc0yqsollem1 45176 itscnhlinecirc02plem1 45196 |
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