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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 11792 | . . 3 ⊢ 3 = (2 + 1) | |
2 | 1 | oveq2i 7193 | . 2 ⊢ (2↑3) = (2↑(2 + 1)) |
3 | 2cn 11803 | . . . 4 ⊢ 2 ∈ ℂ | |
4 | 2nn0 12005 | . . . 4 ⊢ 2 ∈ ℕ0 | |
5 | expp1 13540 | . . . 4 ⊢ ((2 ∈ ℂ ∧ 2 ∈ ℕ0) → (2↑(2 + 1)) = ((2↑2) · 2)) | |
6 | 3, 4, 5 | mp2an 692 | . . 3 ⊢ (2↑(2 + 1)) = ((2↑2) · 2) |
7 | sq2 13664 | . . . . 5 ⊢ (2↑2) = 4 | |
8 | 7 | oveq1i 7192 | . . . 4 ⊢ ((2↑2) · 2) = (4 · 2) |
9 | 4t2e8 11896 | . . . 4 ⊢ (4 · 2) = 8 | |
10 | 8, 9 | eqtri 2762 | . . 3 ⊢ ((2↑2) · 2) = 8 |
11 | 6, 10 | eqtri 2762 | . 2 ⊢ (2↑(2 + 1)) = 8 |
12 | 2, 11 | eqtri 2762 | 1 ⊢ (2↑3) = 8 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2114 (class class class)co 7182 ℂcc 10625 1c1 10628 + caddc 10630 · cmul 10632 2c2 11783 3c3 11784 4c4 11785 8c8 11789 ℕ0cn0 11988 ↑cexp 13533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-n0 11989 df-z 12075 df-uz 12337 df-seq 13473 df-exp 13534 |
This theorem is referenced by: ef01bndlem 15641 2exp5 16534 2exp6 16535 2exp11 16538 2503lem2 16586 quartlem1 25607 chtub 25960 bposlem8 26039 3lexlogpow2ineq1 39718 aks4d1p1 39735 2ap1caineq 39739 lhe4.4ex1a 41525 fmtno3 44584 fmtnoprmfac2lem1 44599 fmtno4sqrt 44604 m3prm 44625 5tcu2e40 44648 41prothprm 44652 ackval3012 45619 |
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