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Mirrors > Home > MPE Home > Th. List > 3exp3 | Structured version Visualization version GIF version |
Description: Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.) |
Ref | Expression |
---|---|
3exp3 | ⊢ (3↑3) = ;27 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn0 12006 | . 2 ⊢ 3 ∈ ℕ0 | |
2 | 2nn0 12005 | . 2 ⊢ 2 ∈ ℕ0 | |
3 | 2p1e3 11870 | . 2 ⊢ (2 + 1) = 3 | |
4 | sq3 13665 | . . . 4 ⊢ (3↑2) = 9 | |
5 | 4 | oveq1i 7192 | . . 3 ⊢ ((3↑2) · 3) = (9 · 3) |
6 | 9t3e27 12314 | . . 3 ⊢ (9 · 3) = ;27 | |
7 | 5, 6 | eqtri 2762 | . 2 ⊢ ((3↑2) · 3) = ;27 |
8 | 1, 2, 3, 7 | numexpp1 16526 | 1 ⊢ (3↑3) = ;27 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7182 · cmul 10632 2c2 11783 3c3 11784 7c7 11788 9c9 11790 ;cdc 12191 ↑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-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-uz 12337 df-seq 13473 df-exp 13534 |
This theorem is referenced by: mcubic 25597 log2ub 25699 3exp7 39713 3lexlogpow2ineq1 39718 aks4d1p1 39735 |
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