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Mirrors > Home > MPE Home > Th. List > expne0d | Structured version Visualization version GIF version |
Description: Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
expcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
sqrecd.1 | ⊢ (𝜑 → 𝐴 ≠ 0) |
expclzd.3 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
Ref | Expression |
---|---|
expne0d | ⊢ (𝜑 → (𝐴↑𝑁) ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | expcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | sqrecd.1 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) | |
3 | expclzd.3 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
4 | expne0i 13457 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ≠ 0) | |
5 | 1, 2, 3, 4 | syl3anc 1368 | 1 ⊢ (𝜑 → (𝐴↑𝑁) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ≠ wne 2987 (class class class)co 7135 ℂcc 10524 0cc0 10526 ℤcz 11969 ↑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-rmo 3114 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-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13365 df-exp 13426 |
This theorem is referenced by: znsqcld 13522 absexpz 14657 0.999... 15229 bitsfzo 15774 bitsmod 15775 bitsinv1lem 15780 bitsuz 15813 pcexp 16186 dvdsprmpweqle 16212 pcaddlem 16214 pcadd 16215 qexpz 16227 dvrecg 24576 dvexp3 24581 plyeq0lem 24807 aareccl 24922 taylthlem2 24969 root1cj 25345 cxpeq 25346 dcubic1lem 25429 dcubic2 25430 cubic2 25434 cubic 25435 lgamgulmlem4 25617 basellem4 25669 basellem8 25673 lgseisenlem1 25959 lgseisenlem2 25960 lgsquadlem1 25964 dya2icoseg 31645 dya2iocucvr 31652 omssubadd 31668 oddpwdc 31722 signsplypnf 31930 signsply0 31931 knoppndvlem7 33970 knoppndvlem17 33980 dffltz 39615 fltne 39616 fltnlta 39619 3cubeslem4 39630 rmxyneg 39861 radcnvrat 41018 dvdivbd 42565 iblsplit 42608 wallispi2lem1 42713 wallispi2lem2 42714 wallispi2 42715 stirlinglem3 42718 stirlinglem4 42719 stirlinglem7 42722 stirlinglem8 42723 stirlinglem10 42725 stirlinglem13 42728 stirlinglem14 42729 stirlinglem15 42730 fourierdlem56 42804 fourierdlem57 42805 elaa2lem 42875 sge0ad2en 43070 ovnsubaddlem1 43209 fldivexpfllog2 44979 nn0digval 45014 dignnld 45017 dig2nn1st 45019 dig2bits 45028 dignn0flhalflem1 45029 dignn0flhalflem2 45030 dignn0ehalf 45031 itsclc0xyqsolr 45183 |
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