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Mirrors > Home > MPE Home > Th. List > expne0d | Structured version Visualization version GIF version |
Description: A nonnegative integer power is nonzero if its base 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 13999 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ≠ 0) | |
5 | 1, 2, 3, 4 | syl3anc 1371 | 1 ⊢ (𝜑 → (𝐴↑𝑁) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ≠ wne 2943 (class class class)co 7356 ℂcc 11048 0cc0 11050 ℤcz 12498 ↑cexp 13966 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-n0 12413 df-z 12499 df-uz 12763 df-seq 13906 df-exp 13967 |
This theorem is referenced by: znsqcld 14066 absexpz 15189 0.999... 15765 bitsfzo 16314 bitsmod 16315 bitsinv1lem 16320 bitsuz 16353 pcexp 16730 dvdsprmpweqle 16757 pcaddlem 16759 pcadd 16760 qexpz 16772 dvrecg 25335 dvexp3 25340 plyeq0lem 25569 aareccl 25684 taylthlem2 25731 root1cj 26107 cxpeq 26108 dcubic1lem 26191 dcubic2 26192 cubic2 26196 cubic 26197 lgamgulmlem4 26379 basellem4 26431 basellem8 26435 lgseisenlem1 26721 lgseisenlem2 26722 lgsquadlem1 26726 dya2icoseg 32817 dya2iocucvr 32824 omssubadd 32840 oddpwdc 32894 signsplypnf 33102 signsply0 33103 knoppndvlem7 34971 knoppndvlem17 34981 dvrelogpow2b 40515 aks4d1p1p6 40520 aks4d1p1p7 40521 aks4d1p1p5 40522 aks4d1p8d3 40533 aks4d1p8 40534 aks6d1c2p2 40539 exp11d 40788 dffltz 40949 fltdiv 40951 fltnlta 40978 3cubeslem4 40989 rmxyneg 41221 radcnvrat 42575 dvdivbd 44135 iblsplit 44178 wallispi2lem1 44283 wallispi2lem2 44284 wallispi2 44285 stirlinglem3 44288 stirlinglem4 44289 stirlinglem7 44292 stirlinglem8 44293 stirlinglem10 44295 stirlinglem13 44298 stirlinglem14 44299 stirlinglem15 44300 fourierdlem56 44374 fourierdlem57 44375 elaa2lem 44445 sge0ad2en 44643 ovnsubaddlem1 44782 fldivexpfllog2 46622 nn0digval 46657 dignnld 46660 dig2nn1st 46662 dig2bits 46671 dignn0flhalflem1 46672 dignn0flhalflem2 46673 dignn0ehalf 46674 itsclc0xyqsolr 46826 |
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