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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 13464 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ≠ 0) | |
5 | 1, 2, 3, 4 | syl3anc 1367 | 1 ⊢ (𝜑 → (𝐴↑𝑁) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 3018 (class class class)co 7158 ℂcc 10537 0cc0 10539 ℤcz 11984 ↑cexp 13432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-seq 13373 df-exp 13433 |
This theorem is referenced by: znsqcld 13529 absexpz 14667 0.999... 15239 bitsfzo 15786 bitsmod 15787 bitsinv1lem 15792 bitsuz 15825 pcexp 16198 dvdsprmpweqle 16224 pcaddlem 16226 pcadd 16227 qexpz 16239 dvrecg 24572 dvexp3 24577 plyeq0lem 24802 aareccl 24917 taylthlem2 24964 root1cj 25339 cxpeq 25340 dcubic1lem 25423 dcubic2 25424 cubic2 25428 cubic 25429 lgamgulmlem4 25611 basellem4 25663 basellem8 25667 lgseisenlem1 25953 lgseisenlem2 25954 lgsquadlem1 25958 dya2icoseg 31537 dya2iocucvr 31544 omssubadd 31560 oddpwdc 31614 signsplypnf 31822 signsply0 31823 knoppndvlem7 33859 knoppndvlem17 33869 dffltz 39278 fltne 39279 fltnlta 39282 3cubeslem4 39293 rmxyneg 39524 radcnvrat 40653 dvdivbd 42215 iblsplit 42258 wallispi2lem1 42363 wallispi2lem2 42364 wallispi2 42365 stirlinglem3 42368 stirlinglem4 42369 stirlinglem7 42372 stirlinglem8 42373 stirlinglem10 42375 stirlinglem13 42378 stirlinglem14 42379 stirlinglem15 42380 fourierdlem56 42454 fourierdlem57 42455 elaa2lem 42525 sge0ad2en 42720 ovnsubaddlem1 42859 fldivexpfllog2 44632 nn0digval 44667 dignnld 44670 dig2nn1st 44672 dig2bits 44681 dignn0flhalflem1 44682 dignn0flhalflem2 44683 dignn0ehalf 44684 itsclc0xyqsolr 44763 |
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