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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 12892 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ≠ 0) | |
5 | 1, 2, 3, 4 | syl3anc 1326 | 1 ⊢ (𝜑 → (𝐴↑𝑁) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ≠ wne 2794 (class class class)co 6650 ℂcc 9934 0cc0 9936 ℤcz 11377 ↑cexp 12860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-exp 12861 |
This theorem is referenced by: absexpz 14045 0.999... 14612 0.999...OLD 14613 bitsfzo 15157 bitsmod 15158 bitsinv1lem 15163 bitsuz 15196 pcexp 15564 dvdsprmpweqle 15590 pcaddlem 15592 pcadd 15593 qexpz 15605 dvrecg 23736 dvexp3 23741 plyeq0lem 23966 aareccl 24081 taylthlem2 24128 root1cj 24497 cxpeq 24498 dcubic1lem 24570 dcubic2 24571 cubic2 24575 cubic 24576 lgamgulmlem4 24758 basellem4 24810 basellem8 24814 lgseisenlem1 25100 lgseisenlem2 25101 lgsquadlem1 25105 znsqcld 29512 dya2icoseg 30339 dya2iocucvr 30346 omssubadd 30362 oddpwdc 30416 signsplypnf 30627 signsply0 30628 knoppndvlem7 32509 knoppndvlem17 32519 rmxyneg 37485 radcnvrat 38513 dvdivbd 40138 iblsplit 40182 wallispi2lem1 40288 wallispi2lem2 40289 wallispi2 40290 stirlinglem3 40293 stirlinglem4 40294 stirlinglem7 40297 stirlinglem8 40298 stirlinglem10 40300 stirlinglem13 40303 stirlinglem14 40304 stirlinglem15 40305 fourierdlem56 40379 fourierdlem57 40380 elaa2lem 40450 sge0ad2en 40648 ovnsubaddlem1 40784 fldivexpfllog2 42359 nn0digval 42394 dignnld 42397 dig2nn1st 42399 dig2bits 42408 dignn0flhalflem1 42409 dignn0flhalflem2 42410 dignn0ehalf 42411 |
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