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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 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 13667 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ≠ 0) | |
5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → (𝐴↑𝑁) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ≠ wne 2940 (class class class)co 7213 ℂcc 10727 0cc0 10729 ℤcz 12176 ↑cexp 13635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-seq 13575 df-exp 13636 |
This theorem is referenced by: znsqcld 13732 absexpz 14869 0.999... 15445 bitsfzo 15994 bitsmod 15995 bitsinv1lem 16000 bitsuz 16033 pcexp 16412 dvdsprmpweqle 16439 pcaddlem 16441 pcadd 16442 qexpz 16454 dvrecg 24870 dvexp3 24875 plyeq0lem 25104 aareccl 25219 taylthlem2 25266 root1cj 25642 cxpeq 25643 dcubic1lem 25726 dcubic2 25727 cubic2 25731 cubic 25732 lgamgulmlem4 25914 basellem4 25966 basellem8 25970 lgseisenlem1 26256 lgseisenlem2 26257 lgsquadlem1 26261 dya2icoseg 31956 dya2iocucvr 31963 omssubadd 31979 oddpwdc 32033 signsplypnf 32241 signsply0 32242 knoppndvlem7 34435 knoppndvlem17 34445 dvrelogpow2b 39809 aks4d1p1p6 39814 aks4d1p1p7 39815 aks4d1p1p5 39816 exp11d 40033 dffltz 40174 fltdiv 40176 fltnlta 40203 3cubeslem4 40214 rmxyneg 40445 radcnvrat 41605 dvdivbd 43139 iblsplit 43182 wallispi2lem1 43287 wallispi2lem2 43288 wallispi2 43289 stirlinglem3 43292 stirlinglem4 43293 stirlinglem7 43296 stirlinglem8 43297 stirlinglem10 43299 stirlinglem13 43302 stirlinglem14 43303 stirlinglem15 43304 fourierdlem56 43378 fourierdlem57 43379 elaa2lem 43449 sge0ad2en 43644 ovnsubaddlem1 43783 fldivexpfllog2 45584 nn0digval 45619 dignnld 45622 dig2nn1st 45624 dig2bits 45633 dignn0flhalflem1 45634 dignn0flhalflem2 45635 dignn0ehalf 45636 itsclc0xyqsolr 45788 |
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