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Mirrors > Home > MPE Home > Th. List > exp0d | Structured version Visualization version GIF version |
Description: Value of a complex number raised to the 0th power. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
expcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
exp0d | ⊢ (𝜑 → (𝐴↑0) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | expcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | exp0 13639 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴↑0) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 (class class class)co 7213 ℂcc 10727 0cc0 10729 1c1 10730 ↑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-pr 5322 ax-1cn 10787 ax-addrcl 10790 ax-rnegex 10800 ax-cnre 10802 |
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-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 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-iota 6338 df-fun 6382 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-neg 11065 df-z 12177 df-seq 13575 df-exp 13636 |
This theorem is referenced by: faclbnd4lem3 13861 faclbnd4lem4 13862 faclbnd6 13865 hashmap 14002 absexp 14868 binom 15394 geoser 15431 pwdif 15432 cvgrat 15447 efexp 15662 pwp1fsum 15952 prmdvdsexpr 16274 rpexp1i 16280 phiprm 16330 odzdvds 16348 pclem 16391 pcpre1 16395 pcexp 16412 dvdsprmpweqnn 16438 prmpwdvds 16457 pgp0 18985 sylow2alem2 19007 ablfac1eu 19460 pgpfac1lem3a 19463 plyeq0lem 25104 plyco 25135 vieta1 25205 abelthlem9 25332 advlogexp 25543 cxpmul2 25577 nnlogbexp 25664 ftalem5 25959 0sgm 26026 1sgmprm 26080 dchrptlem2 26146 bposlem5 26169 lgsval2lem 26188 lgsmod 26204 lgsdilem2 26214 lgsne0 26216 chebbnd1lem1 26350 dchrisum0flblem1 26389 qabvexp 26507 ostth2lem2 26515 ostth3 26519 rusgrnumwwlk 28059 nexple 31689 faclim 33430 faclim2 33432 knoppndvlem14 34442 lcmineqlem12 39782 nn0rppwr 40041 nn0expgcd 40043 flt0 40177 fltnltalem 40202 mzpexpmpt 40270 pell14qrexpclnn0 40391 pellfund14 40423 rmxy0 40448 jm2.17a 40485 jm2.17b 40486 jm2.18 40513 jm2.23 40521 expdioph 40548 cnsrexpcl 40693 binomcxplemnotnn0 41647 dvnxpaek 43158 wallispilem2 43282 etransclem24 43474 etransclem25 43475 etransclem35 43485 lighneallem3 44732 lighneallem4 44735 altgsumbcALT 45362 expnegico01 45532 digexp 45626 dig1 45627 |
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