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Mirrors > Home > MPE Home > Th. List > efcl | Structured version Visualization version GIF version |
Description: Closure law for the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.) |
Ref | Expression |
---|---|
efcl | ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eff 15218 | . 2 ⊢ exp:ℂ⟶ℂ | |
2 | 1 | ffvelrni 6624 | 1 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ‘cfv 6137 ℂcc 10272 expce 15198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-pm 8145 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-n0 11647 df-z 11733 df-uz 11997 df-rp 12142 df-ico 12497 df-fz 12648 df-fzo 12789 df-fl 12916 df-seq 13124 df-exp 13183 df-fac 13383 df-hash 13440 df-shft 14218 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-limsup 14614 df-clim 14631 df-rlim 14632 df-sum 14829 df-ef 15204 |
This theorem is referenced by: fprodefsum 15231 efne0 15233 efneg 15234 eff2 15235 efsub 15236 efexp 15237 ef4p 15249 sinf 15260 cosf 15261 tanval2 15269 tanval3 15270 resinval 15271 recosval 15272 resincl 15276 recoscl 15277 sinneg 15282 cosneg 15283 efival 15288 sinhval 15290 coshval 15291 absef 15333 efieq1re 15335 dveflem 24183 dvef 24184 dvsincos 24185 reeff1o 24642 efper 24673 pige3 24711 sineq0 24715 efeq1 24717 efif1olem4 24733 efifo 24735 eff1olem 24736 eflogeq 24789 dvloglem 24835 logf1o2 24837 efopn 24845 cxpcl 24861 dvcxp1 24925 dvcxp2 24926 dvcncxp1 24928 sinasin 25071 asinsin 25074 efiatan2 25099 atantan 25105 efrlim 25152 efcld 31275 iprodefisumlem 32224 iprodefisum 32225 expgrowthi 39498 expgrowth 39500 sineq0ALT 40116 sinhpcosh 43599 |
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