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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 16058 | . 2 ⊢ exp:ℂ⟶ℂ | |
2 | 1 | ffvelcdmi 7093 | 1 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ‘cfv 6548 ℂcc 11137 expce 16038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-ico 13363 df-fz 13518 df-fzo 13661 df-fl 13790 df-seq 14000 df-exp 14060 df-fac 14266 df-hash 14323 df-shft 15047 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 df-sum 15666 df-ef 16044 |
This theorem is referenced by: efcld 16060 fprodefsum 16072 efne0 16074 efneg 16075 eff2 16076 efsub 16077 efexp 16078 ef4p 16090 sinf 16101 cosf 16102 tanval2 16110 tanval3 16111 resinval 16112 recosval 16113 resincl 16117 recoscl 16118 sinneg 16123 cosneg 16124 efival 16129 sinhval 16131 coshval 16132 absef 16174 efieq1re 16176 dveflem 25924 dvef 25925 dvsincos 25926 reeff1o 26397 efper 26427 pige3ALT 26467 sineq0 26471 efeq1 26475 efif1olem4 26492 efifo 26494 eff1olem 26495 eflogeq 26549 dvloglem 26595 logf1o2 26597 efopn 26605 cxpcl 26621 dvcxp1 26687 dvcxp2 26688 dvcncxp1 26690 sinasin 26834 asinsin 26837 efiatan2 26862 atantan 26868 efrlim 26914 efrlimOLD 26915 iprodefisumlem 35334 iprodefisum 35335 expgrowthi 43770 expgrowth 43772 sineq0ALT 44376 sinhpcosh 48171 |
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