![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ere | Structured version Visualization version GIF version |
Description: Euler's constant e = 2.71828... is a real number. (Contributed by NM, 19-Mar-2005.) (Revised by Steve Rodriguez, 8-Mar-2006.) |
Ref | Expression |
---|---|
ere | ⊢ e ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-e 16009 | . 2 ⊢ e = (exp‘1) | |
2 | 1re 11211 | . . 3 ⊢ 1 ∈ ℝ | |
3 | reefcl 16027 | . . 3 ⊢ (1 ∈ ℝ → (exp‘1) ∈ ℝ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (exp‘1) ∈ ℝ |
5 | 1, 4 | eqeltri 2821 | 1 ⊢ e ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ‘cfv 6533 ℝcr 11105 1c1 11107 expce 16002 eceu 16003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-ico 13327 df-fz 13482 df-fzo 13625 df-fl 13754 df-seq 13964 df-exp 14025 df-fac 14231 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-e 16009 |
This theorem is referenced by: ege2le3 16030 eirrlem 16144 egt2lt3 16146 epos 16147 epr 16148 ene0 16149 ene1 16150 logdivlti 26470 logdivlt 26471 logdivle 26472 ecxp 26523 elogb 26618 logblog 26640 cxploglim2 26827 harmonicbnd3 26856 bposlem7 27139 bposlem9 27141 chebbnd1lem2 27319 chebbnd1lem3 27320 chebbnd1 27321 dchrvmasumlema 27349 logdivsum 27382 mulog2sumlem2 27384 selberg3lem1 27406 pntpbnd1a 27434 pntpbnd2 27436 pntlemb 27446 pntlemj 27452 pntlemk 27455 subfaclim 34668 subfacval3 34669 aks4d1p1p7 41432 stirlinglem3 45277 stirlinglem4 45278 stirlinglem13 45287 stirlinglem15 45289 stirlingr 45291 etransclem18 45453 etransclem23 45458 etransclem46 45481 etransclem47 45482 etransclem48 45483 etransc 45484 |
Copyright terms: Public domain | W3C validator |