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Mirrors > Home > MPE Home > Th. List > efgt0 | Structured version Visualization version GIF version |
Description: The exponential function of a real number is greater than 0. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
efgt0 | ⊢ (𝐴 ∈ ℝ → 0 < (exp‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reefcl 15023 | . 2 ⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ) | |
2 | rehalfcl 11460 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ) | |
3 | 2 | reefcld 15024 | . . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘(𝐴 / 2)) ∈ ℝ) |
4 | 3 | sqge0d 13243 | . . 3 ⊢ (𝐴 ∈ ℝ → 0 ≤ ((exp‘(𝐴 / 2))↑2)) |
5 | 2 | recnd 10270 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℂ) |
6 | 2z 11611 | . . . . 5 ⊢ 2 ∈ ℤ | |
7 | efexp 15037 | . . . . 5 ⊢ (((𝐴 / 2) ∈ ℂ ∧ 2 ∈ ℤ) → (exp‘(2 · (𝐴 / 2))) = ((exp‘(𝐴 / 2))↑2)) | |
8 | 5, 6, 7 | sylancl 574 | . . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘(2 · (𝐴 / 2))) = ((exp‘(𝐴 / 2))↑2)) |
9 | recn 10228 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
10 | 2cn 11293 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
11 | 2ne0 11315 | . . . . . . 7 ⊢ 2 ≠ 0 | |
12 | divcan2 10895 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (2 · (𝐴 / 2)) = 𝐴) | |
13 | 10, 11, 12 | mp3an23 1564 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (2 · (𝐴 / 2)) = 𝐴) |
14 | 9, 13 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (2 · (𝐴 / 2)) = 𝐴) |
15 | 14 | fveq2d 6336 | . . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘(2 · (𝐴 / 2))) = (exp‘𝐴)) |
16 | 8, 15 | eqtr3d 2807 | . . 3 ⊢ (𝐴 ∈ ℝ → ((exp‘(𝐴 / 2))↑2) = (exp‘𝐴)) |
17 | 4, 16 | breqtrd 4812 | . 2 ⊢ (𝐴 ∈ ℝ → 0 ≤ (exp‘𝐴)) |
18 | efne0 15033 | . . 3 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ≠ 0) | |
19 | 9, 18 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ≠ 0) |
20 | 1, 17, 19 | ne0gt0d 10376 | 1 ⊢ (𝐴 ∈ ℝ → 0 < (exp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 class class class wbr 4786 ‘cfv 6031 (class class class)co 6793 ℂcc 10136 ℝcr 10137 0cc0 10138 · cmul 10143 < clt 10276 ≤ cle 10277 / cdiv 10886 2c2 11272 ℤcz 11579 ↑cexp 13067 expce 14998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 ax-addf 10217 ax-mulf 10218 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-pm 8012 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8504 df-inf 8505 df-oi 8571 df-card 8965 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-n0 11495 df-z 11580 df-uz 11889 df-rp 12036 df-ico 12386 df-fz 12534 df-fzo 12674 df-fl 12801 df-seq 13009 df-exp 13068 df-fac 13265 df-bc 13294 df-hash 13322 df-shft 14015 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-limsup 14410 df-clim 14427 df-rlim 14428 df-sum 14625 df-ef 15004 |
This theorem is referenced by: rpefcl 15040 eflt 15053 tanhlt1 15096 absef 15133 efieq1re 15135 rpcxpcl 24643 asinsinlem 24839 birthdaylem3 24901 pntpbnd1 25496 pntpbnd2 25497 xrge0iifcnv 30319 |
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