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Mirrors > Home > MPE Home > Th. List > efgt0 | Structured version Visualization version GIF version |
Description: The exponential 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 15433 | . 2 ⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ) | |
2 | rehalfcl 11857 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ) | |
3 | 2 | reefcld 15434 | . . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘(𝐴 / 2)) ∈ ℝ) |
4 | 3 | sqge0d 13609 | . . 3 ⊢ (𝐴 ∈ ℝ → 0 ≤ ((exp‘(𝐴 / 2))↑2)) |
5 | 2 | recnd 10662 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℂ) |
6 | 2z 12008 | . . . . 5 ⊢ 2 ∈ ℤ | |
7 | efexp 15447 | . . . . 5 ⊢ (((𝐴 / 2) ∈ ℂ ∧ 2 ∈ ℤ) → (exp‘(2 · (𝐴 / 2))) = ((exp‘(𝐴 / 2))↑2)) | |
8 | 5, 6, 7 | sylancl 588 | . . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘(2 · (𝐴 / 2))) = ((exp‘(𝐴 / 2))↑2)) |
9 | recn 10620 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
10 | 2cn 11706 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
11 | 2ne0 11735 | . . . . . . 7 ⊢ 2 ≠ 0 | |
12 | divcan2 11299 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (2 · (𝐴 / 2)) = 𝐴) | |
13 | 10, 11, 12 | mp3an23 1448 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (2 · (𝐴 / 2)) = 𝐴) |
14 | 9, 13 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (2 · (𝐴 / 2)) = 𝐴) |
15 | 14 | fveq2d 6667 | . . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘(2 · (𝐴 / 2))) = (exp‘𝐴)) |
16 | 8, 15 | eqtr3d 2857 | . . 3 ⊢ (𝐴 ∈ ℝ → ((exp‘(𝐴 / 2))↑2) = (exp‘𝐴)) |
17 | 4, 16 | breqtrd 5085 | . 2 ⊢ (𝐴 ∈ ℝ → 0 ≤ (exp‘𝐴)) |
18 | efne0 15443 | . . 3 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ≠ 0) | |
19 | 9, 18 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ≠ 0) |
20 | 1, 17, 19 | ne0gt0d 10770 | 1 ⊢ (𝐴 ∈ ℝ → 0 < (exp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 class class class wbr 5059 ‘cfv 6348 (class class class)co 7149 ℂcc 10528 ℝcr 10529 0cc0 10530 · cmul 10535 < clt 10668 ≤ cle 10669 / cdiv 11290 2c2 11686 ℤcz 11975 ↑cexp 13426 expce 15408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-pm 8402 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-inf 8900 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-ico 12738 df-fz 12890 df-fzo 13031 df-fl 13159 df-seq 13367 df-exp 13427 df-fac 13631 df-bc 13660 df-hash 13688 df-shft 14419 df-cj 14451 df-re 14452 df-im 14453 df-sqrt 14587 df-abs 14588 df-limsup 14821 df-clim 14838 df-rlim 14839 df-sum 15036 df-ef 15414 |
This theorem is referenced by: rpefcl 15450 eflt 15463 tanhlt1 15506 absef 15543 efieq1re 15545 rpcxpcl 25255 asinsinlem 25465 birthdaylem3 25527 pntpbnd1 26158 pntpbnd2 26159 xrge0iifcnv 31195 cxpgt0d 39263 |
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