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Mirrors > Home > MPE Home > Th. List > efne0 | Structured version Visualization version GIF version |
Description: The exponential of a complex number is nonzero. Corollary 15-4.3 of [Gleason] p. 309. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 29-Apr-2014.) |
Ref | Expression |
---|---|
efne0 | ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1ne0 11229 | . 2 ⊢ 1 ≠ 0 | |
2 | oveq1 7433 | . . . 4 ⊢ ((exp‘𝐴) = 0 → ((exp‘𝐴) · (exp‘-𝐴)) = (0 · (exp‘-𝐴))) | |
3 | efcan 16100 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘𝐴) · (exp‘-𝐴)) = 1) | |
4 | negcl 11512 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
5 | efcl 16086 | . . . . . . 7 ⊢ (-𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) |
7 | 6 | mul02d 11464 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (0 · (exp‘-𝐴)) = 0) |
8 | 3, 7 | eqeq12d 2742 | . . . 4 ⊢ (𝐴 ∈ ℂ → (((exp‘𝐴) · (exp‘-𝐴)) = (0 · (exp‘-𝐴)) ↔ 1 = 0)) |
9 | 2, 8 | imbitrid 243 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘𝐴) = 0 → 1 = 0)) |
10 | 9 | necon3d 2951 | . 2 ⊢ (𝐴 ∈ ℂ → (1 ≠ 0 → (exp‘𝐴) ≠ 0)) |
11 | 1, 10 | mpi 20 | 1 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ‘cfv 6556 (class class class)co 7426 ℂcc 11158 0cc0 11160 1c1 11161 · cmul 11165 -cneg 11497 expce 16065 |
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 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-inf2 9686 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-pre-sup 11238 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-se 5640 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-isom 6565 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-er 8736 df-pm 8860 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-sup 9487 df-inf 9488 df-oi 9555 df-card 9984 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12613 df-uz 12877 df-rp 13031 df-ico 13386 df-fz 13541 df-fzo 13684 df-fl 13814 df-seq 14024 df-exp 14084 df-fac 14293 df-bc 14322 df-hash 14350 df-shft 15074 df-cj 15106 df-re 15107 df-im 15108 df-sqrt 15242 df-abs 15243 df-limsup 15475 df-clim 15492 df-rlim 15493 df-sum 15693 df-ef 16071 |
This theorem is referenced by: efneg 16102 eff2 16103 efsub 16104 efgt0 16107 tanval3 16138 reeff1o 26480 efeq1 26558 efif1olem4 26575 eff1olem 26578 eflogeq 26632 dvloglem 26678 logf1o2 26680 efopn 26688 cxpne0 26707 atantan 26954 cxploglim 27009 gamne0 27077 iprodefisum 35565 expgrowth 44027 |
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