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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 10452 | . 2 ⊢ 1 ≠ 0 | |
2 | oveq1 7023 | . . . 4 ⊢ ((exp‘𝐴) = 0 → ((exp‘𝐴) · (exp‘-𝐴)) = (0 · (exp‘-𝐴))) | |
3 | efcan 15282 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘𝐴) · (exp‘-𝐴)) = 1) | |
4 | negcl 10733 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
5 | efcl 15269 | . . . . . . 7 ⊢ (-𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) |
7 | 6 | mul02d 10685 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (0 · (exp‘-𝐴)) = 0) |
8 | 3, 7 | eqeq12d 2810 | . . . 4 ⊢ (𝐴 ∈ ℂ → (((exp‘𝐴) · (exp‘-𝐴)) = (0 · (exp‘-𝐴)) ↔ 1 = 0)) |
9 | 2, 8 | syl5ib 245 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘𝐴) = 0 → 1 = 0)) |
10 | 9 | necon3d 3005 | . 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 1522 ∈ wcel 2081 ≠ wne 2984 ‘cfv 6225 (class class class)co 7016 ℂcc 10381 0cc0 10383 1c1 10384 · cmul 10388 -cneg 10718 expce 15248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-pm 8259 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-inf 8753 df-oi 8820 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-n0 11746 df-z 11830 df-uz 12094 df-rp 12240 df-ico 12594 df-fz 12743 df-fzo 12884 df-fl 13012 df-seq 13220 df-exp 13280 df-fac 13484 df-bc 13513 df-hash 13541 df-shft 14260 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-sum 14877 df-ef 15254 |
This theorem is referenced by: efneg 15284 eff2 15285 efsub 15286 efgt0 15289 tanval3 15320 reeff1o 24718 efeq1 24794 efif1olem4 24810 eff1olem 24813 eflogeq 24866 dvloglem 24912 logf1o2 24914 efopn 24922 cxpne0 24941 atantan 25182 cxploglim 25237 gamne0 25305 iprodefisum 32581 expgrowth 40205 |
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