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Mirrors > Home > MPE Home > Th. List > efmival | Structured version Visualization version GIF version |
Description: The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006.) |
Ref | Expression |
---|---|
efmival | β’ (π΄ β β β (expβ(-i Β· π΄)) = ((cosβπ΄) β (i Β· (sinβπ΄)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 11168 | . . . 4 β’ i β β | |
2 | mulneg12 11653 | . . . 4 β’ ((i β β β§ π΄ β β) β (-i Β· π΄) = (i Β· -π΄)) | |
3 | 1, 2 | mpan 687 | . . 3 β’ (π΄ β β β (-i Β· π΄) = (i Β· -π΄)) |
4 | 3 | fveq2d 6888 | . 2 β’ (π΄ β β β (expβ(-i Β· π΄)) = (expβ(i Β· -π΄))) |
5 | negcl 11461 | . . . 4 β’ (π΄ β β β -π΄ β β) | |
6 | efival 16099 | . . . 4 β’ (-π΄ β β β (expβ(i Β· -π΄)) = ((cosβ-π΄) + (i Β· (sinβ-π΄)))) | |
7 | 5, 6 | syl 17 | . . 3 β’ (π΄ β β β (expβ(i Β· -π΄)) = ((cosβ-π΄) + (i Β· (sinβ-π΄)))) |
8 | cosneg 16094 | . . . . 5 β’ (π΄ β β β (cosβ-π΄) = (cosβπ΄)) | |
9 | sinneg 16093 | . . . . . . 7 β’ (π΄ β β β (sinβ-π΄) = -(sinβπ΄)) | |
10 | 9 | oveq2d 7420 | . . . . . 6 β’ (π΄ β β β (i Β· (sinβ-π΄)) = (i Β· -(sinβπ΄))) |
11 | sincl 16073 | . . . . . . 7 β’ (π΄ β β β (sinβπ΄) β β) | |
12 | mulneg2 11652 | . . . . . . 7 β’ ((i β β β§ (sinβπ΄) β β) β (i Β· -(sinβπ΄)) = -(i Β· (sinβπ΄))) | |
13 | 1, 11, 12 | sylancr 586 | . . . . . 6 β’ (π΄ β β β (i Β· -(sinβπ΄)) = -(i Β· (sinβπ΄))) |
14 | 10, 13 | eqtrd 2766 | . . . . 5 β’ (π΄ β β β (i Β· (sinβ-π΄)) = -(i Β· (sinβπ΄))) |
15 | 8, 14 | oveq12d 7422 | . . . 4 β’ (π΄ β β β ((cosβ-π΄) + (i Β· (sinβ-π΄))) = ((cosβπ΄) + -(i Β· (sinβπ΄)))) |
16 | coscl 16074 | . . . . 5 β’ (π΄ β β β (cosβπ΄) β β) | |
17 | mulcl 11193 | . . . . . 6 β’ ((i β β β§ (sinβπ΄) β β) β (i Β· (sinβπ΄)) β β) | |
18 | 1, 11, 17 | sylancr 586 | . . . . 5 β’ (π΄ β β β (i Β· (sinβπ΄)) β β) |
19 | 16, 18 | negsubd 11578 | . . . 4 β’ (π΄ β β β ((cosβπ΄) + -(i Β· (sinβπ΄))) = ((cosβπ΄) β (i Β· (sinβπ΄)))) |
20 | 15, 19 | eqtrd 2766 | . . 3 β’ (π΄ β β β ((cosβ-π΄) + (i Β· (sinβ-π΄))) = ((cosβπ΄) β (i Β· (sinβπ΄)))) |
21 | 7, 20 | eqtrd 2766 | . 2 β’ (π΄ β β β (expβ(i Β· -π΄)) = ((cosβπ΄) β (i Β· (sinβπ΄)))) |
22 | 4, 21 | eqtrd 2766 | 1 β’ (π΄ β β β (expβ(-i Β· π΄)) = ((cosβπ΄) β (i Β· (sinβπ΄)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6536 (class class class)co 7404 βcc 11107 ici 11111 + caddc 11112 Β· cmul 11114 β cmin 11445 -cneg 11446 expce 16008 sincsin 16010 cosccos 16011 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-ico 13333 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14030 df-fac 14236 df-hash 14293 df-shft 15017 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-limsup 15418 df-clim 15435 df-rlim 15436 df-sum 15636 df-ef 16014 df-sin 16016 df-cos 16017 |
This theorem is referenced by: sinadd 16111 cosadd 16112 |
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