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Mirrors > Home > MPE Home > Th. List > resinval | Structured version Visualization version GIF version |
Description: The sine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.) |
Ref | Expression |
---|---|
resinval | β’ (π΄ β β β (sinβπ΄) = (ββ(expβ(i Β· π΄)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 11207 | . . . . . . . 8 β’ i β β | |
2 | recn 11238 | . . . . . . . 8 β’ (π΄ β β β π΄ β β) | |
3 | cjmul 15131 | . . . . . . . 8 β’ ((i β β β§ π΄ β β) β (ββ(i Β· π΄)) = ((ββi) Β· (ββπ΄))) | |
4 | 1, 2, 3 | sylancr 585 | . . . . . . 7 β’ (π΄ β β β (ββ(i Β· π΄)) = ((ββi) Β· (ββπ΄))) |
5 | cji 15148 | . . . . . . . . 9 β’ (ββi) = -i | |
6 | 5 | oveq1i 7436 | . . . . . . . 8 β’ ((ββi) Β· (ββπ΄)) = (-i Β· (ββπ΄)) |
7 | cjre 15128 | . . . . . . . . 9 β’ (π΄ β β β (ββπ΄) = π΄) | |
8 | 7 | oveq2d 7442 | . . . . . . . 8 β’ (π΄ β β β (-i Β· (ββπ΄)) = (-i Β· π΄)) |
9 | 6, 8 | eqtrid 2780 | . . . . . . 7 β’ (π΄ β β β ((ββi) Β· (ββπ΄)) = (-i Β· π΄)) |
10 | 4, 9 | eqtrd 2768 | . . . . . 6 β’ (π΄ β β β (ββ(i Β· π΄)) = (-i Β· π΄)) |
11 | 10 | fveq2d 6906 | . . . . 5 β’ (π΄ β β β (expβ(ββ(i Β· π΄))) = (expβ(-i Β· π΄))) |
12 | mulcl 11232 | . . . . . . 7 β’ ((i β β β§ π΄ β β) β (i Β· π΄) β β) | |
13 | 1, 2, 12 | sylancr 585 | . . . . . 6 β’ (π΄ β β β (i Β· π΄) β β) |
14 | efcj 16078 | . . . . . 6 β’ ((i Β· π΄) β β β (expβ(ββ(i Β· π΄))) = (ββ(expβ(i Β· π΄)))) | |
15 | 13, 14 | syl 17 | . . . . 5 β’ (π΄ β β β (expβ(ββ(i Β· π΄))) = (ββ(expβ(i Β· π΄)))) |
16 | 11, 15 | eqtr3d 2770 | . . . 4 β’ (π΄ β β β (expβ(-i Β· π΄)) = (ββ(expβ(i Β· π΄)))) |
17 | 16 | oveq2d 7442 | . . 3 β’ (π΄ β β β ((expβ(i Β· π΄)) β (expβ(-i Β· π΄))) = ((expβ(i Β· π΄)) β (ββ(expβ(i Β· π΄))))) |
18 | 17 | oveq1d 7441 | . 2 β’ (π΄ β β β (((expβ(i Β· π΄)) β (expβ(-i Β· π΄))) / (2 Β· i)) = (((expβ(i Β· π΄)) β (ββ(expβ(i Β· π΄)))) / (2 Β· i))) |
19 | sinval 16108 | . . 3 β’ (π΄ β β β (sinβπ΄) = (((expβ(i Β· π΄)) β (expβ(-i Β· π΄))) / (2 Β· i))) | |
20 | 2, 19 | syl 17 | . 2 β’ (π΄ β β β (sinβπ΄) = (((expβ(i Β· π΄)) β (expβ(-i Β· π΄))) / (2 Β· i))) |
21 | efcl 16068 | . . 3 β’ ((i Β· π΄) β β β (expβ(i Β· π΄)) β β) | |
22 | imval2 15140 | . . 3 β’ ((expβ(i Β· π΄)) β β β (ββ(expβ(i Β· π΄))) = (((expβ(i Β· π΄)) β (ββ(expβ(i Β· π΄)))) / (2 Β· i))) | |
23 | 13, 21, 22 | 3syl 18 | . 2 β’ (π΄ β β β (ββ(expβ(i Β· π΄))) = (((expβ(i Β· π΄)) β (ββ(expβ(i Β· π΄)))) / (2 Β· i))) |
24 | 18, 20, 23 | 3eqtr4d 2778 | 1 β’ (π΄ β β β (sinβπ΄) = (ββ(expβ(i Β· π΄)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 βcc 11146 βcr 11147 ici 11150 Β· cmul 11153 β cmin 11484 -cneg 11485 / cdiv 11911 2c2 12307 βccj 15085 βcim 15087 expce 16047 sincsin 16049 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-pm 8856 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-sup 9475 df-inf 9476 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-n0 12513 df-z 12599 df-uz 12863 df-rp 13017 df-ico 13372 df-fz 13527 df-fzo 13670 df-fl 13799 df-seq 14009 df-exp 14069 df-fac 14275 df-hash 14332 df-shft 15056 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-limsup 15457 df-clim 15474 df-rlim 15475 df-sum 15675 df-ef 16053 df-sin 16055 |
This theorem is referenced by: resin4p 16124 resincl 16126 argimgt0 26574 |
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