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Mirrors > Home > MPE Home > Th. List > absefi | Structured version Visualization version GIF version |
Description: The absolute value of the exponential of an imaginary number is one. Equation 48 of [Rudin] p. 167. (Contributed by Jason Orendorff, 9-Feb-2007.) |
Ref | Expression |
---|---|
absefi | β’ (π΄ β β β (absβ(expβ(i Β· π΄))) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 11236 | . . . 4 β’ (π΄ β β β π΄ β β) | |
2 | efival 16136 | . . . 4 β’ (π΄ β β β (expβ(i Β· π΄)) = ((cosβπ΄) + (i Β· (sinβπ΄)))) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π΄ β β β (expβ(i Β· π΄)) = ((cosβπ΄) + (i Β· (sinβπ΄)))) |
4 | 3 | fveq2d 6906 | . 2 β’ (π΄ β β β (absβ(expβ(i Β· π΄))) = (absβ((cosβπ΄) + (i Β· (sinβπ΄))))) |
5 | recoscl 16125 | . . . 4 β’ (π΄ β β β (cosβπ΄) β β) | |
6 | resincl 16124 | . . . 4 β’ (π΄ β β β (sinβπ΄) β β) | |
7 | absreim 15280 | . . . 4 β’ (((cosβπ΄) β β β§ (sinβπ΄) β β) β (absβ((cosβπ΄) + (i Β· (sinβπ΄)))) = (ββ(((cosβπ΄)β2) + ((sinβπ΄)β2)))) | |
8 | 5, 6, 7 | syl2anc 582 | . . 3 β’ (π΄ β β β (absβ((cosβπ΄) + (i Β· (sinβπ΄)))) = (ββ(((cosβπ΄)β2) + ((sinβπ΄)β2)))) |
9 | 5 | resqcld 14129 | . . . . . . . 8 β’ (π΄ β β β ((cosβπ΄)β2) β β) |
10 | 9 | recnd 11280 | . . . . . . 7 β’ (π΄ β β β ((cosβπ΄)β2) β β) |
11 | 6 | resqcld 14129 | . . . . . . . 8 β’ (π΄ β β β ((sinβπ΄)β2) β β) |
12 | 11 | recnd 11280 | . . . . . . 7 β’ (π΄ β β β ((sinβπ΄)β2) β β) |
13 | 10, 12 | addcomd 11454 | . . . . . 6 β’ (π΄ β β β (((cosβπ΄)β2) + ((sinβπ΄)β2)) = (((sinβπ΄)β2) + ((cosβπ΄)β2))) |
14 | sincossq 16160 | . . . . . . 7 β’ (π΄ β β β (((sinβπ΄)β2) + ((cosβπ΄)β2)) = 1) | |
15 | 1, 14 | syl 17 | . . . . . 6 β’ (π΄ β β β (((sinβπ΄)β2) + ((cosβπ΄)β2)) = 1) |
16 | 13, 15 | eqtrd 2768 | . . . . 5 β’ (π΄ β β β (((cosβπ΄)β2) + ((sinβπ΄)β2)) = 1) |
17 | 16 | fveq2d 6906 | . . . 4 β’ (π΄ β β β (ββ(((cosβπ΄)β2) + ((sinβπ΄)β2))) = (ββ1)) |
18 | sqrt1 15258 | . . . 4 β’ (ββ1) = 1 | |
19 | 17, 18 | eqtrdi 2784 | . . 3 β’ (π΄ β β β (ββ(((cosβπ΄)β2) + ((sinβπ΄)β2))) = 1) |
20 | 8, 19 | eqtrd 2768 | . 2 β’ (π΄ β β β (absβ((cosβπ΄) + (i Β· (sinβπ΄)))) = 1) |
21 | 4, 20 | eqtrd 2768 | 1 β’ (π΄ β β β (absβ(expβ(i Β· π΄))) = 1) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 βcc 11144 βcr 11145 1c1 11147 ici 11148 + caddc 11149 Β· cmul 11151 2c2 12305 βcexp 14066 βcsqrt 15220 abscabs 15221 expce 16045 sincsin 16047 cosccos 16048 |
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 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
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 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-ico 13370 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-sin 16053 df-cos 16054 |
This theorem is referenced by: absef 16181 efieq1re 16183 pige3ALT 26474 efif1olem4 26499 efifo 26501 |
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