![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > efieq | Structured version Visualization version GIF version |
Description: The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.) |
Ref | Expression |
---|---|
efieq | β’ ((π΄ β β β§ π΅ β β) β ((expβ(i Β· π΄)) = (expβ(i Β· π΅)) β ((cosβπ΄) = (cosβπ΅) β§ (sinβπ΄) = (sinβπ΅)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 11228 | . . 3 β’ (π΄ β β β π΄ β β) | |
2 | recn 11228 | . . 3 β’ (π΅ β β β π΅ β β) | |
3 | efival 16128 | . . . 4 β’ (π΄ β β β (expβ(i Β· π΄)) = ((cosβπ΄) + (i Β· (sinβπ΄)))) | |
4 | efival 16128 | . . . 4 β’ (π΅ β β β (expβ(i Β· π΅)) = ((cosβπ΅) + (i Β· (sinβπ΅)))) | |
5 | 3, 4 | eqeqan12d 2742 | . . 3 β’ ((π΄ β β β§ π΅ β β) β ((expβ(i Β· π΄)) = (expβ(i Β· π΅)) β ((cosβπ΄) + (i Β· (sinβπ΄))) = ((cosβπ΅) + (i Β· (sinβπ΅))))) |
6 | 1, 2, 5 | syl2an 595 | . 2 β’ ((π΄ β β β§ π΅ β β) β ((expβ(i Β· π΄)) = (expβ(i Β· π΅)) β ((cosβπ΄) + (i Β· (sinβπ΄))) = ((cosβπ΅) + (i Β· (sinβπ΅))))) |
7 | recoscl 16117 | . . . 4 β’ (π΄ β β β (cosβπ΄) β β) | |
8 | resincl 16116 | . . . 4 β’ (π΄ β β β (sinβπ΄) β β) | |
9 | 7, 8 | jca 511 | . . 3 β’ (π΄ β β β ((cosβπ΄) β β β§ (sinβπ΄) β β)) |
10 | recoscl 16117 | . . . 4 β’ (π΅ β β β (cosβπ΅) β β) | |
11 | resincl 16116 | . . . 4 β’ (π΅ β β β (sinβπ΅) β β) | |
12 | 10, 11 | jca 511 | . . 3 β’ (π΅ β β β ((cosβπ΅) β β β§ (sinβπ΅) β β)) |
13 | cru 12234 | . . 3 β’ ((((cosβπ΄) β β β§ (sinβπ΄) β β) β§ ((cosβπ΅) β β β§ (sinβπ΅) β β)) β (((cosβπ΄) + (i Β· (sinβπ΄))) = ((cosβπ΅) + (i Β· (sinβπ΅))) β ((cosβπ΄) = (cosβπ΅) β§ (sinβπ΄) = (sinβπ΅)))) | |
14 | 9, 12, 13 | syl2an 595 | . 2 β’ ((π΄ β β β§ π΅ β β) β (((cosβπ΄) + (i Β· (sinβπ΄))) = ((cosβπ΅) + (i Β· (sinβπ΅))) β ((cosβπ΄) = (cosβπ΅) β§ (sinβπ΄) = (sinβπ΅)))) |
15 | 6, 14 | bitrd 279 | 1 β’ ((π΄ β β β§ π΅ β β) β ((expβ(i Β· π΄)) = (expβ(i Β· π΅)) β ((cosβπ΄) = (cosβπ΅) β§ (sinβπ΄) = (sinβπ΅)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 βcfv 6548 (class class class)co 7420 βcc 11136 βcr 11137 ici 11140 + caddc 11141 Β· cmul 11143 expce 16037 sincsin 16039 cosccos 16040 |
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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-ico 13362 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-fac 14265 df-hash 14322 df-shft 15046 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-sum 15665 df-ef 16043 df-sin 16045 df-cos 16046 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |