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Mirrors > Home > MPE Home > Th. List > cosargd | Structured version Visualization version GIF version |
Description: The cosine of the argument is the quotient of the real part and the absolute value. Compare to efiarg 26561. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
cosargd.1 | โข (๐ โ ๐ โ โ) |
cosargd.2 | โข (๐ โ ๐ โ 0) |
Ref | Expression |
---|---|
cosargd | โข (๐ โ (cosโ(โโ(logโ๐))) = ((โโ๐) / (absโ๐))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cosargd.1 | . . . 4 โข (๐ โ ๐ โ โ) | |
2 | 1 | cjcld 15183 | . . . 4 โข (๐ โ (โโ๐) โ โ) |
3 | 1, 2 | addcld 11271 | . . 3 โข (๐ โ (๐ + (โโ๐)) โ โ) |
4 | 1 | abscld 15423 | . . . 4 โข (๐ โ (absโ๐) โ โ) |
5 | 4 | recnd 11280 | . . 3 โข (๐ โ (absโ๐) โ โ) |
6 | 2cnd 12328 | . . 3 โข (๐ โ 2 โ โ) | |
7 | cosargd.2 | . . . 4 โข (๐ โ ๐ โ 0) | |
8 | 1, 7 | absne0d 15434 | . . 3 โข (๐ โ (absโ๐) โ 0) |
9 | 2ne0 12354 | . . . 4 โข 2 โ 0 | |
10 | 9 | a1i 11 | . . 3 โข (๐ โ 2 โ 0) |
11 | 3, 5, 6, 8, 10 | divdiv32d 12053 | . 2 โข (๐ โ (((๐ + (โโ๐)) / (absโ๐)) / 2) = (((๐ + (โโ๐)) / 2) / (absโ๐))) |
12 | 1, 7 | logcld 26524 | . . . . . 6 โข (๐ โ (logโ๐) โ โ) |
13 | 12 | imcld 15182 | . . . . 5 โข (๐ โ (โโ(logโ๐)) โ โ) |
14 | 13 | recnd 11280 | . . . 4 โข (๐ โ (โโ(logโ๐)) โ โ) |
15 | cosval 16107 | . . . 4 โข ((โโ(logโ๐)) โ โ โ (cosโ(โโ(logโ๐))) = (((expโ(i ยท (โโ(logโ๐)))) + (expโ(-i ยท (โโ(logโ๐))))) / 2)) | |
16 | 14, 15 | syl 17 | . . 3 โข (๐ โ (cosโ(โโ(logโ๐))) = (((expโ(i ยท (โโ(logโ๐)))) + (expโ(-i ยท (โโ(logโ๐))))) / 2)) |
17 | efiarg 26561 | . . . . . . 7 โข ((๐ โ โ โง ๐ โ 0) โ (expโ(i ยท (โโ(logโ๐)))) = (๐ / (absโ๐))) | |
18 | 1, 7, 17 | syl2anc 582 | . . . . . 6 โข (๐ โ (expโ(i ยท (โโ(logโ๐)))) = (๐ / (absโ๐))) |
19 | ax-icn 11205 | . . . . . . . . . . 11 โข i โ โ | |
20 | 19 | a1i 11 | . . . . . . . . . 10 โข (๐ โ i โ โ) |
21 | 20, 14 | mulcld 11272 | . . . . . . . . 9 โข (๐ โ (i ยท (โโ(logโ๐))) โ โ) |
22 | efcj 16076 | . . . . . . . . 9 โข ((i ยท (โโ(logโ๐))) โ โ โ (expโ(โโ(i ยท (โโ(logโ๐))))) = (โโ(expโ(i ยท (โโ(logโ๐)))))) | |
23 | 21, 22 | syl 17 | . . . . . . . 8 โข (๐ โ (expโ(โโ(i ยท (โโ(logโ๐))))) = (โโ(expโ(i ยท (โโ(logโ๐)))))) |
24 | 20, 14 | cjmuld 15208 | . . . . . . . . . 10 โข (๐ โ (โโ(i ยท (โโ(logโ๐)))) = ((โโi) ยท (โโ(โโ(logโ๐))))) |
25 | cji 15146 | . . . . . . . . . . . 12 โข (โโi) = -i | |
26 | 25 | a1i 11 | . . . . . . . . . . 11 โข (๐ โ (โโi) = -i) |
27 | 13 | cjred 15213 | . . . . . . . . . . 11 โข (๐ โ (โโ(โโ(logโ๐))) = (โโ(logโ๐))) |
28 | 26, 27 | oveq12d 7444 | . . . . . . . . . 10 โข (๐ โ ((โโi) ยท (โโ(โโ(logโ๐)))) = (-i ยท (โโ(logโ๐)))) |
29 | 24, 28 | eqtrd 2768 | . . . . . . . . 9 โข (๐ โ (โโ(i ยท (โโ(logโ๐)))) = (-i ยท (โโ(logโ๐)))) |
30 | 29 | fveq2d 6906 | . . . . . . . 8 โข (๐ โ (expโ(โโ(i ยท (โโ(logโ๐))))) = (expโ(-i ยท (โโ(logโ๐))))) |
31 | 18 | fveq2d 6906 | . . . . . . . 8 โข (๐ โ (โโ(expโ(i ยท (โโ(logโ๐))))) = (โโ(๐ / (absโ๐)))) |
32 | 23, 30, 31 | 3eqtr3d 2776 | . . . . . . 7 โข (๐ โ (expโ(-i ยท (โโ(logโ๐)))) = (โโ(๐ / (absโ๐)))) |
33 | 1, 5, 8 | cjdivd 15210 | . . . . . . 7 โข (๐ โ (โโ(๐ / (absโ๐))) = ((โโ๐) / (โโ(absโ๐)))) |
34 | 4 | cjred 15213 | . . . . . . . 8 โข (๐ โ (โโ(absโ๐)) = (absโ๐)) |
35 | 34 | oveq2d 7442 | . . . . . . 7 โข (๐ โ ((โโ๐) / (โโ(absโ๐))) = ((โโ๐) / (absโ๐))) |
36 | 32, 33, 35 | 3eqtrd 2772 | . . . . . 6 โข (๐ โ (expโ(-i ยท (โโ(logโ๐)))) = ((โโ๐) / (absโ๐))) |
37 | 18, 36 | oveq12d 7444 | . . . . 5 โข (๐ โ ((expโ(i ยท (โโ(logโ๐)))) + (expโ(-i ยท (โโ(logโ๐))))) = ((๐ / (absโ๐)) + ((โโ๐) / (absโ๐)))) |
38 | 1, 2, 5, 8 | divdird 12066 | . . . . 5 โข (๐ โ ((๐ + (โโ๐)) / (absโ๐)) = ((๐ / (absโ๐)) + ((โโ๐) / (absโ๐)))) |
39 | 37, 38 | eqtr4d 2771 | . . . 4 โข (๐ โ ((expโ(i ยท (โโ(logโ๐)))) + (expโ(-i ยท (โโ(logโ๐))))) = ((๐ + (โโ๐)) / (absโ๐))) |
40 | 39 | oveq1d 7441 | . . 3 โข (๐ โ (((expโ(i ยท (โโ(logโ๐)))) + (expโ(-i ยท (โโ(logโ๐))))) / 2) = (((๐ + (โโ๐)) / (absโ๐)) / 2)) |
41 | 16, 40 | eqtrd 2768 | . 2 โข (๐ โ (cosโ(โโ(logโ๐))) = (((๐ + (โโ๐)) / (absโ๐)) / 2)) |
42 | reval 15093 | . . . 4 โข (๐ โ โ โ (โโ๐) = ((๐ + (โโ๐)) / 2)) | |
43 | 1, 42 | syl 17 | . . 3 โข (๐ โ (โโ๐) = ((๐ + (โโ๐)) / 2)) |
44 | 43 | oveq1d 7441 | . 2 โข (๐ โ ((โโ๐) / (absโ๐)) = (((๐ + (โโ๐)) / 2) / (absโ๐))) |
45 | 11, 41, 44 | 3eqtr4d 2778 | 1 โข (๐ โ (cosโ(โโ(logโ๐))) = ((โโ๐) / (absโ๐))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โ wne 2937 โcfv 6553 (class class class)co 7426 โcc 11144 0cc0 11146 ici 11148 + caddc 11149 ยท cmul 11151 -cneg 11483 / cdiv 11909 2c2 12305 โccj 15083 โcre 15084 โcim 15085 abscabs 15221 expce 16045 cosccos 16048 logclog 26508 |
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 ax-addf 11225 |
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-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 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-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 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-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 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 df-pi 16056 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24818 df-limc 25815 df-dv 25816 df-log 26510 |
This theorem is referenced by: cosarg0d 26563 cosangneg2d 26759 |
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