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Mirrors > Home > MPE Home > Th. List > cos0 | Structured version Visualization version GIF version |
Description: Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.) |
Ref | Expression |
---|---|
cos0 | ⊢ (cos‘0) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 11241 | . . 3 ⊢ 0 ∈ ℝ | |
2 | recosval 16107 | . . 3 ⊢ (0 ∈ ℝ → (cos‘0) = (ℜ‘(exp‘(i · 0)))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (cos‘0) = (ℜ‘(exp‘(i · 0))) |
4 | it0e0 12459 | . . . . . 6 ⊢ (i · 0) = 0 | |
5 | 4 | fveq2i 6895 | . . . . 5 ⊢ (exp‘(i · 0)) = (exp‘0) |
6 | ef0 16062 | . . . . 5 ⊢ (exp‘0) = 1 | |
7 | 5, 6 | eqtri 2756 | . . . 4 ⊢ (exp‘(i · 0)) = 1 |
8 | 7 | fveq2i 6895 | . . 3 ⊢ (ℜ‘(exp‘(i · 0))) = (ℜ‘1) |
9 | re1 15128 | . . 3 ⊢ (ℜ‘1) = 1 | |
10 | 8, 9 | eqtri 2756 | . 2 ⊢ (ℜ‘(exp‘(i · 0))) = 1 |
11 | 3, 10 | eqtri 2756 | 1 ⊢ (cos‘0) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ‘cfv 6543 (class class class)co 7415 ℝcr 11132 0cc0 11133 1c1 11134 ici 11135 · cmul 11138 ℜcre 15071 expce 16032 cosccos 16035 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
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 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-pm 8842 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-ico 13357 df-fz 13512 df-fzo 13655 df-fl 13784 df-seq 13994 df-exp 14054 df-fac 14260 df-hash 14317 df-shft 15041 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-limsup 15442 df-clim 15459 df-rlim 15460 df-sum 15660 df-ef 16038 df-cos 16041 |
This theorem is referenced by: tan0 16122 sincossq 16147 demoivreALT 16172 cos2kpi 26413 coseq00topi 26431 recosf1o 26463 ex-co 30242 tan2h 37080 cosknegpi 45248 itgsin0pilem1 45329 fourierdlem62 45547 fourierdlem83 45568 sqwvfoura 45607 sqwvfourb 45608 sec0 48182 |
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