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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 10253 | . . 3 ⊢ 0 ∈ ℝ | |
2 | recosval 15086 | . . 3 ⊢ (0 ∈ ℝ → (cos‘0) = (ℜ‘(exp‘(i · 0)))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (cos‘0) = (ℜ‘(exp‘(i · 0))) |
4 | it0e0 11467 | . . . . . 6 ⊢ (i · 0) = 0 | |
5 | 4 | fveq2i 6357 | . . . . 5 ⊢ (exp‘(i · 0)) = (exp‘0) |
6 | ef0 15041 | . . . . 5 ⊢ (exp‘0) = 1 | |
7 | 5, 6 | eqtri 2783 | . . . 4 ⊢ (exp‘(i · 0)) = 1 |
8 | 7 | fveq2i 6357 | . . 3 ⊢ (ℜ‘(exp‘(i · 0))) = (ℜ‘1) |
9 | re1 14114 | . . 3 ⊢ (ℜ‘1) = 1 | |
10 | 8, 9 | eqtri 2783 | . 2 ⊢ (ℜ‘(exp‘(i · 0))) = 1 |
11 | 3, 10 | eqtri 2783 | 1 ⊢ (cos‘0) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∈ wcel 2140 ‘cfv 6050 (class class class)co 6815 ℝcr 10148 0cc0 10149 1c1 10150 ici 10151 · cmul 10154 ℜcre 14057 expce 15012 cosccos 15015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-rep 4924 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 ax-inf2 8714 ax-cnex 10205 ax-resscn 10206 ax-1cn 10207 ax-icn 10208 ax-addcl 10209 ax-addrcl 10210 ax-mulcl 10211 ax-mulrcl 10212 ax-mulcom 10213 ax-addass 10214 ax-mulass 10215 ax-distr 10216 ax-i2m1 10217 ax-1ne0 10218 ax-1rid 10219 ax-rnegex 10220 ax-rrecex 10221 ax-cnre 10222 ax-pre-lttri 10223 ax-pre-lttrn 10224 ax-pre-ltadd 10225 ax-pre-mulgt0 10226 ax-pre-sup 10227 ax-addf 10228 ax-mulf 10229 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-uni 4590 df-int 4629 df-iun 4675 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-se 5227 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-isom 6059 df-riota 6776 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-om 7233 df-1st 7335 df-2nd 7336 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-1o 7731 df-oadd 7735 df-er 7914 df-pm 8029 df-en 8125 df-dom 8126 df-sdom 8127 df-fin 8128 df-sup 8516 df-inf 8517 df-oi 8583 df-card 8976 df-pnf 10289 df-mnf 10290 df-xr 10291 df-ltxr 10292 df-le 10293 df-sub 10481 df-neg 10482 df-div 10898 df-nn 11234 df-2 11292 df-3 11293 df-n0 11506 df-z 11591 df-uz 11901 df-rp 12047 df-ico 12395 df-fz 12541 df-fzo 12681 df-fl 12808 df-seq 13017 df-exp 13076 df-fac 13276 df-hash 13333 df-shft 14027 df-cj 14059 df-re 14060 df-im 14061 df-sqrt 14195 df-abs 14196 df-limsup 14422 df-clim 14439 df-rlim 14440 df-sum 14637 df-ef 15018 df-cos 15021 |
This theorem is referenced by: tan0 15101 sincossq 15126 demoivreALT 15151 cos2kpi 24457 coseq00topi 24475 recosf1o 24502 ex-co 27628 tan2h 33733 cosknegpi 40602 itgsin0pilem1 40687 fourierdlem62 40907 fourierdlem83 40928 sqwvfoura 40967 sqwvfourb 40968 sec0 43033 |
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