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Mirrors > Home > ILE Home > Th. List > recos4p | Unicode version |
Description: Separate out the first four terms of the infinite series expansion of the cosine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
efi4p.1 |
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Ref | Expression |
---|---|
recos4p |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recosval 11723 |
. 2
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2 | recn 7943 |
. . . . 5
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3 | efi4p.1 |
. . . . . 6
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4 | 3 | efi4p 11724 |
. . . . 5
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5 | 2, 4 | syl 14 |
. . . 4
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6 | 5 | fveq2d 5519 |
. . 3
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7 | 1re 7955 |
. . . . . . 7
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8 | resqcl 10587 |
. . . . . . . 8
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9 | 8 | rehalfcld 9164 |
. . . . . . 7
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10 | resubcl 8220 |
. . . . . . 7
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11 | 7, 9, 10 | sylancr 414 |
. . . . . 6
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12 | 11 | recnd 7985 |
. . . . 5
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13 | ax-icn 7905 |
. . . . . 6
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14 | 3nn0 9193 |
. . . . . . . . . 10
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15 | reexpcl 10536 |
. . . . . . . . . 10
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16 | 14, 15 | mpan2 425 |
. . . . . . . . 9
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17 | 6re 8999 |
. . . . . . . . . 10
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18 | 6pos 9019 |
. . . . . . . . . . 11
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19 | 17, 18 | gt0ap0ii 8584 |
. . . . . . . . . 10
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20 | redivclap 8687 |
. . . . . . . . . 10
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21 | 17, 19, 20 | mp3an23 1329 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 16, 21 | syl 14 |
. . . . . . . 8
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23 | resubcl 8220 |
. . . . . . . 8
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24 | 22, 23 | mpdan 421 |
. . . . . . 7
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25 | 24 | recnd 7985 |
. . . . . 6
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26 | mulcl 7937 |
. . . . . 6
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27 | 13, 25, 26 | sylancr 414 |
. . . . 5
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28 | 12, 27 | addcld 7976 |
. . . 4
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29 | mulcl 7937 |
. . . . . 6
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30 | 13, 2, 29 | sylancr 414 |
. . . . 5
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31 | 4nn0 9194 |
. . . . 5
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32 | 3 | eftlcl 11695 |
. . . . 5
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33 | 30, 31, 32 | sylancl 413 |
. . . 4
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34 | 28, 33 | readdd 10967 |
. . 3
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35 | 11, 24 | crred 10984 |
. . . 4
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36 | 35 | oveq1d 5889 |
. . 3
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37 | 6, 34, 36 | 3eqtrd 2214 |
. 2
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38 | 1, 37 | eqtrd 2210 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-frec 6391 df-1o 6416 df-oadd 6420 df-er 6534 df-en 6740 df-dom 6741 df-fin 6742 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-inn 8919 df-2 8977 df-3 8978 df-4 8979 df-5 8980 df-6 8981 df-n0 9176 df-z 9253 df-uz 9528 df-q 9619 df-rp 9653 df-ico 9893 df-fz 10008 df-fzo 10142 df-seqfrec 10445 df-exp 10519 df-fac 10705 df-ihash 10755 df-cj 10850 df-re 10851 df-im 10852 df-rsqrt 11006 df-abs 11007 df-clim 11286 df-sumdc 11361 df-ef 11655 df-cos 11658 |
This theorem is referenced by: cos01bnd 11765 |
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