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Mirrors > Home > ILE Home > Th. List > sincossq | Unicode version |
Description: Sine squared plus cosine squared is 1. Equation 17 of [Gleason] p. 311. Note that this holds for non-real arguments, even though individually each term is unbounded. (Contributed by NM, 15-Jan-2006.) |
Ref | Expression |
---|---|
sincossq |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negcl 8182 |
. . 3
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2 | cosadd 11772 |
. . 3
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3 | 1, 2 | mpdan 421 |
. 2
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4 | negid 8229 |
. . . 4
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5 | 4 | fveq2d 5535 |
. . 3
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6 | cos0 11765 |
. . 3
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7 | 5, 6 | eqtrdi 2238 |
. 2
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8 | sincl 11741 |
. . . . 5
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9 | 8 | sqcld 10678 |
. . . 4
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10 | coscl 11742 |
. . . . 5
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11 | 10 | sqcld 10678 |
. . . 4
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12 | 9, 11 | addcomd 8133 |
. . 3
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13 | 10 | sqvald 10677 |
. . . . 5
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14 | cosneg 11762 |
. . . . . 6
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15 | 14 | oveq2d 5908 |
. . . . 5
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16 | 13, 15 | eqtr4d 2225 |
. . . 4
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17 | 8 | sqvald 10677 |
. . . . . 6
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18 | sinneg 11761 |
. . . . . . . . 9
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19 | 18 | negeqd 8177 |
. . . . . . . 8
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20 | 8 | negnegd 8284 |
. . . . . . . 8
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21 | 19, 20 | eqtrd 2222 |
. . . . . . 7
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22 | 21 | oveq2d 5908 |
. . . . . 6
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23 | 17, 22 | eqtr4d 2225 |
. . . . 5
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24 | 1 | sincld 11745 |
. . . . . 6
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25 | 8, 24 | mulneg2d 8394 |
. . . . 5
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26 | 23, 25 | eqtrd 2222 |
. . . 4
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27 | 16, 26 | oveq12d 5910 |
. . 3
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28 | 1 | coscld 11746 |
. . . . 5
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29 | 10, 28 | mulcld 8003 |
. . . 4
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30 | 8, 24 | mulcld 8003 |
. . . 4
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31 | 29, 30 | negsubd 8299 |
. . 3
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32 | 12, 27, 31 | 3eqtrrd 2227 |
. 2
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33 | 3, 7, 32 | 3eqtr3rd 2231 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-mulrcl 7935 ax-addcom 7936 ax-mulcom 7937 ax-addass 7938 ax-mulass 7939 ax-distr 7940 ax-i2m1 7941 ax-0lt1 7942 ax-1rid 7943 ax-0id 7944 ax-rnegex 7945 ax-precex 7946 ax-cnre 7947 ax-pre-ltirr 7948 ax-pre-ltwlin 7949 ax-pre-lttrn 7950 ax-pre-apti 7951 ax-pre-ltadd 7952 ax-pre-mulgt0 7953 ax-pre-mulext 7954 ax-arch 7955 ax-caucvg 7956 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-disj 3996 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-isom 5241 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-recs 6325 df-irdg 6390 df-frec 6411 df-1o 6436 df-oadd 6440 df-er 6554 df-en 6762 df-dom 6763 df-fin 6764 df-sup 7008 df-pnf 8019 df-mnf 8020 df-xr 8021 df-ltxr 8022 df-le 8023 df-sub 8155 df-neg 8156 df-reap 8557 df-ap 8564 df-div 8655 df-inn 8945 df-2 9003 df-3 9004 df-4 9005 df-n0 9202 df-z 9279 df-uz 9554 df-q 9645 df-rp 9679 df-ico 9919 df-fz 10034 df-fzo 10168 df-seqfrec 10472 df-exp 10546 df-fac 10733 df-bc 10755 df-ihash 10783 df-cj 10878 df-re 10879 df-im 10880 df-rsqrt 11034 df-abs 11035 df-clim 11314 df-sumdc 11389 df-ef 11683 df-sin 11685 df-cos 11686 |
This theorem is referenced by: cos2t 11785 cos2tsin 11786 sinbnd 11787 cosbnd 11788 absefi 11803 sinhalfpilem 14649 sincos6thpi 14700 |
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