Theorem sincossq 12534

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.)

Assertion

Ref	Expression
sincossq	|- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = 1 )

Proof of Theorem sincossq

Step	Hyp	Ref	Expression
1		negcl 8528	. . 3 |- ( A e. CC -> -u A e. CC )
2		cosadd 12523	. . 3 |- ( ( A e. CC /\ -u A e. CC ) -> ( cos ` ( A + -u A ) ) = ( ( ( cos ` A ) x. ( cos ` -u A ) ) - ( ( sin ` A ) x. ( sin ` -u A ) ) ) )
3	1, 2	mpdan 425	. 2 |- ( A e. CC -> ( cos ` ( A + -u A ) ) = ( ( ( cos ` A ) x. ( cos ` -u A ) ) - ( ( sin ` A ) x. ( sin ` -u A ) ) ) )
4		negid 8575	. . . 4 |- ( A e. CC -> ( A + -u A ) = 0 )
5	4	fveq2d 5699	. . 3 |- ( A e. CC -> ( cos ` ( A + -u A ) ) = ( cos ` 0 ) )
6		cos0 12516	. . 3 |- ( cos ` 0 ) = 1
7	5, 6	eqtrdi 2287	. 2 |- ( A e. CC -> ( cos ` ( A + -u A ) ) = 1 )
8		sincl 12492	. . . . 5 |- ( A e. CC -> ( sin ` A ) e. CC )
9	8	sqcld 11124	. . . 4 |- ( A e. CC -> ( ( sin ` A ) ^ 2 ) e. CC )
10		coscl 12493	. . . . 5 |- ( A e. CC -> ( cos ` A ) e. CC )
11	10	sqcld 11124	. . . 4 |- ( A e. CC -> ( ( cos ` A ) ^ 2 ) e. CC )
12	9, 11	addcomd 8479	. . 3 |- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = ( ( ( cos ` A ) ^ 2 ) + ( ( sin ` A ) ^ 2 ) ) )
13	10	sqvald 11123	. . . . 5 |- ( A e. CC -> ( ( cos ` A ) ^ 2 ) = ( ( cos ` A ) x. ( cos ` A ) ) )
14		cosneg 12513	. . . . . 6 |- ( A e. CC -> ( cos ` -u A ) = ( cos ` A ) )
15	14	oveq2d 6101	. . . . 5 |- ( A e. CC -> ( ( cos ` A ) x. ( cos ` -u A ) ) = ( ( cos ` A ) x. ( cos ` A ) ) )
16	13, 15	eqtr4d 2274	. . . 4 |- ( A e. CC -> ( ( cos ` A ) ^ 2 ) = ( ( cos ` A ) x. ( cos ` -u A ) ) )
17	8	sqvald 11123	. . . . . 6 |- ( A e. CC -> ( ( sin ` A ) ^ 2 ) = ( ( sin ` A ) x. ( sin ` A ) ) )
18		sinneg 12512	. . . . . . . . 9 |- ( A e. CC -> ( sin ` -u A ) = -u ( sin ` A ) )
19	18	negeqd 8523	. . . . . . . 8 |- ( A e. CC -> -u ( sin ` -u A ) = -u -u ( sin ` A ) )
20	8	negnegd 8630	. . . . . . . 8 |- ( A e. CC -> -u -u ( sin ` A ) = ( sin ` A ) )
21	19, 20	eqtrd 2271	. . . . . . 7 |- ( A e. CC -> -u ( sin ` -u A ) = ( sin ` A ) )
22	21	oveq2d 6101	. . . . . 6 |- ( A e. CC -> ( ( sin ` A ) x. -u ( sin ` -u A ) ) = ( ( sin ` A ) x. ( sin ` A ) ) )
23	17, 22	eqtr4d 2274	. . . . 5 |- ( A e. CC -> ( ( sin ` A ) ^ 2 ) = ( ( sin ` A ) x. -u ( sin ` -u A ) ) )
24	1	sincld 12496	. . . . . 6 |- ( A e. CC -> ( sin ` -u A ) e. CC )
25	8, 24	mulneg2d 8741	. . . . 5 |- ( A e. CC -> ( ( sin ` A ) x. -u ( sin ` -u A ) ) = -u ( ( sin ` A ) x. ( sin ` -u A ) ) )
26	23, 25	eqtrd 2271	. . . 4 |- ( A e. CC -> ( ( sin ` A ) ^ 2 ) = -u ( ( sin ` A ) x. ( sin ` -u A ) ) )
27	16, 26	oveq12d 6103	. . 3 |- ( A e. CC -> ( ( ( cos ` A ) ^ 2 ) + ( ( sin ` A ) ^ 2 ) ) = ( ( ( cos ` A ) x. ( cos ` -u A ) ) + -u ( ( sin ` A ) x. ( sin ` -u A ) ) ) )
28	1	coscld 12497	. . . . 5 |- ( A e. CC -> ( cos ` -u A ) e. CC )
29	10, 28	mulcld 8347	. . . 4 |- ( A e. CC -> ( ( cos ` A ) x. ( cos ` -u A ) ) e. CC )
30	8, 24	mulcld 8347	. . . 4 |- ( A e. CC -> ( ( sin ` A ) x. ( sin ` -u A ) ) e. CC )
31	29, 30	negsubd 8645	. . 3 |- ( A e. CC -> ( ( ( cos ` A ) x. ( cos ` -u A ) ) + -u ( ( sin ` A ) x. ( sin ` -u A ) ) ) = ( ( ( cos ` A ) x. ( cos ` -u A ) ) - ( ( sin ` A ) x. ( sin ` -u A ) ) ) )
32	12, 27, 31	3eqtrrd 2276	. 2 |- ( A e. CC -> ( ( ( cos ` A ) x. ( cos ` -u A ) ) - ( ( sin ` A ) x. ( sin ` -u A ) ) ) = ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) )
33	3, 7, 32	3eqtr3rd 2280	1 |- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = 1 )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   CCcc 8178   0cc0 8180   1c1 8181   + caddc 8183   x. cmul 8185   - cmin 8499   -ucneg 8500   2c2 9358   ^cexp 10990   sincsin 12430   cosccos 12431

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299  ax-caucvg 8300

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7325  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-n0 9569  df-z 9650  df-uz 9932  df-q 10030  df-rp 10066  df-ico 10307  df-fz 10423  df-fzo 10561  df-seqfrec 10900  df-exp 10991  df-fac 11180  df-bc 11202  df-ihash 11231  df-cj 11623  df-re 11624  df-im 11625  df-rsqrt 11780  df-abs 11781  df-clim 12064  df-sumdc 12139  df-ef 12434  df-sin 12436  df-cos 12437

This theorem is used by:  cos2t  12536  cos2tsin  12537  sinbnd  12538  cosbnd  12539  absefi  12555  sinhalfpilem  15946  sincos6thpi  15997