Theorem sincossq 12496

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 8519	. . 3 |- ( A e. CC -> -u A e. CC )
2		cosadd 12485	. . 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 8566	. . . 4 |- ( A e. CC -> ( A + -u A ) = 0 )
5	4	fveq2d 5697	. . 3 |- ( A e. CC -> ( cos ` ( A + -u A ) ) = ( cos ` 0 ) )
6		cos0 12478	. . 3 |- ( cos ` 0 ) = 1
7	5, 6	eqtrdi 2287	. 2 |- ( A e. CC -> ( cos ` ( A + -u A ) ) = 1 )
8		sincl 12454	. . . . 5 |- ( A e. CC -> ( sin ` A ) e. CC )
9	8	sqcld 11090	. . . 4 |- ( A e. CC -> ( ( sin ` A ) ^ 2 ) e. CC )
10		coscl 12455	. . . . 5 |- ( A e. CC -> ( cos ` A ) e. CC )
11	10	sqcld 11090	. . . 4 |- ( A e. CC -> ( ( cos ` A ) ^ 2 ) e. CC )
12	9, 11	addcomd 8470	. . 3 |- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = ( ( ( cos ` A ) ^ 2 ) + ( ( sin ` A ) ^ 2 ) ) )
13	10	sqvald 11089	. . . . 5 |- ( A e. CC -> ( ( cos ` A ) ^ 2 ) = ( ( cos ` A ) x. ( cos ` A ) ) )
14		cosneg 12475	. . . . . 6 |- ( A e. CC -> ( cos ` -u A ) = ( cos ` A ) )
15	14	oveq2d 6094	. . . . 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 11089	. . . . . 6 |- ( A e. CC -> ( ( sin ` A ) ^ 2 ) = ( ( sin ` A ) x. ( sin ` A ) ) )
18		sinneg 12474	. . . . . . . . 9 |- ( A e. CC -> ( sin ` -u A ) = -u ( sin ` A ) )
19	18	negeqd 8514	. . . . . . . 8 |- ( A e. CC -> -u ( sin ` -u A ) = -u -u ( sin ` A ) )
20	8	negnegd 8621	. . . . . . . 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 6094	. . . . . 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 12458	. . . . . 6 |- ( A e. CC -> ( sin ` -u A ) e. CC )
25	8, 24	mulneg2d 8732	. . . . 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 6096	. . 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 12459	. . . . 5 |- ( A e. CC -> ( cos ` -u A ) e. CC )
29	10, 28	mulcld 8339	. . . 4 |- ( A e. CC -> ( ( cos ` A ) x. ( cos ` -u A ) ) e. CC )
30	8, 24	mulcld 8339	. . . 4 |- ( A e. CC -> ( ( sin ` A ) x. ( sin ` -u A ) ) e. CC )
31	29, 30	negsubd 8636	. . 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 )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   ` cfv 5375  (class class class)co 6078   CCcc 8170   0cc0 8172   1c1 8173   + caddc 8175   x. cmul 8177   - cmin 8490   -ucneg 8491   2c2 9337   ^cexp 10956   sincsin 12392   cosccos 12393

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 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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-mulrcl 8271  ax-addcom 8272  ax-mulcom 8273  ax-addass 8274  ax-mulass 8275  ax-distr 8276  ax-i2m1 8277  ax-0lt1 8278  ax-1rid 8279  ax-0id 8280  ax-rnegex 8281  ax-precex 8282  ax-cnre 8283  ax-pre-ltirr 8284  ax-pre-ltwlin 8285  ax-pre-lttrn 8286  ax-pre-apti 8287  ax-pre-ltadd 8288  ax-pre-mulgt0 8289  ax-pre-mulext 8290  ax-arch 8291  ax-caucvg 8292

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-disj 4105  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-recs 6569  df-irdg 6634  df-frec 6655  df-1o 6680  df-oadd 6684  df-er 6800  df-en 7016  df-dom 7017  df-fin 7018  df-sup 7317  df-pnf 8355  df-mnf 8356  df-xr 8357  df-ltxr 8358  df-le 8359  df-sub 8492  df-neg 8493  df-reap 8896  df-ap 8903  df-div 8996  df-inn 9287  df-2 9345  df-3 9346  df-4 9347  df-n0 9546  df-z 9627  df-uz 9904  df-q 10002  df-rp 10037  df-ico 10278  df-fz 10394  df-fzo 10531  df-seqfrec 10866  df-exp 10957  df-fac 11145  df-bc 11167  df-ihash 11196  df-cj 11588  df-re 11589  df-im 11590  df-rsqrt 11745  df-abs 11746  df-clim 12026  df-sumdc 12101  df-ef 12396  df-sin 12398  df-cos 12399

This theorem is referenced by:  cos2t  12498  cos2tsin  12499  sinbnd  12500  cosbnd  12501  absefi  12517  sinhalfpilem  15818  sincos6thpi  15869