Theorem sincossq 12438

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 8475	. . 3 |- ( A e. CC -> -u A e. CC )
2		cosadd 12427	. . 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 421	. 2 |- ( A e. CC -> ( cos ` ( A + -u A ) ) = ( ( ( cos ` A ) x. ( cos ` -u A ) ) - ( ( sin ` A ) x. ( sin ` -u A ) ) ) )
4		negid 8522	. . . 4 |- ( A e. CC -> ( A + -u A ) = 0 )
5	4	fveq2d 5676	. . 3 |- ( A e. CC -> ( cos ` ( A + -u A ) ) = ( cos ` 0 ) )
6		cos0 12420	. . 3 |- ( cos ` 0 ) = 1
7	5, 6	eqtrdi 2283	. 2 |- ( A e. CC -> ( cos ` ( A + -u A ) ) = 1 )
8		sincl 12396	. . . . 5 |- ( A e. CC -> ( sin ` A ) e. CC )
9	8	sqcld 11037	. . . 4 |- ( A e. CC -> ( ( sin ` A ) ^ 2 ) e. CC )
10		coscl 12397	. . . . 5 |- ( A e. CC -> ( cos ` A ) e. CC )
11	10	sqcld 11037	. . . 4 |- ( A e. CC -> ( ( cos ` A ) ^ 2 ) e. CC )
12	9, 11	addcomd 8426	. . 3 |- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = ( ( ( cos ` A ) ^ 2 ) + ( ( sin ` A ) ^ 2 ) ) )
13	10	sqvald 11036	. . . . 5 |- ( A e. CC -> ( ( cos ` A ) ^ 2 ) = ( ( cos ` A ) x. ( cos ` A ) ) )
14		cosneg 12417	. . . . . 6 |- ( A e. CC -> ( cos ` -u A ) = ( cos ` A ) )
15	14	oveq2d 6068	. . . . 5 |- ( A e. CC -> ( ( cos ` A ) x. ( cos ` -u A ) ) = ( ( cos ` A ) x. ( cos ` A ) ) )
16	13, 15	eqtr4d 2270	. . . 4 |- ( A e. CC -> ( ( cos ` A ) ^ 2 ) = ( ( cos ` A ) x. ( cos ` -u A ) ) )
17	8	sqvald 11036	. . . . . 6 |- ( A e. CC -> ( ( sin ` A ) ^ 2 ) = ( ( sin ` A ) x. ( sin ` A ) ) )
18		sinneg 12416	. . . . . . . . 9 |- ( A e. CC -> ( sin ` -u A ) = -u ( sin ` A ) )
19	18	negeqd 8470	. . . . . . . 8 |- ( A e. CC -> -u ( sin ` -u A ) = -u -u ( sin ` A ) )
20	8	negnegd 8577	. . . . . . . 8 |- ( A e. CC -> -u -u ( sin ` A ) = ( sin ` A ) )
21	19, 20	eqtrd 2267	. . . . . . 7 |- ( A e. CC -> -u ( sin ` -u A ) = ( sin ` A ) )
22	21	oveq2d 6068	. . . . . 6 |- ( A e. CC -> ( ( sin ` A ) x. -u ( sin ` -u A ) ) = ( ( sin ` A ) x. ( sin ` A ) ) )
23	17, 22	eqtr4d 2270	. . . . 5 |- ( A e. CC -> ( ( sin ` A ) ^ 2 ) = ( ( sin ` A ) x. -u ( sin ` -u A ) ) )
24	1	sincld 12400	. . . . . 6 |- ( A e. CC -> ( sin ` -u A ) e. CC )
25	8, 24	mulneg2d 8687	. . . . 5 |- ( A e. CC -> ( ( sin ` A ) x. -u ( sin ` -u A ) ) = -u ( ( sin ` A ) x. ( sin ` -u A ) ) )
26	23, 25	eqtrd 2267	. . . 4 |- ( A e. CC -> ( ( sin ` A ) ^ 2 ) = -u ( ( sin ` A ) x. ( sin ` -u A ) ) )
27	16, 26	oveq12d 6070	. . 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 12401	. . . . 5 |- ( A e. CC -> ( cos ` -u A ) e. CC )
29	10, 28	mulcld 8296	. . . 4 |- ( A e. CC -> ( ( cos ` A ) x. ( cos ` -u A ) ) e. CC )
30	8, 24	mulcld 8296	. . . 4 |- ( A e. CC -> ( ( sin ` A ) x. ( sin ` -u A ) ) e. CC )
31	29, 30	negsubd 8592	. . 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 2272	. 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 2276	1 |- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = 1 )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   ` cfv 5354  (class class class)co 6052   CCcc 8127   0cc0 8129   1c1 8130   + caddc 8132   x. cmul 8134   - cmin 8446   -ucneg 8447   2c2 9290   ^cexp 10904   sincsin 12334   cosccos 12335

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248  ax-caucvg 8249

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-disj 4088  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-sup 7277  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-ico 10230  df-fz 10346  df-fzo 10481  df-seqfrec 10814  df-exp 10905  df-fac 11092  df-bc 11114  df-ihash 11143  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-clim 11968  df-sumdc 12043  df-ef 12338  df-sin 12340  df-cos 12341

This theorem is referenced by:  cos2t  12440  cos2tsin  12441  sinbnd  12442  cosbnd  12443  absefi  12459  sinhalfpilem  15673  sincos6thpi  15724