Theorem sincos4thpi 15530

Description: The sine and cosine of pi / 4. (Contributed by Paul Chapman, 25-Jan-2008.)

Assertion

Ref	Expression
sincos4thpi	|- ( ( sin ` ( pi / 4 ) ) = ( 1 / ( sqr ` 2 ) ) /\ ( cos ` ( pi / 4 ) ) = ( 1 / ( sqr ` 2 ) ) )

Proof of Theorem sincos4thpi

Step	Hyp	Ref	Expression
1		halfcn 9336	. . . . . . . . . 10 |- ( 1 / 2 ) e. CC
2		ax-1cn 8103	. . . . . . . . . . 11 |- 1 e. CC
3		2halves 9351	. . . . . . . . . . 11 |- ( 1 e. CC -> ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1 )
4	2, 3	ax-mp 5	. . . . . . . . . 10 |- ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1
5		sincosq1eq 15529	. . . . . . . . . 10 |- ( ( ( 1 / 2 ) e. CC /\ ( 1 / 2 ) e. CC /\ ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1 ) -> ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) = ( cos ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) )
6	1, 1, 4, 5	mp3an 1371	. . . . . . . . 9 |- ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) = ( cos ` ( ( 1 / 2 ) x. ( pi / 2 ) ) )
7	6	oveq2i 6018	. . . . . . . 8 |- ( ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) x. ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) = ( ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) x. ( cos ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) )
8	7	oveq2i 6018	. . . . . . 7 |- ( 2 x. ( ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) x. ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) ) = ( 2 x. ( ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) x. ( cos ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) )
9		2cn 9192	. . . . . . . . . . . 12 |- 2 e. CC
10		pire 15476	. . . . . . . . . . . . 13 |- pi e. RR
11	10	recni 8169	. . . . . . . . . . . 12 |- pi e. CC
12		2ap0 9214	. . . . . . . . . . . 12 |- 2 # 0
13	2, 9, 11, 9, 12, 12	divmuldivapi 8930	. . . . . . . . . . 11 |- ( ( 1 / 2 ) x. ( pi / 2 ) ) = ( ( 1 x. pi ) / ( 2 x. 2 ) )
14	11	mullidi 8160	. . . . . . . . . . . 12 |- ( 1 x. pi ) = pi
15		2t2e4 9276	. . . . . . . . . . . 12 |- ( 2 x. 2 ) = 4
16	14, 15	oveq12i 6019	. . . . . . . . . . 11 |- ( ( 1 x. pi ) / ( 2 x. 2 ) ) = ( pi / 4 )
17	13, 16	eqtri 2250	. . . . . . . . . 10 |- ( ( 1 / 2 ) x. ( pi / 2 ) ) = ( pi / 4 )
18	17	fveq2i 5632	. . . . . . . . 9 |- ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) = ( sin ` ( pi / 4 ) )
19	18, 18	oveq12i 6019	. . . . . . . 8 |- ( ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) x. ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) = ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) )
20	19	oveq2i 6018	. . . . . . 7 |- ( 2 x. ( ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) x. ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) ) = ( 2 x. ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) )
21	9, 12	recidapi 8901	. . . . . . . . . . 11 |- ( 2 x. ( 1 / 2 ) ) = 1
22	21	oveq1i 6017	. . . . . . . . . 10 |- ( ( 2 x. ( 1 / 2 ) ) x. ( pi / 2 ) ) = ( 1 x. ( pi / 2 ) )
23		2re 9191	. . . . . . . . . . . . 13 |- 2 e. RR
24	10, 23, 12	redivclapi 8937	. . . . . . . . . . . 12 |- ( pi / 2 ) e. RR
25	24	recni 8169	. . . . . . . . . . 11 |- ( pi / 2 ) e. CC
26	9, 1, 25	mulassi 8166	. . . . . . . . . 10 |- ( ( 2 x. ( 1 / 2 ) ) x. ( pi / 2 ) ) = ( 2 x. ( ( 1 / 2 ) x. ( pi / 2 ) ) )
27	25	mullidi 8160	. . . . . . . . . 10 |- ( 1 x. ( pi / 2 ) ) = ( pi / 2 )
28	22, 26, 27	3eqtr3i 2258	. . . . . . . . 9 |- ( 2 x. ( ( 1 / 2 ) x. ( pi / 2 ) ) ) = ( pi / 2 )
29	28	fveq2i 5632	. . . . . . . 8 |- ( sin ` ( 2 x. ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) = ( sin ` ( pi / 2 ) )
30	1, 25	mulcli 8162	. . . . . . . . 9 |- ( ( 1 / 2 ) x. ( pi / 2 ) ) e. CC
31		sin2t 12276	. . . . . . . . 9 |- ( ( ( 1 / 2 ) x. ( pi / 2 ) ) e. CC -> ( sin ` ( 2 x. ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) = ( 2 x. ( ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) x. ( cos ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) ) )
32	30, 31	ax-mp 5	. . . . . . . 8 |- ( sin ` ( 2 x. ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) = ( 2 x. ( ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) x. ( cos ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) )
33		sinhalfpi 15486	. . . . . . . 8 |- ( sin ` ( pi / 2 ) ) = 1
34	29, 32, 33	3eqtr3i 2258	. . . . . . 7 |- ( 2 x. ( ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) x. ( cos ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) ) = 1
35	8, 20, 34	3eqtr3i 2258	. . . . . 6 |- ( 2 x. ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) = 1
36	35	fveq2i 5632	. . . . 5 |- ( sqr ` ( 2 x. ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) ) = ( sqr ` 1 )
37		4re 9198	. . . . . . . . 9 |- 4 e. RR
38		4ap0 9220	. . . . . . . . 9 |- 4 # 0
39	10, 37, 38	redivclapi 8937	. . . . . . . 8 |- ( pi / 4 ) e. RR
40		resincl 12247	. . . . . . . 8 |- ( ( pi / 4 ) e. RR -> ( sin ` ( pi / 4 ) ) e. RR )
41	39, 40	ax-mp 5	. . . . . . 7 |- ( sin ` ( pi / 4 ) ) e. RR
42	41, 41	remulcli 8171	. . . . . 6 |- ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) e. RR
43		0le2 9211	. . . . . 6 |- 0 <_ 2
44	41	msqge0i 8775	. . . . . 6 |- 0 <_ ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) )
45	23, 42, 43, 44	sqrtmulii 11661	. . . . 5 |- ( sqr ` ( 2 x. ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) ) = ( ( sqr ` 2 ) x. ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) )
46		sqrt1 11573	. . . . 5 |- ( sqr ` 1 ) = 1
47	36, 45, 46	3eqtr3ri 2259	. . . 4 |- 1 = ( ( sqr ` 2 ) x. ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) )
48	42	sqrtcli 11647	. . . . . . 7 |- ( 0 <_ ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) -> ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) e. RR )
49	44, 48	ax-mp 5	. . . . . 6 |- ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) e. RR
50	49	recni 8169	. . . . 5 |- ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) e. CC
51		sqrt2re 12701	. . . . . . 7 |- ( sqr ` 2 ) e. RR
52	51	recni 8169	. . . . . 6 |- ( sqr ` 2 ) e. CC
53		2pos 9212	. . . . . . . 8 |- 0 < 2
54	23, 53	sqrtgt0ii 11658	. . . . . . 7 |- 0 < ( sqr ` 2 )
55	51, 54	gt0ap0ii 8786	. . . . . 6 |- ( sqr ` 2 ) # 0
56	52, 55	pm3.2i 272	. . . . 5 |- ( ( sqr ` 2 ) e. CC /\ ( sqr ` 2 ) # 0 )
57		divmulap2 8834	. . . . 5 |- ( ( 1 e. CC /\ ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) e. CC /\ ( ( sqr ` 2 ) e. CC /\ ( sqr ` 2 ) # 0 ) ) -> ( ( 1 / ( sqr ` 2 ) ) = ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) <-> 1 = ( ( sqr ` 2 ) x. ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) ) ) )
58	2, 50, 56, 57	mp3an 1371	. . . 4 |- ( ( 1 / ( sqr ` 2 ) ) = ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) <-> 1 = ( ( sqr ` 2 ) x. ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) ) )
59	47, 58	mpbir 146	. . 3 |- ( 1 / ( sqr ` 2 ) ) = ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) )
60		0re 8157	. . . . 5 |- 0 e. RR
61		pipos 15478	. . . . . . . 8 |- 0 < pi
62		4pos 9218	. . . . . . . 8 |- 0 < 4
63	10, 37, 61, 62	divgt0ii 9077	. . . . . . 7 |- 0 < ( pi / 4 )
64		1re 8156	. . . . . . . 8 |- 1 e. RR
65		pigt2lt4 15474	. . . . . . . . . . 11 |- ( 2 < pi /\ pi < 4 )
66	65	simpri 113	. . . . . . . . . 10 |- pi < 4
67	10, 37, 37, 62	ltdiv1ii 9087	. . . . . . . . . 10 |- ( pi < 4 <-> ( pi / 4 ) < ( 4 / 4 ) )
68	66, 67	mpbi 145	. . . . . . . . 9 |- ( pi / 4 ) < ( 4 / 4 )
69	37	recni 8169	. . . . . . . . . 10 |- 4 e. CC
70	69, 38	dividapi 8903	. . . . . . . . 9 |- ( 4 / 4 ) = 1
71	68, 70	breqtri 4108	. . . . . . . 8 |- ( pi / 4 ) < 1
72	39, 64, 71	ltleii 8260	. . . . . . 7 |- ( pi / 4 ) <_ 1
73		0xr 8204	. . . . . . . 8 |- 0 e. RR*
74		elioc2 10144	. . . . . . . 8 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( ( pi / 4 ) e. ( 0 (,] 1 ) <-> ( ( pi / 4 ) e. RR /\ 0 < ( pi / 4 ) /\ ( pi / 4 ) <_ 1 ) ) )
75	73, 64, 74	mp2an 426	. . . . . . 7 |- ( ( pi / 4 ) e. ( 0 (,] 1 ) <-> ( ( pi / 4 ) e. RR /\ 0 < ( pi / 4 ) /\ ( pi / 4 ) <_ 1 ) )
76	39, 63, 72, 75	mpbir3an 1203	. . . . . 6 |- ( pi / 4 ) e. ( 0 (,] 1 )
77		sin01gt0 12289	. . . . . 6 |- ( ( pi / 4 ) e. ( 0 (,] 1 ) -> 0 < ( sin ` ( pi / 4 ) ) )
78	76, 77	ax-mp 5	. . . . 5 |- 0 < ( sin ` ( pi / 4 ) )
79	60, 41, 78	ltleii 8260	. . . 4 |- 0 <_ ( sin ` ( pi / 4 ) )
80	41	sqrtmsqi 11649	. . . 4 |- ( 0 <_ ( sin ` ( pi / 4 ) ) -> ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) = ( sin ` ( pi / 4 ) ) )
81	79, 80	ax-mp 5	. . 3 |- ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) = ( sin ` ( pi / 4 ) )
82	59, 81	eqtr2i 2251	. 2 |- ( sin ` ( pi / 4 ) ) = ( 1 / ( sqr ` 2 ) )
83	59, 81	eqtri 2250	. . 3 |- ( 1 / ( sqr ` 2 ) ) = ( sin ` ( pi / 4 ) )
84	17	fveq2i 5632	. . . 4 |- ( cos ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) = ( cos ` ( pi / 4 ) )
85	6, 18, 84	3eqtr3i 2258	. . 3 |- ( sin ` ( pi / 4 ) ) = ( cos ` ( pi / 4 ) )
86	83, 85	eqtr2i 2251	. 2 |- ( cos ` ( pi / 4 ) ) = ( 1 / ( sqr ` 2 ) )
87	82, 86	pm3.2i 272	1 |- ( ( sin ` ( pi / 4 ) ) = ( 1 / ( sqr ` 2 ) ) /\ ( cos ` ( pi / 4 ) ) = ( 1 / ( sqr ` 2 ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   CCcc 8008   RRcr 8009   0cc0 8010   1c1 8011   + caddc 8013   x. cmul 8015   RR*cxr 8191   < clt 8192   <_ cle 8193   # cap 8739   / cdiv 8830   2c2 9172   4c4 9174   (,]cioc 10097   sqrcsqrt 11523   sincsin 12171   cosccos 12172   picpi 12174

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130  ax-pre-suploc 8131  ax-addf 8132  ax-mulf 8133

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-disj 4060  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-of 6224  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-oadd 6572  df-er 6688  df-map 6805  df-pm 6806  df-en 6896  df-dom 6897  df-fin 6898  df-sup 7162  df-inf 7163  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-xneg 9980  df-xadd 9981  df-ioo 10100  df-ioc 10101  df-ico 10102  df-icc 10103  df-fz 10217  df-fzo 10351  df-seqfrec 10682  df-exp 10773  df-fac 10960  df-bc 10982  df-ihash 11010  df-shft 11342  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-clim 11806  df-sumdc 11881  df-ef 12175  df-sin 12177  df-cos 12178  df-pi 12180  df-rest 13290  df-topgen 13309  df-psmet 14523  df-xmet 14524  df-met 14525  df-bl 14526  df-mopn 14527  df-top 14688  df-topon 14701  df-bases 14733  df-ntr 14786  df-cn 14878  df-cnp 14879  df-tx 14943  df-cncf 15261  df-limced 15346  df-dvap 15347

This theorem is referenced by:  tan4thpi  15531