Theorem sincos4thpi 15651

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 9417	. . . . . . . . . 10 |- ( 1 / 2 ) e. CC
2		ax-1cn 8185	. . . . . . . . . . 11 |- 1 e. CC
3		2halves 9432	. . . . . . . . . . 11 |- ( 1 e. CC -> ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1 )
4	2, 3	ax-mp 5	. . . . . . . . . 10 |- ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1
5		sincosq1eq 15650	. . . . . . . . . 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 1374	. . . . . . . . 9 |- ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) = ( cos ` ( ( 1 / 2 ) x. ( pi / 2 ) ) )
7	6	oveq2i 6039	. . . . . . . 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 6039	. . . . . . 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 9273	. . . . . . . . . . . 12 |- 2 e. CC
10		pire 15597	. . . . . . . . . . . . 13 |- pi e. RR
11	10	recni 8251	. . . . . . . . . . . 12 |- pi e. CC
12		2ap0 9295	. . . . . . . . . . . 12 |- 2 # 0
13	2, 9, 11, 9, 12, 12	divmuldivapi 9011	. . . . . . . . . . 11 |- ( ( 1 / 2 ) x. ( pi / 2 ) ) = ( ( 1 x. pi ) / ( 2 x. 2 ) )
14	11	mullidi 8242	. . . . . . . . . . . 12 |- ( 1 x. pi ) = pi
15		2t2e4 9357	. . . . . . . . . . . 12 |- ( 2 x. 2 ) = 4
16	14, 15	oveq12i 6040	. . . . . . . . . . 11 |- ( ( 1 x. pi ) / ( 2 x. 2 ) ) = ( pi / 4 )
17	13, 16	eqtri 2252	. . . . . . . . . 10 |- ( ( 1 / 2 ) x. ( pi / 2 ) ) = ( pi / 4 )
18	17	fveq2i 5651	. . . . . . . . 9 |- ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) = ( sin ` ( pi / 4 ) )
19	18, 18	oveq12i 6040	. . . . . . . 8 |- ( ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) x. ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) = ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) )
20	19	oveq2i 6039	. . . . . . 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 8982	. . . . . . . . . . 11 |- ( 2 x. ( 1 / 2 ) ) = 1
22	21	oveq1i 6038	. . . . . . . . . 10 |- ( ( 2 x. ( 1 / 2 ) ) x. ( pi / 2 ) ) = ( 1 x. ( pi / 2 ) )
23		2re 9272	. . . . . . . . . . . . 13 |- 2 e. RR
24	10, 23, 12	redivclapi 9018	. . . . . . . . . . . 12 |- ( pi / 2 ) e. RR
25	24	recni 8251	. . . . . . . . . . 11 |- ( pi / 2 ) e. CC
26	9, 1, 25	mulassi 8248	. . . . . . . . . 10 |- ( ( 2 x. ( 1 / 2 ) ) x. ( pi / 2 ) ) = ( 2 x. ( ( 1 / 2 ) x. ( pi / 2 ) ) )
27	25	mullidi 8242	. . . . . . . . . 10 |- ( 1 x. ( pi / 2 ) ) = ( pi / 2 )
28	22, 26, 27	3eqtr3i 2260	. . . . . . . . 9 |- ( 2 x. ( ( 1 / 2 ) x. ( pi / 2 ) ) ) = ( pi / 2 )
29	28	fveq2i 5651	. . . . . . . 8 |- ( sin ` ( 2 x. ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) = ( sin ` ( pi / 2 ) )
30	1, 25	mulcli 8244	. . . . . . . . 9 |- ( ( 1 / 2 ) x. ( pi / 2 ) ) e. CC
31		sin2t 12390	. . . . . . . . 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 15607	. . . . . . . 8 |- ( sin ` ( pi / 2 ) ) = 1
34	29, 32, 33	3eqtr3i 2260	. . . . . . 7 |- ( 2 x. ( ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) x. ( cos ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) ) = 1
35	8, 20, 34	3eqtr3i 2260	. . . . . 6 |- ( 2 x. ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) = 1
36	35	fveq2i 5651	. . . . 5 |- ( sqr ` ( 2 x. ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) ) = ( sqr ` 1 )
37		4re 9279	. . . . . . . . 9 |- 4 e. RR
38		4ap0 9301	. . . . . . . . 9 |- 4 # 0
39	10, 37, 38	redivclapi 9018	. . . . . . . 8 |- ( pi / 4 ) e. RR
40		resincl 12361	. . . . . . . 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 8253	. . . . . 6 |- ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) e. RR
43		0le2 9292	. . . . . 6 |- 0 <_ 2
44	41	msqge0i 8856	. . . . . 6 |- 0 <_ ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) )
45	23, 42, 43, 44	sqrtmulii 11774	. . . . 5 |- ( sqr ` ( 2 x. ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) ) = ( ( sqr ` 2 ) x. ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) )
46		sqrt1 11686	. . . . 5 |- ( sqr ` 1 ) = 1
47	36, 45, 46	3eqtr3ri 2261	. . . 4 |- 1 = ( ( sqr ` 2 ) x. ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) )
48	42	sqrtcli 11760	. . . . . . 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 8251	. . . . 5 |- ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) e. CC
51		sqrt2re 12815	. . . . . . 7 |- ( sqr ` 2 ) e. RR
52	51	recni 8251	. . . . . 6 |- ( sqr ` 2 ) e. CC
53		2pos 9293	. . . . . . . 8 |- 0 < 2
54	23, 53	sqrtgt0ii 11771	. . . . . . 7 |- 0 < ( sqr ` 2 )
55	51, 54	gt0ap0ii 8867	. . . . . 6 |- ( sqr ` 2 ) # 0
56	52, 55	pm3.2i 272	. . . . 5 |- ( ( sqr ` 2 ) e. CC /\ ( sqr ` 2 ) # 0 )
57		divmulap2 8915	. . . . 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 1374	. . . 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 8239	. . . . 5 |- 0 e. RR
61		pipos 15599	. . . . . . . 8 |- 0 < pi
62		4pos 9299	. . . . . . . 8 |- 0 < 4
63	10, 37, 61, 62	divgt0ii 9158	. . . . . . 7 |- 0 < ( pi / 4 )
64		1re 8238	. . . . . . . 8 |- 1 e. RR
65		pigt2lt4 15595	. . . . . . . . . . 11 |- ( 2 < pi /\ pi < 4 )
66	65	simpri 113	. . . . . . . . . 10 |- pi < 4
67	10, 37, 37, 62	ltdiv1ii 9168	. . . . . . . . . 10 |- ( pi < 4 <-> ( pi / 4 ) < ( 4 / 4 ) )
68	66, 67	mpbi 145	. . . . . . . . 9 |- ( pi / 4 ) < ( 4 / 4 )
69	37	recni 8251	. . . . . . . . . 10 |- 4 e. CC
70	69, 38	dividapi 8984	. . . . . . . . 9 |- ( 4 / 4 ) = 1
71	68, 70	breqtri 4118	. . . . . . . 8 |- ( pi / 4 ) < 1
72	39, 64, 71	ltleii 8341	. . . . . . 7 |- ( pi / 4 ) <_ 1
73		0xr 8285	. . . . . . . 8 |- 0 e. RR*
74		elioc2 10232	. . . . . . . 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 1206	. . . . . 6 |- ( pi / 4 ) e. ( 0 (,] 1 )
77		sin01gt0 12403	. . . . . 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 8341	. . . 4 |- 0 <_ ( sin ` ( pi / 4 ) )
80	41	sqrtmsqi 11762	. . . 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 2253	. 2 |- ( sin ` ( pi / 4 ) ) = ( 1 / ( sqr ` 2 ) )
83	59, 81	eqtri 2252	. . 3 |- ( 1 / ( sqr ` 2 ) ) = ( sin ` ( pi / 4 ) )
84	17	fveq2i 5651	. . . 4 |- ( cos ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) = ( cos ` ( pi / 4 ) )
85	6, 18, 84	3eqtr3i 2260	. . 3 |- ( sin ` ( pi / 4 ) ) = ( cos ` ( pi / 4 ) )
86	83, 85	eqtr2i 2253	. 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 1005   = wceq 1398   e. wcel 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8090   RRcr 8091   0cc0 8092   1c1 8093   + caddc 8095   x. cmul 8097   RR*cxr 8272   < clt 8273   <_ cle 8274   # cap 8820   / cdiv 8911   2c2 9253   4c4 9255   (,]cioc 10185   sqrcsqrt 11636   sincsin 12285   cosccos 12286   picpi 12288

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212  ax-pre-suploc 8213  ax-addf 8214  ax-mulf 8215

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-disj 4070  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-of 6244  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-map 6862  df-pm 6863  df-en 6953  df-dom 6954  df-fin 6955  df-sup 7243  df-inf 7244  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-5 9264  df-6 9265  df-7 9266  df-8 9267  df-9 9268  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-xneg 10068  df-xadd 10069  df-ioo 10188  df-ioc 10189  df-ico 10190  df-icc 10191  df-fz 10306  df-fzo 10440  df-seqfrec 10773  df-exp 10864  df-fac 11051  df-bc 11073  df-ihash 11101  df-shft 11455  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919  df-sumdc 11994  df-ef 12289  df-sin 12291  df-cos 12292  df-pi 12294  df-rest 13404  df-topgen 13423  df-psmet 14639  df-xmet 14640  df-met 14641  df-bl 14642  df-mopn 14643  df-top 14809  df-topon 14822  df-bases 14854  df-ntr 14907  df-cn 14999  df-cnp 15000  df-tx 15064  df-cncf 15382  df-limced 15467  df-dvap 15468

This theorem is referenced by:  tan4thpi  15652