Theorem sincos4thpi 16033

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 9524	. . . . . . . . . 10 |- ( 1 / 2 ) e. CC
2		ax-1cn 8273	. . . . . . . . . . 11 |- 1 e. CC
3		2halves 9539	. . . . . . . . . . 11 |- ( 1 e. CC -> ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1 )
4	2, 3	ax-mp 5	. . . . . . . . . 10 |- ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1
5		sincosq1eq 16032	. . . . . . . . . 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 1378	. . . . . . . . 9 |- ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) = ( cos ` ( ( 1 / 2 ) x. ( pi / 2 ) ) )
7	6	oveq2i 6096	. . . . . . . 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 6096	. . . . . . 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 9378	. . . . . . . . . . . 12 |- 2 e. CC
10		pire 15979	. . . . . . . . . . . . 13 |- pi e. RR
11	10	recni 8339	. . . . . . . . . . . 12 |- pi e. CC
12		2ap0 9400	. . . . . . . . . . . 12 |- 2 # 0
13	2, 9, 11, 9, 12, 12	divmuldivapi 9105	. . . . . . . . . . 11 |- ( ( 1 / 2 ) x. ( pi / 2 ) ) = ( ( 1 x. pi ) / ( 2 x. 2 ) )
14	11	mullidi 8330	. . . . . . . . . . . 12 |- ( 1 x. pi ) = pi
15		2t2e4 9462	. . . . . . . . . . . 12 |- ( 2 x. 2 ) = 4
16	14, 15	oveq12i 6097	. . . . . . . . . . 11 |- ( ( 1 x. pi ) / ( 2 x. 2 ) ) = ( pi / 4 )
17	13, 16	eqtri 2259	. . . . . . . . . 10 |- ( ( 1 / 2 ) x. ( pi / 2 ) ) = ( pi / 4 )
18	17	fveq2i 5698	. . . . . . . . 9 |- ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) = ( sin ` ( pi / 4 ) )
19	18, 18	oveq12i 6097	. . . . . . . 8 |- ( ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) x. ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) = ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) )
20	19	oveq2i 6096	. . . . . . 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 9076	. . . . . . . . . . 11 |- ( 2 x. ( 1 / 2 ) ) = 1
22	21	oveq1i 6095	. . . . . . . . . 10 |- ( ( 2 x. ( 1 / 2 ) ) x. ( pi / 2 ) ) = ( 1 x. ( pi / 2 ) )
23		2re 9377	. . . . . . . . . . . . 13 |- 2 e. RR
24	10, 23, 12	redivclapi 9112	. . . . . . . . . . . 12 |- ( pi / 2 ) e. RR
25	24	recni 8339	. . . . . . . . . . 11 |- ( pi / 2 ) e. CC
26	9, 1, 25	mulassi 8336	. . . . . . . . . 10 |- ( ( 2 x. ( 1 / 2 ) ) x. ( pi / 2 ) ) = ( 2 x. ( ( 1 / 2 ) x. ( pi / 2 ) ) )
27	25	mullidi 8330	. . . . . . . . . 10 |- ( 1 x. ( pi / 2 ) ) = ( pi / 2 )
28	22, 26, 27	3eqtr3i 2267	. . . . . . . . 9 |- ( 2 x. ( ( 1 / 2 ) x. ( pi / 2 ) ) ) = ( pi / 2 )
29	28	fveq2i 5698	. . . . . . . 8 |- ( sin ` ( 2 x. ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) = ( sin ` ( pi / 2 ) )
30	1, 25	mulcli 8332	. . . . . . . . 9 |- ( ( 1 / 2 ) x. ( pi / 2 ) ) e. CC
31		sin2t 12535	. . . . . . . . 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 15989	. . . . . . . 8 |- ( sin ` ( pi / 2 ) ) = 1
34	29, 32, 33	3eqtr3i 2267	. . . . . . 7 |- ( 2 x. ( ( sin ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) x. ( cos ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) ) ) = 1
35	8, 20, 34	3eqtr3i 2267	. . . . . 6 |- ( 2 x. ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) = 1
36	35	fveq2i 5698	. . . . 5 |- ( sqr ` ( 2 x. ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) ) = ( sqr ` 1 )
37		4re 9384	. . . . . . . . 9 |- 4 e. RR
38		4ap0 9406	. . . . . . . . 9 |- 4 # 0
39	10, 37, 38	redivclapi 9112	. . . . . . . 8 |- ( pi / 4 ) e. RR
40		resincl 12506	. . . . . . . 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 8341	. . . . . 6 |- ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) e. RR
43		0le2 9397	. . . . . 6 |- 0 <_ 2
44	41	msqge0i 8948	. . . . . 6 |- 0 <_ ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) )
45	23, 42, 43, 44	sqrtmulii 11917	. . . . 5 |- ( sqr ` ( 2 x. ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) ) = ( ( sqr ` 2 ) x. ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) )
46		sqrt1 11828	. . . . 5 |- ( sqr ` 1 ) = 1
47	36, 45, 46	3eqtr3ri 2268	. . . 4 |- 1 = ( ( sqr ` 2 ) x. ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) )
48	42	sqrtcli 11903	. . . . . . 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 8339	. . . . 5 |- ( sqr ` ( ( sin ` ( pi / 4 ) ) x. ( sin ` ( pi / 4 ) ) ) ) e. CC
51		sqrt2re 12961	. . . . . . 7 |- ( sqr ` 2 ) e. RR
52	51	recni 8339	. . . . . 6 |- ( sqr ` 2 ) e. CC
53		2pos 9398	. . . . . . . 8 |- 0 < 2
54	23, 53	sqrtgt0ii 11914	. . . . . . 7 |- 0 < ( sqr ` 2 )
55	51, 54	gt0ap0ii 8959	. . . . . 6 |- ( sqr ` 2 ) # 0
56	52, 55	pm3.2i 272	. . . . 5 |- ( ( sqr ` 2 ) e. CC /\ ( sqr ` 2 ) # 0 )
57		divmulap2 9009	. . . . 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 1378	. . . 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 8327	. . . . 5 |- 0 e. RR
61		pipos 15981	. . . . . . . 8 |- 0 < pi
62		4pos 9404	. . . . . . . 8 |- 0 < 4
63	10, 37, 61, 62	divgt0ii 9252	. . . . . . 7 |- 0 < ( pi / 4 )
64		1re 8326	. . . . . . . 8 |- 1 e. RR
65		pigt2lt4 15977	. . . . . . . . . . 11 |- ( 2 < pi /\ pi < 4 )
66	65	simpri 113	. . . . . . . . . 10 |- pi < 4
67	10, 37, 37, 62	ltdiv1ii 9262	. . . . . . . . . 10 |- ( pi < 4 <-> ( pi / 4 ) < ( 4 / 4 ) )
68	66, 67	mpbi 145	. . . . . . . . 9 |- ( pi / 4 ) < ( 4 / 4 )
69	37	recni 8339	. . . . . . . . . 10 |- 4 e. CC
70	69, 38	dividapi 9078	. . . . . . . . 9 |- ( 4 / 4 ) = 1
71	68, 70	breqtri 4155	. . . . . . . 8 |- ( pi / 4 ) < 1
72	39, 64, 71	ltleii 8430	. . . . . . 7 |- ( pi / 4 ) <_ 1
73		0xr 8373	. . . . . . . 8 |- 0 e. RR*
74		elioc2 10349	. . . . . . . 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 430	. . . . . . 7 |- ( ( pi / 4 ) e. ( 0 (,] 1 ) <-> ( ( pi / 4 ) e. RR /\ 0 < ( pi / 4 ) /\ ( pi / 4 ) <_ 1 ) )
76	39, 63, 72, 75	mpbir3an 1210	. . . . . 6 |- ( pi / 4 ) e. ( 0 (,] 1 )
77		sin01gt0 12548	. . . . . 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 8430	. . . 4 |- 0 <_ ( sin ` ( pi / 4 ) )
80	41	sqrtmsqi 11905	. . . 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 2260	. 2 |- ( sin ` ( pi / 4 ) ) = ( 1 / ( sqr ` 2 ) )
83	59, 81	eqtri 2259	. . 3 |- ( 1 / ( sqr ` 2 ) ) = ( sin ` ( pi / 4 ) )
84	17	fveq2i 5698	. . . 4 |- ( cos ` ( ( 1 / 2 ) x. ( pi / 2 ) ) ) = ( cos ` ( pi / 4 ) )
85	6, 18, 84	3eqtr3i 2267	. . 3 |- ( sin ` ( pi / 4 ) ) = ( cos ` ( pi / 4 ) )
86	83, 85	eqtr2i 2260	. 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 ) ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8178   RRcr 8179   0cc0 8180   1c1 8181   + caddc 8183   x. cmul 8185   RR*cxr 8360   < clt 8361   <_ cle 8362   # cap 8912   / cdiv 9005   2c2 9358   4c4 9360   (,]cioc 10302   sqrcsqrt 11778   sincsin 12430   cosccos 12431   picpi 12433

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  ax-pre-suploc 8301  ax-addf 8302  ax-mulf 8303

This proof depends on definitions:  df-bi 117  df-stab 843  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-of 6302  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-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7325  df-inf 7326  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-5 9369  df-6 9370  df-7 9371  df-8 9372  df-9 9373  df-n0 9569  df-z 9650  df-uz 9932  df-q 10030  df-rp 10066  df-xneg 10185  df-xadd 10186  df-ioo 10305  df-ioc 10306  df-ico 10307  df-icc 10308  df-fz 10423  df-fzo 10561  df-seqfrec 10900  df-exp 10991  df-fac 11180  df-bc 11202  df-ihash 11231  df-shft 11596  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  df-pi 12439  df-rest 13648  df-topgen 13667  df-psmet 14964  df-xmet 14965  df-met 14966  df-bl 14967  df-mopn 14968  df-top 15190  df-topon 15203  df-bases 15235  df-ntr 15288  df-cn 15380  df-cnp 15381  df-tx 15445  df-cncf 15763  df-limced 15848  df-dvap 15849

This theorem is used by:  tan4thpi  16034