Theorem taupi 16400

Description: Relationship between tau and pi. This can be seen as connecting the ratio of a circle's circumference to its radius and the ratio of a circle's circumference to its diameter. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.)

Assertion

Ref	Expression
taupi	|- tau = ( 2 x. pi )

Proof of Theorem taupi

Dummy variables f g x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-tau 12282	. 2 |- tau = inf ( ( RR+ i^i ( `' cos " { 1 } ) ) , RR , < )
2		lttri3 8222	. . . . 5 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
3	2	adantl 277	. . . 4 |- ( ( T. /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
4		2re 9176	. . . . . 6 |- 2 e. RR
5		pire 15454	. . . . . 6 |- pi e. RR
6	4, 5	remulcli 8156	. . . . 5 |- ( 2 x. pi ) e. RR
7	6	a1i 9	. . . 4 |- ( T. -> ( 2 x. pi ) e. RR )
8		2rp 9850	. . . . . . 7 |- 2 e. RR+
9		pirp 15457	. . . . . . 7 |- pi e. RR+
10		rpmulcl 9870	. . . . . . 7 |- ( ( 2 e. RR+ /\ pi e. RR+ ) -> ( 2 x. pi ) e. RR+ )
11	8, 9, 10	mp2an 426	. . . . . 6 |- ( 2 x. pi ) e. RR+
12	6	recni 8154	. . . . . . 7 |- ( 2 x. pi ) e. CC
13		cos2pi 15472	. . . . . . 7 |- ( cos ` ( 2 x. pi ) ) = 1
14		cosf 12211	. . . . . . . . 9 |- cos : CC --> CC
15		ffn 5472	. . . . . . . . 9 |- ( cos : CC --> CC -> cos Fn CC )
16	14, 15	ax-mp 5	. . . . . . . 8 |- cos Fn CC
17		fniniseg 5754	. . . . . . . 8 |- ( cos Fn CC -> ( ( 2 x. pi ) e. ( `' cos " { 1 } ) <-> ( ( 2 x. pi ) e. CC /\ ( cos ` ( 2 x. pi ) ) = 1 ) ) )
18	16, 17	ax-mp 5	. . . . . . 7 |- ( ( 2 x. pi ) e. ( `' cos " { 1 } ) <-> ( ( 2 x. pi ) e. CC /\ ( cos ` ( 2 x. pi ) ) = 1 ) )
19	12, 13, 18	mpbir2an 948	. . . . . 6 |- ( 2 x. pi ) e. ( `' cos " { 1 } )
20	11, 19	elini 3388	. . . . 5 |- ( 2 x. pi ) e. ( RR+ i^i ( `' cos " { 1 } ) )
21	20	a1i 9	. . . 4 |- ( T. -> ( 2 x. pi ) e. ( RR+ i^i ( `' cos " { 1 } ) ) )
22		elinel2 3391	. . . . . . . . . 10 |- ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) -> x e. ( `' cos " { 1 } ) )
23		fniniseg 5754	. . . . . . . . . . 11 |- ( cos Fn CC -> ( x e. ( `' cos " { 1 } ) <-> ( x e. CC /\ ( cos ` x ) = 1 ) ) )
24	16, 23	ax-mp 5	. . . . . . . . . 10 |- ( x e. ( `' cos " { 1 } ) <-> ( x e. CC /\ ( cos ` x ) = 1 ) )
25	22, 24	sylib 122	. . . . . . . . 9 |- ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) -> ( x e. CC /\ ( cos ` x ) = 1 ) )
26	25	simprd 114	. . . . . . . 8 |- ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) -> ( cos ` x ) = 1 )
27	26	adantr 276	. . . . . . 7 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> ( cos ` x ) = 1 )
28		elinel1 3390	. . . . . . . . . . 11 |- ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) -> x e. RR+ )
29	28	rpred 9888	. . . . . . . . . 10 |- ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) -> x e. RR )
30	29	adantr 276	. . . . . . . . 9 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> x e. RR )
31	28	rpgt0d 9891	. . . . . . . . . 10 |- ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) -> 0 < x )
32	31	adantr 276	. . . . . . . . 9 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> 0 < x )
33		simpr 110	. . . . . . . . 9 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> x < ( 2 x. pi ) )
34		0xr 8189	. . . . . . . . . 10 |- 0 e. RR*
35	6	rexri 8200	. . . . . . . . . 10 |- ( 2 x. pi ) e. RR*
36		elioo2 10113	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ ( 2 x. pi ) e. RR* ) -> ( x e. ( 0 (,) ( 2 x. pi ) ) <-> ( x e. RR /\ 0 < x /\ x < ( 2 x. pi ) ) ) )
37	34, 35, 36	mp2an 426	. . . . . . . . 9 |- ( x e. ( 0 (,) ( 2 x. pi ) ) <-> ( x e. RR /\ 0 < x /\ x < ( 2 x. pi ) ) )
38	30, 32, 33, 37	syl3anbrc 1205	. . . . . . . 8 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> x e. ( 0 (,) ( 2 x. pi ) ) )
39		cos02pilt1 15519	. . . . . . . 8 |- ( x e. ( 0 (,) ( 2 x. pi ) ) -> ( cos ` x ) < 1 )
40	38, 39	syl 14	. . . . . . 7 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> ( cos ` x ) < 1 )
41	27, 40	eqbrtrrd 4106	. . . . . 6 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> 1 < 1 )
42		1red 8157	. . . . . . 7 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> 1 e. RR )
43	42	ltnrd 8254	. . . . . 6 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> -. 1 < 1 )
44	41, 43	pm2.65da 665	. . . . 5 |- ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) -> -. x < ( 2 x. pi ) )
45	44	adantl 277	. . . 4 |- ( ( T. /\ x e. ( RR+ i^i ( `' cos " { 1 } ) ) ) -> -. x < ( 2 x. pi ) )
46	3, 7, 21, 45	infminti 7190	. . 3 |- ( T. -> inf ( ( RR+ i^i ( `' cos " { 1 } ) ) , RR , < ) = ( 2 x. pi ) )
47	46	mptru 1404	. 2 |- inf ( ( RR+ i^i ( `' cos " { 1 } ) ) , RR , < ) = ( 2 x. pi )
48	1, 47	eqtri 2250	1 |- tau = ( 2 x. pi )

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   T. wtru 1396   e. wcel 2200   i^i cin 3196   {csn 3666   class class class wbr 4082   `'ccnv 4717   "cima 4721   Fn wfn 5312   -->wf 5313   ` cfv 5317  (class class class)co 6000  infcinf 7146   CCcc 7993   RRcr 7994   0cc0 7995   1c1 7996   x. cmul 8000   RR*cxr 8176   < clt 8177   2c2 9157   RR+crp 9845   (,)cioo 10080   cosccos 12151   picpi 12153   tauctau 12281

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115  ax-pre-suploc 8116  ax-addf 8117  ax-mulf 8118

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 3888  df-int 3923  df-iun 3966  df-disj 4059  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-of 6216  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-frec 6535  df-1o 6560  df-oadd 6564  df-er 6678  df-map 6795  df-pm 6796  df-en 6886  df-dom 6887  df-fin 6888  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-5 9168  df-6 9169  df-7 9170  df-8 9171  df-9 9172  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-xneg 9964  df-xadd 9965  df-ioo 10084  df-ioc 10085  df-ico 10086  df-icc 10087  df-fz 10201  df-fzo 10335  df-seqfrec 10665  df-exp 10756  df-fac 10943  df-bc 10965  df-ihash 10993  df-shft 11321  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-clim 11785  df-sumdc 11860  df-ef 12154  df-sin 12156  df-cos 12157  df-pi 12159  df-tau 12282  df-rest 13269  df-topgen 13288  df-psmet 14501  df-xmet 14502  df-met 14503  df-bl 14504  df-mopn 14505  df-top 14666  df-topon 14679  df-bases 14711  df-ntr 14764  df-cn 14856  df-cnp 14857  df-tx 14921  df-cncf 15239  df-limced 15324  df-dvap 15325

This theorem is referenced by: (None)