Theorem taupi 16789

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 12400	. 2 |- tau = inf ( ( RR+ i^i ( `' cos " { 1 } ) ) , RR , < )
2		lttri3 8301	. . . . 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 9255	. . . . . 6 |- 2 e. RR
5		pire 15580	. . . . . 6 |- pi e. RR
6	4, 5	remulcli 8236	. . . . 5 |- ( 2 x. pi ) e. RR
7	6	a1i 9	. . . 4 |- ( T. -> ( 2 x. pi ) e. RR )
8		2rp 9937	. . . . . . 7 |- 2 e. RR+
9		pirp 15583	. . . . . . 7 |- pi e. RR+
10		rpmulcl 9957	. . . . . . 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 8234	. . . . . . 7 |- ( 2 x. pi ) e. CC
13		cos2pi 15598	. . . . . . 7 |- ( cos ` ( 2 x. pi ) ) = 1
14		cosf 12329	. . . . . . . . 9 |- cos : CC --> CC
15		ffn 5489	. . . . . . . . 9 |- ( cos : CC --> CC -> cos Fn CC )
16	14, 15	ax-mp 5	. . . . . . . 8 |- cos Fn CC
17		fniniseg 5776	. . . . . . . 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 951	. . . . . 6 |- ( 2 x. pi ) e. ( `' cos " { 1 } )
20	11, 19	elini 3393	. . . . 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 3396	. . . . . . . . . 10 |- ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) -> x e. ( `' cos " { 1 } ) )
23		fniniseg 5776	. . . . . . . . . . 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 3395	. . . . . . . . . . 11 |- ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) -> x e. RR+ )
29	28	rpred 9975	. . . . . . . . . 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 9978	. . . . . . . . . 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 8268	. . . . . . . . . 10 |- 0 e. RR*
35	6	rexri 8279	. . . . . . . . . 10 |- ( 2 x. pi ) e. RR*
36		elioo2 10200	. . . . . . . . . 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 1208	. . . . . . . 8 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> x e. ( 0 (,) ( 2 x. pi ) ) )
39		cos02pilt1 15645	. . . . . . . 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 4117	. . . . . 6 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> 1 < 1 )
42		1red 8237	. . . . . . 7 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> 1 e. RR )
43	42	ltnrd 8333	. . . . . 6 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> -. 1 < 1 )
44	41, 43	pm2.65da 667	. . . . 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 7269	. . 3 |- ( T. -> inf ( ( RR+ i^i ( `' cos " { 1 } ) ) , RR , < ) = ( 2 x. pi ) )
47	46	mptru 1407	. 2 |- inf ( ( RR+ i^i ( `' cos " { 1 } ) ) , RR , < ) = ( 2 x. pi )
48	1, 47	eqtri 2252	1 |- tau = ( 2 x. pi )

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   T. wtru 1399   e. wcel 2202   i^i cin 3200   {csn 3673   class class class wbr 4093   `'ccnv 4730   "cima 4734   Fn wfn 5328   -->wf 5329   ` cfv 5333  (class class class)co 6028  infcinf 7225   CCcc 8073   RRcr 8074   0cc0 8075   1c1 8076   x. cmul 8080   RR*cxr 8255   < clt 8256   2c2 9236   RR+crp 9932   (,)cioo 10167   cosccos 12269   picpi 12271   tauctau 12399

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 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195  ax-pre-suploc 8196  ax-addf 8197  ax-mulf 8198

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 7226  df-inf 7227  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-xneg 10051  df-xadd 10052  df-ioo 10171  df-ioc 10172  df-ico 10173  df-icc 10174  df-fz 10289  df-fzo 10423  df-seqfrec 10756  df-exp 10847  df-fac 11034  df-bc 11056  df-ihash 11084  df-shft 11438  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-clim 11902  df-sumdc 11977  df-ef 12272  df-sin 12274  df-cos 12275  df-pi 12277  df-tau 12400  df-rest 13387  df-topgen 13406  df-psmet 14622  df-xmet 14623  df-met 14624  df-bl 14625  df-mopn 14626  df-top 14792  df-topon 14805  df-bases 14837  df-ntr 14890  df-cn 14982  df-cnp 14983  df-tx 15047  df-cncf 15365  df-limced 15450  df-dvap 15451

This theorem is referenced by: (None)