Theorem taupi 16985

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 12487	. 2 |- tau = inf ( ( RR+ i^i ( `' cos " { 1 } ) ) , RR , < )
2		lttri3 8369	. . . . 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 9324	. . . . . 6 |- 2 e. RR
5		pire 15777	. . . . . 6 |- pi e. RR
6	4, 5	remulcli 8304	. . . . 5 |- ( 2 x. pi ) e. RR
7	6	a1i 9	. . . 4 |- ( T. -> ( 2 x. pi ) e. RR )
8		2rp 10009	. . . . . . 7 |- 2 e. RR+
9		pirp 15780	. . . . . . 7 |- pi e. RR+
10		rpmulcl 10029	. . . . . . 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 8302	. . . . . . 7 |- ( 2 x. pi ) e. CC
13		cos2pi 15795	. . . . . . 7 |- ( cos ` ( 2 x. pi ) ) = 1
14		cosf 12416	. . . . . . . . 9 |- cos : CC --> CC
15		ffn 5513	. . . . . . . . 9 |- ( cos : CC --> CC -> cos Fn CC )
16	14, 15	ax-mp 5	. . . . . . . 8 |- cos Fn CC
17		fniniseg 5803	. . . . . . . 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 3407	. . . . 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 3410	. . . . . . . . . 10 |- ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) -> x e. ( `' cos " { 1 } ) )
23		fniniseg 5803	. . . . . . . . . . 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 3409	. . . . . . . . . . 11 |- ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) -> x e. RR+ )
29	28	rpred 10047	. . . . . . . . . 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 10050	. . . . . . . . . 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 8336	. . . . . . . . . 10 |- 0 e. RR*
35	6	rexri 8347	. . . . . . . . . 10 |- ( 2 x. pi ) e. RR*
36		elioo2 10273	. . . . . . . . . 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 15842	. . . . . . . 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 4138	. . . . . 6 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> 1 < 1 )
42		1red 8305	. . . . . . 7 |- ( ( x e. ( RR+ i^i ( `' cos " { 1 } ) ) /\ x < ( 2 x. pi ) ) -> 1 e. RR )
43	42	ltnrd 8401	. . . . . 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 7331	. . 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 2255	1 |- tau = ( 2 x. pi )

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   T. wtru 1399   e. wcel 2205   i^i cin 3213   {csn 3694   class class class wbr 4114   `'ccnv 4753   "cima 4757   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058  infcinf 7287   CCcc 8141   RRcr 8142   0cc0 8143   1c1 8144   x. cmul 8148   RR*cxr 8323   < clt 8324   2c2 9305   RR+crp 10004   (,)cioo 10240   cosccos 12356   picpi 12358   tauctau 12486

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263  ax-pre-suploc 8264  ax-addf 8265  ax-mulf 8266

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-disj 4091  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-map 6897  df-pm 6898  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-xneg 10124  df-xadd 10125  df-ioo 10244  df-ioc 10245  df-ico 10246  df-icc 10247  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-fac 11113  df-bc 11135  df-ihash 11164  df-shft 11525  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064  df-ef 12359  df-sin 12361  df-cos 12362  df-pi 12364  df-tau 12487  df-rest 13538  df-topgen 13557  df-psmet 14817  df-xmet 14818  df-met 14819  df-bl 14820  df-mopn 14821  df-top 14989  df-topon 15002  df-bases 15034  df-ntr 15087  df-cn 15179  df-cnp 15180  df-tx 15244  df-cncf 15562  df-limced 15647  df-dvap 15648

This theorem is referenced by: (None)