Theorem birthdaylog2 16228

Description: The Birthday Problem. There is a more than even chance that out of 23 people in a room, at least two of them have the same birthday. Mathematically, this is asserting that for K = 2 3 and N = 3 6 5, fewer than half of the set of all functions from 1 ... K to 1 ... N are injective.

The presence of the hypothesis giving a series which converges to ( log ` 2 ) is a temporary measure until it can be proved as log2cnv .

Although this is Metamath 100 proof #93, we cannot consider it proved until we prove the missing log2cnv piece (or prove the theorem another way which does not require it). (Contributed by Mario Carneiro, 17-Apr-2015.)

Hypotheses

Ref	Expression
birthday.s	|- S = { f | f : ( 1 ... K ) --> ( 1 ... N ) }
birthday.t	|- T = { f | f : ( 1 ... K ) -1-1-> ( 1 ... N ) }
birthday.k	|- K = ; 2 3
birthday.n	|- N = ;; 3 6 5
birthdaylog2.log2cnv	|- seq 0 ( + , ( k e. NN0 |-> ( 2 / ( ( 3 x. ( ( 2 x. k ) + 1 ) ) x. ( 9 ^ k ) ) ) ) ) ~~> ( log ` 2 )

Assertion

Ref	Expression
birthdaylog2	|- ( (♯ ` T ) / (♯ ` S ) ) < ( 1 / 2 )

Distinct variable groups:   f, k, K   f, N, k

Allowed substitution hints:   S( f, k)   T( f, k)

Proof of Theorem birthdaylog2

Step	Hyp	Ref	Expression
1		birthday.k	. . . 4 |- K = ; 2 3
2		2nn0 9591	. . . . 5 |- 2 e. NN0
3		3nn0 9592	. . . . 5 |- 3 e. NN0
4	2, 3	deccl 9802	. . . 4 |- ; 2 3 e. NN0
5	1, 4	eqeltri 2311	. . 3 |- K e. NN0
6		birthday.n	. . . 4 |- N = ;; 3 6 5
7		6nn0 9595	. . . . . 6 |- 6 e. NN0
8	3, 7	deccl 9802	. . . . 5 |- ; 3 6 e. NN0
9		5nn 9480	. . . . 5 |- 5 e. NN
10	8, 9	decnncl 9811	. . . 4 |- ;; 3 6 5 e. NN
11	6, 10	eqeltri 2311	. . 3 |- N e. NN
12		birthday.s	. . . 4 |- S = { f | f : ( 1 ... K ) --> ( 1 ... N ) }
13		birthday.t	. . . 4 |- T = { f | f : ( 1 ... K ) -1-1-> ( 1 ... N ) }
14	12, 13	birthdaylem3 16227	. . 3 |- ( ( K e. NN0 /\ N e. NN ) -> ( (♯ ` T ) / (♯ ` S ) ) <_ ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) )
15	5, 11, 14	mp2an 430	. 2 |- ( (♯ ` T ) / (♯ ` S ) ) <_ ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) )
16		birthdaylog2.log2cnv	. . . . . . 7 |- seq 0 ( + , ( k e. NN0 |-> ( 2 / ( ( 3 x. ( ( 2 x. k ) + 1 ) ) x. ( 9 ^ k ) ) ) ) ) ~~> ( log ` 2 )
17	16	log2ublog2 16224	. . . . . 6 |- ( log ` 2 ) < (;; 2 5 3 / ;; 3 6 5 )
18	5	nn0cni 9586	. . . . . . . . . . . 12 |- K e. CC
19	18	sqvali 11077	. . . . . . . . . . 11 |- ( K ^ 2 ) = ( K x. K )
20	18	mulridi 8334	. . . . . . . . . . . 12 |- ( K x. 1 ) = K
21	20	eqcomi 2242	. . . . . . . . . . 11 |- K = ( K x. 1 )
22	19, 21	oveq12i 6100	. . . . . . . . . 10 |- ( ( K ^ 2 ) - K ) = ( ( K x. K ) - ( K x. 1 ) )
23		ax-1cn 8278	. . . . . . . . . . 11 |- 1 e. CC
24	18, 18, 23	subdii 8741	. . . . . . . . . 10 |- ( K x. ( K - 1 ) ) = ( ( K x. K ) - ( K x. 1 ) )
25	22, 24	eqtr4i 2262	. . . . . . . . 9 |- ( ( K ^ 2 ) - K ) = ( K x. ( K - 1 ) )
26	25	oveq1i 6098	. . . . . . . 8 |- ( ( ( K ^ 2 ) - K ) / 2 ) = ( ( K x. ( K - 1 ) ) / 2 )
27	18, 23	subcli 8609	. . . . . . . . 9 |- ( K - 1 ) e. CC
28		2cn 9384	. . . . . . . . 9 |- 2 e. CC
29		2ap0 9406	. . . . . . . . 9 |- 2 # 0
30	18, 27, 28, 29	divassapi 9106	. . . . . . . 8 |- ( ( K x. ( K - 1 ) ) / 2 ) = ( K x. ( ( K - 1 ) / 2 ) )
31		1nn0 9590	. . . . . . . . 9 |- 1 e. NN0
32	2, 2	deccl 9802	. . . . . . . . . . . . 13 |- ; 2 2 e. NN0
33	32	nn0cni 9586	. . . . . . . . . . . 12 |- ; 2 2 e. CC
34		2p1e3 9447	. . . . . . . . . . . . . 14 |- ( 2 + 1 ) = 3
35		eqid 2238	. . . . . . . . . . . . . 14 |- ; 2 2 = ; 2 2
36	2, 2, 34, 35	decsuc 9823	. . . . . . . . . . . . 13 |- (; 2 2 + 1 ) = ; 2 3
37	1, 36	eqtr4i 2262	. . . . . . . . . . . 12 |- K = (; 2 2 + 1 )
38	33, 23, 37	mvrraddi 8550	. . . . . . . . . . 11 |- ( K - 1 ) = ; 2 2
39	38	oveq1i 6098	. . . . . . . . . 10 |- ( ( K - 1 ) / 2 ) = (; 2 2 / 2 )
40	2	11multnc 9860	. . . . . . . . . . 11 |- ( 2 x. ; 1 1 ) = ; 2 2
41	31, 31	deccl 9802	. . . . . . . . . . . . 13 |- ; 1 1 e. NN0
42	41	nn0cni 9586	. . . . . . . . . . . 12 |- ; 1 1 e. CC
43	33, 28, 42, 29	divmulapi 9104	. . . . . . . . . . 11 |- ( (; 2 2 / 2 ) = ; 1 1 <-> ( 2 x. ; 1 1 ) = ; 2 2 )
44	40, 43	mpbir 146	. . . . . . . . . 10 |- (; 2 2 / 2 ) = ; 1 1
45	39, 44	eqtri 2259	. . . . . . . . 9 |- ( ( K - 1 ) / 2 ) = ; 1 1
46	20, 1	eqtri 2259	. . . . . . . . . 10 |- ( K x. 1 ) = ; 2 3
47		3p2e5 9455	. . . . . . . . . 10 |- ( 3 + 2 ) = 5
48	2, 3, 2, 46, 47	decaddi 9852	. . . . . . . . 9 |- ( ( K x. 1 ) + 2 ) = ; 2 5
49	5, 31, 31, 45, 3, 2, 48, 46	decmul2c 9858	. . . . . . . 8 |- ( K x. ( ( K - 1 ) / 2 ) ) = ;; 2 5 3
50	26, 30, 49	3eqtri 2263	. . . . . . 7 |- ( ( ( K ^ 2 ) - K ) / 2 ) = ;; 2 5 3
51	50, 6	oveq12i 6100	. . . . . 6 |- ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) = (;; 2 5 3 / ;; 3 6 5 )
52	17, 51	breqtrri 4158	. . . . 5 |- ( log ` 2 ) < ( ( ( ( K ^ 2 ) - K ) / 2 ) / N )
53		2rp 10076	. . . . . . 7 |- 2 e. RR+
54		relogcl 16091	. . . . . . 7 |- ( 2 e. RR+ -> ( log ` 2 ) e. RR )
55	53, 54	ax-mp 5	. . . . . 6 |- ( log ` 2 ) e. RR
56		5nn0 9594	. . . . . . . . . . 11 |- 5 e. NN0
57	2, 56	deccl 9802	. . . . . . . . . 10 |- ; 2 5 e. NN0
58	57, 3	deccl 9802	. . . . . . . . 9 |- ;; 2 5 3 e. NN0
59	50, 58	eqeltri 2311	. . . . . . . 8 |- ( ( ( K ^ 2 ) - K ) / 2 ) e. NN0
60	59	nn0rei 9585	. . . . . . 7 |- ( ( ( K ^ 2 ) - K ) / 2 ) e. RR
61		nndivre 9349	. . . . . . 7 |- ( ( ( ( ( K ^ 2 ) - K ) / 2 ) e. RR /\ N e. NN ) -> ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) e. RR )
62	60, 11, 61	mp2an 430	. . . . . 6 |- ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) e. RR
63	55, 62	ltnegi 8828	. . . . 5 |- ( ( log ` 2 ) < ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) <-> -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) < -u ( log ` 2 ) )
64	52, 63	mpbi 145	. . . 4 |- -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) < -u ( log ` 2 )
65	62	renegcli 8595	. . . . 5 |- -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) e. RR
66	55	renegcli 8595	. . . . 5 |- -u ( log ` 2 ) e. RR
67		eflt 16001	. . . . 5 |- ( ( -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) e. RR /\ -u ( log ` 2 ) e. RR ) -> ( -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) < -u ( log ` 2 ) <-> ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) < ( exp ` -u ( log ` 2 ) ) ) )
68	65, 66, 67	mp2an 430	. . . 4 |- ( -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) < -u ( log ` 2 ) <-> ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) < ( exp ` -u ( log ` 2 ) ) )
69	64, 68	mpbi 145	. . 3 |- ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) < ( exp ` -u ( log ` 2 ) )
70	55	recni 8344	. . . . 5 |- ( log ` 2 ) e. CC
71		efneg 12471	. . . . 5 |- ( ( log ` 2 ) e. CC -> ( exp ` -u ( log ` 2 ) ) = ( 1 / ( exp ` ( log ` 2 ) ) ) )
72	70, 71	ax-mp 5	. . . 4 |- ( exp ` -u ( log ` 2 ) ) = ( 1 / ( exp ` ( log ` 2 ) ) )
73		reeflog 16092	. . . . . 6 |- ( 2 e. RR+ -> ( exp ` ( log ` 2 ) ) = 2 )
74	53, 73	ax-mp 5	. . . . 5 |- ( exp ` ( log ` 2 ) ) = 2
75	74	oveq2i 6099	. . . 4 |- ( 1 / ( exp ` ( log ` 2 ) ) ) = ( 1 / 2 )
76	72, 75	eqtri 2259	. . 3 |- ( exp ` -u ( log ` 2 ) ) = ( 1 / 2 )
77	69, 76	breqtri 4156	. 2 |- ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) < ( 1 / 2 )
78	31	nn0zi 9677	. . . . . . . . 9 |- 1 e. ZZ
79	5	nn0zi 9677	. . . . . . . . 9 |- K e. ZZ
80		fzfig 10888	. . . . . . . . 9 |- ( ( 1 e. ZZ /\ K e. ZZ ) -> ( 1 ... K ) e. Fin )
81	78, 79, 80	mp2an 430	. . . . . . . 8 |- ( 1 ... K ) e. Fin
82	11	nnzi 9676	. . . . . . . . 9 |- N e. ZZ
83		fzfig 10888	. . . . . . . . 9 |- ( ( 1 e. ZZ /\ N e. ZZ ) -> ( 1 ... N ) e. Fin )
84	78, 82, 83	mp2an 430	. . . . . . . 8 |- ( 1 ... N ) e. Fin
85		f1setfi 7322	. . . . . . . 8 |- ( ( ( 1 ... K ) e. Fin /\ ( 1 ... N ) e. Fin ) -> { f | f : ( 1 ... K ) -1-1-> ( 1 ... N ) } e. Fin )
86	81, 84, 85	mp2an 430	. . . . . . 7 |- { f | f : ( 1 ... K ) -1-1-> ( 1 ... N ) } e. Fin
87	13, 86	eqeltri 2311	. . . . . 6 |- T e. Fin
88		hashcl 11242	. . . . . 6 |- ( T e. Fin -> (♯ ` T ) e. NN0 )
89	87, 88	ax-mp 5	. . . . 5 |- (♯ ` T ) e. NN0
90	89	nn0rei 9585	. . . 4 |- (♯ ` T ) e. RR
91	12, 13	birthdaylem1g 16225	. . . . . . 7 |- ( ( K e. NN0 /\ N e. NN ) -> ( T C_ S /\ S e. Fin /\ S =/= (/) ) )
92	5, 11, 91	mp2an 430	. . . . . 6 |- ( T C_ S /\ S e. Fin /\ S =/= (/) )
93	92	simp3i 1039	. . . . 5 |- S =/= (/)
94	92	simp2i 1038	. . . . . 6 |- S e. Fin
95		hashnncl 11256	. . . . . 6 |- ( S e. Fin -> ( (♯ ` S ) e. NN <-> S =/= (/) ) )
96	94, 95	ax-mp 5	. . . . 5 |- ( (♯ ` S ) e. NN <-> S =/= (/) )
97	93, 96	mpbir 146	. . . 4 |- (♯ ` S ) e. NN
98		nndivre 9349	. . . 4 |- ( ( (♯ ` T ) e. RR /\ (♯ ` S ) e. NN ) -> ( (♯ ` T ) / (♯ ` S ) ) e. RR )
99	90, 97, 98	mp2an 430	. . 3 |- ( (♯ ` T ) / (♯ ` S ) ) e. RR
100		reefcl 12460	. . . 4 |- ( -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) e. RR -> ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) e. RR )
101	65, 100	ax-mp 5	. . 3 |- ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) e. RR
102		halfre 9529	. . 3 |- ( 1 / 2 ) e. RR
103	99, 101, 102	lelttri 8438	. 2 |- ( ( ( (♯ ` T ) / (♯ ` S ) ) <_ ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) /\ ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) < ( 1 / 2 ) ) -> ( (♯ ` T ) / (♯ ` S ) ) < ( 1 / 2 ) )
104	15, 77, 103	mp2an 430	1 |- ( (♯ ` T ) / (♯ ` S ) ) < ( 1 / 2 )

This proof depends on syntax axioms:   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cab 2224   =/= wne 2420   C_ wss 3220   (/)c0 3520   class class class wbr 4131   |-> cmpt 4193   -->wf 5376   -1-1->wf1 5377   ` cfv 5380  (class class class)co 6088   Fincfn 7026   CCcc 8183   RRcr 8184   0cc0 8185   1c1 8186   + caddc 8188   x. cmul 8190   < clt 8366   <_ cle 8367   - cmin 8504   -ucneg 8505   / cdiv 9010   NNcn 9313   2c2 9364   3c3 9365   5c5 9367   6c6 9368   9c9 9371   NN0cn0 9574   ZZcz 9655  ;cdc 9788   RR+crp 10071   ...cfz 10428   seqcseq 10905   ^cexp 10996  ♯chash 11236   ~~> cli 12069   expce 12434   logclog 16085

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305  ax-pre-suploc 8306  ax-addf 8307  ax-mulf 8308

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-map 6928  df-pm 6929  df-en 7027  df-dom 7028  df-fin 7029  df-sup 7330  df-inf 7331  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9656  df-dec 9789  df-uz 9938  df-q 10036  df-rp 10072  df-xneg 10191  df-xadd 10192  df-ioo 10311  df-ico 10313  df-icc 10314  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-exp 10997  df-fac 11186  df-bc 11208  df-ihash 11237  df-shft 11602  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070  df-sumdc 12145  df-ef 12440  df-e 12441  df-rest 13654  df-topgen 13673  df-psmet 14989  df-xmet 14990  df-met 14991  df-bl 14992  df-mopn 14993  df-top 15224  df-topon 15237  df-bases 15269  df-ntr 15322  df-cn 15414  df-cnp 15415  df-tx 15479  df-cncf 15797  df-limced 15882  df-dvap 15883  df-relog 16087

This theorem is used by: (None)