Theorem birthday 24681

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. This is Metamath 100 proof #93. (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

Assertion

Ref	Expression
birthday	|- ( ( # ` T ) / ( # ` S ) ) < ( 1 / 2 )

Distinct variable groups:   f, K   f, N

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

Proof of Theorem birthday

Step	Hyp	Ref	Expression
1		birthday.k	. . . 4 |- K = ; 2 3
2		2nn0 11309	. . . . 5 |- 2 e. NN0
3		3nn0 11310	. . . . 5 |- 3 e. NN0
4	2, 3	deccl 11512	. . . 4 |- ; 2 3 e. NN0
5	1, 4	eqeltri 2697	. . 3 |- K e. NN0
6		birthday.n	. . . 4 |- N = ;; 3 6 5
7		6nn0 11313	. . . . . 6 |- 6 e. NN0
8	3, 7	deccl 11512	. . . . 5 |- ; 3 6 e. NN0
9		5nn 11188	. . . . 5 |- 5 e. NN
10	8, 9	decnncl 11518	. . . 4 |- ;; 3 6 5 e. NN
11	6, 10	eqeltri 2697	. . 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 24680	. . 3 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( # ` T ) / ( # ` S ) ) <_ ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) )
15	5, 11, 14	mp2an 708	. 2 |- ( ( # ` T ) / ( # ` S ) ) <_ ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) )
16		log2ub 24676	. . . . . 6 |- ( log ` 2 ) < (;; 2 5 3 / ;; 3 6 5 )
17	5	nn0cni 11304	. . . . . . . . . . . 12 |- K e. CC
18	17	sqvali 12943	. . . . . . . . . . 11 |- ( K ^ 2 ) = ( K x. K )
19	17	mulid1i 10042	. . . . . . . . . . . 12 |- ( K x. 1 ) = K
20	19	eqcomi 2631	. . . . . . . . . . 11 |- K = ( K x. 1 )
21	18, 20	oveq12i 6662	. . . . . . . . . 10 |- ( ( K ^ 2 ) - K ) = ( ( K x. K ) - ( K x. 1 ) )
22		ax-1cn 9994	. . . . . . . . . . 11 |- 1 e. CC
23	17, 17, 22	subdii 10479	. . . . . . . . . 10 |- ( K x. ( K - 1 ) ) = ( ( K x. K ) - ( K x. 1 ) )
24	21, 23	eqtr4i 2647	. . . . . . . . 9 |- ( ( K ^ 2 ) - K ) = ( K x. ( K - 1 ) )
25	24	oveq1i 6660	. . . . . . . 8 |- ( ( ( K ^ 2 ) - K ) / 2 ) = ( ( K x. ( K - 1 ) ) / 2 )
26	17, 22	subcli 10357	. . . . . . . . . 10 |- ( K - 1 ) e. CC
27		2cn 11091	. . . . . . . . . 10 |- 2 e. CC
28		2ne0 11113	. . . . . . . . . 10 |- 2 =/= 0
29	17, 26, 27, 28	divassi 10781	. . . . . . . . 9 |- ( ( K x. ( K - 1 ) ) / 2 ) = ( K x. ( ( K - 1 ) / 2 ) )
30		1nn0 11308	. . . . . . . . . 10 |- 1 e. NN0
31		2p1e3 11151	. . . . . . . . . . . . . . . 16 |- ( 2 + 1 ) = 3
32		eqid 2622	. . . . . . . . . . . . . . . 16 |- ; 2 2 = ; 2 2
33	2, 2, 31, 32	decsuc 11535	. . . . . . . . . . . . . . 15 |- (; 2 2 + 1 ) = ; 2 3
34	1, 33	eqtr4i 2647	. . . . . . . . . . . . . 14 |- K = (; 2 2 + 1 )
35	34	oveq1i 6660	. . . . . . . . . . . . 13 |- ( K - 1 ) = ( (; 2 2 + 1 ) - 1 )
36	2, 2	deccl 11512	. . . . . . . . . . . . . . 15 |- ; 2 2 e. NN0
37	36	nn0cni 11304	. . . . . . . . . . . . . 14 |- ; 2 2 e. CC
38	37, 22	pncan3oi 10297	. . . . . . . . . . . . 13 |- ( (; 2 2 + 1 ) - 1 ) = ; 2 2
39	35, 38	eqtri 2644	. . . . . . . . . . . 12 |- ( K - 1 ) = ; 2 2
40	39	oveq1i 6660	. . . . . . . . . . 11 |- ( ( K - 1 ) / 2 ) = (; 2 2 / 2 )
41		eqid 2622	. . . . . . . . . . . . 13 |- ; 1 1 = ; 1 1
42		0nn0 11307	. . . . . . . . . . . . 13 |- 0 e. NN0
43	27	mulid1i 10042	. . . . . . . . . . . . . . 15 |- ( 2 x. 1 ) = 2
44	43	oveq1i 6660	. . . . . . . . . . . . . 14 |- ( ( 2 x. 1 ) + 0 ) = ( 2 + 0 )
45	27	addid1i 10223	. . . . . . . . . . . . . 14 |- ( 2 + 0 ) = 2
46	44, 45	eqtri 2644	. . . . . . . . . . . . 13 |- ( ( 2 x. 1 ) + 0 ) = 2
47	2	dec0h 11522	. . . . . . . . . . . . . 14 |- 2 = ; 0 2
48	43, 47	eqtri 2644	. . . . . . . . . . . . 13 |- ( 2 x. 1 ) = ; 0 2
49	2, 30, 30, 41, 2, 42, 46, 48	decmul2c 11589	. . . . . . . . . . . 12 |- ( 2 x. ; 1 1 ) = ; 2 2
50	30, 30	deccl 11512	. . . . . . . . . . . . . 14 |- ; 1 1 e. NN0
51	50	nn0cni 11304	. . . . . . . . . . . . 13 |- ; 1 1 e. CC
52	37, 27, 51, 28	divmuli 10779	. . . . . . . . . . . 12 |- ( (; 2 2 / 2 ) = ; 1 1 <-> ( 2 x. ; 1 1 ) = ; 2 2 )
53	49, 52	mpbir 221	. . . . . . . . . . 11 |- (; 2 2 / 2 ) = ; 1 1
54	40, 53	eqtri 2644	. . . . . . . . . 10 |- ( ( K - 1 ) / 2 ) = ; 1 1
55	19, 1	eqtri 2644	. . . . . . . . . . 11 |- ( K x. 1 ) = ; 2 3
56		3p2e5 11160	. . . . . . . . . . 11 |- ( 3 + 2 ) = 5
57	2, 3, 2, 55, 56	decaddi 11579	. . . . . . . . . 10 |- ( ( K x. 1 ) + 2 ) = ; 2 5
58	5, 30, 30, 54, 3, 2, 57, 55	decmul2c 11589	. . . . . . . . 9 |- ( K x. ( ( K - 1 ) / 2 ) ) = ;; 2 5 3
59	29, 58	eqtri 2644	. . . . . . . 8 |- ( ( K x. ( K - 1 ) ) / 2 ) = ;; 2 5 3
60	25, 59	eqtri 2644	. . . . . . 7 |- ( ( ( K ^ 2 ) - K ) / 2 ) = ;; 2 5 3
61	60, 6	oveq12i 6662	. . . . . 6 |- ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) = (;; 2 5 3 / ;; 3 6 5 )
62	16, 61	breqtrri 4680	. . . . 5 |- ( log ` 2 ) < ( ( ( ( K ^ 2 ) - K ) / 2 ) / N )
63		2rp 11837	. . . . . . 7 |- 2 e. RR+
64		relogcl 24322	. . . . . . 7 |- ( 2 e. RR+ -> ( log ` 2 ) e. RR )
65	63, 64	ax-mp 5	. . . . . 6 |- ( log ` 2 ) e. RR
66		5nn0 11312	. . . . . . . . . . 11 |- 5 e. NN0
67	2, 66	deccl 11512	. . . . . . . . . 10 |- ; 2 5 e. NN0
68	67, 3	deccl 11512	. . . . . . . . 9 |- ;; 2 5 3 e. NN0
69	60, 68	eqeltri 2697	. . . . . . . 8 |- ( ( ( K ^ 2 ) - K ) / 2 ) e. NN0
70	69	nn0rei 11303	. . . . . . 7 |- ( ( ( K ^ 2 ) - K ) / 2 ) e. RR
71		nndivre 11056	. . . . . . 7 |- ( ( ( ( ( K ^ 2 ) - K ) / 2 ) e. RR /\ N e. NN ) -> ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) e. RR )
72	70, 11, 71	mp2an 708	. . . . . 6 |- ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) e. RR
73	65, 72	ltnegi 10572	. . . . 5 |- ( ( log ` 2 ) < ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) <-> -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) < -u ( log ` 2 ) )
74	62, 73	mpbi 220	. . . 4 |- -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) < -u ( log ` 2 )
75	72	renegcli 10342	. . . . 5 |- -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) e. RR
76	65	renegcli 10342	. . . . 5 |- -u ( log ` 2 ) e. RR
77		eflt 14847	. . . . 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 ) ) ) )
78	75, 76, 77	mp2an 708	. . . 4 |- ( -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) < -u ( log ` 2 ) <-> ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) < ( exp ` -u ( log ` 2 ) ) )
79	74, 78	mpbi 220	. . 3 |- ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) < ( exp ` -u ( log ` 2 ) )
80	65	recni 10052	. . . . 5 |- ( log ` 2 ) e. CC
81		efneg 14828	. . . . 5 |- ( ( log ` 2 ) e. CC -> ( exp ` -u ( log ` 2 ) ) = ( 1 / ( exp ` ( log ` 2 ) ) ) )
82	80, 81	ax-mp 5	. . . 4 |- ( exp ` -u ( log ` 2 ) ) = ( 1 / ( exp ` ( log ` 2 ) ) )
83		reeflog 24327	. . . . . 6 |- ( 2 e. RR+ -> ( exp ` ( log ` 2 ) ) = 2 )
84	63, 83	ax-mp 5	. . . . 5 |- ( exp ` ( log ` 2 ) ) = 2
85	84	oveq2i 6661	. . . 4 |- ( 1 / ( exp ` ( log ` 2 ) ) ) = ( 1 / 2 )
86	82, 85	eqtri 2644	. . 3 |- ( exp ` -u ( log ` 2 ) ) = ( 1 / 2 )
87	79, 86	breqtri 4678	. 2 |- ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) < ( 1 / 2 )
88	12, 13	birthdaylem1 24678	. . . . . . . 8 |- ( T C_ S /\ S e. Fin /\ ( N e. NN -> S =/= (/) ) )
89	88	simp2i 1071	. . . . . . 7 |- S e. Fin
90	88	simp1i 1070	. . . . . . 7 |- T C_ S
91		ssfi 8180	. . . . . . 7 |- ( ( S e. Fin /\ T C_ S ) -> T e. Fin )
92	89, 90, 91	mp2an 708	. . . . . 6 |- T e. Fin
93		hashcl 13147	. . . . . 6 |- ( T e. Fin -> ( # ` T ) e. NN0 )
94	92, 93	ax-mp 5	. . . . 5 |- ( # ` T ) e. NN0
95	94	nn0rei 11303	. . . 4 |- ( # ` T ) e. RR
96	88	simp3i 1072	. . . . . 6 |- ( N e. NN -> S =/= (/) )
97	11, 96	ax-mp 5	. . . . 5 |- S =/= (/)
98		hashnncl 13157	. . . . . 6 |- ( S e. Fin -> ( ( # ` S ) e. NN <-> S =/= (/) ) )
99	89, 98	ax-mp 5	. . . . 5 |- ( ( # ` S ) e. NN <-> S =/= (/) )
100	97, 99	mpbir 221	. . . 4 |- ( # ` S ) e. NN
101		nndivre 11056	. . . 4 |- ( ( ( # ` T ) e. RR /\ ( # ` S ) e. NN ) -> ( ( # ` T ) / ( # ` S ) ) e. RR )
102	95, 100, 101	mp2an 708	. . 3 |- ( ( # ` T ) / ( # ` S ) ) e. RR
103		reefcl 14817	. . . 4 |- ( -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) e. RR -> ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) e. RR )
104	75, 103	ax-mp 5	. . 3 |- ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) e. RR
105		halfre 11246	. . 3 |- ( 1 / 2 ) e. RR
106	102, 104, 105	lelttri 10164	. 2 |- ( ( ( ( # ` T ) / ( # ` S ) ) <_ ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) /\ ( exp ` -u ( ( ( ( K ^ 2 ) - K ) / 2 ) / N ) ) < ( 1 / 2 ) ) -> ( ( # ` T ) / ( # ` S ) ) < ( 1 / 2 ) )
107	15, 87, 106	mp2an 708	1 |- ( ( # ` T ) / ( # ` S ) ) < ( 1 / 2 )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   C_ wss 3574   (/)c0 3915   class class class wbr 4653   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   5c5 11073   6c6 11074   NN0cn0 11292  ;cdc 11493   RR+crp 11832   ...cfz 12326   ^cexp 12860   #chash 13117   expce 14792   logclog 24301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  df-dvds 14984  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-ulm 24131  df-log 24303  df-atan 24594

This theorem is referenced by: (None)