Theorem birthdaylog2 16090

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 = 23 and 𝑁 = 365, 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 = ;23
birthday.n	⊢ 𝑁 = ;;365
birthdaylog2.log2cnv	⊢ seq0( + , (𝑘 ∈ ℕ0 ↦ (2 / ((3 · ((2 · 𝑘) + 1)) · (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 = ;23
2		2nn0 9580	. . . . 5 ⊢ 2 ∈ ℕ0
3		3nn0 9581	. . . . 5 ⊢ 3 ∈ ℕ0
4	2, 3	deccl 9791	. . . 4 ⊢ ;23 ∈ ℕ0
5	1, 4	eqeltri 2311	. . 3 ⊢ 𝐾 ∈ ℕ0
6		birthday.n	. . . 4 ⊢ 𝑁 = ;;365
7		6nn0 9584	. . . . . 6 ⊢ 6 ∈ ℕ0
8	3, 7	deccl 9791	. . . . 5 ⊢ ;36 ∈ ℕ0
9		5nn 9469	. . . . 5 ⊢ 5 ∈ ℕ
10	8, 9	decnncl 9796	. . . 4 ⊢ ;;365 ∈ ℕ
11	6, 10	eqeltri 2311	. . 3 ⊢ 𝑁 ∈ ℕ
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 16089	. . 3 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → ((♯‘𝑇) / (♯‘𝑆)) ≤ (exp‘-((((𝐾↑2) − 𝐾) / 2) / 𝑁)))
15	5, 11, 14	mp2an 430	. 2 ⊢ ((♯‘𝑇) / (♯‘𝑆)) ≤ (exp‘-((((𝐾↑2) − 𝐾) / 2) / 𝑁))
16		birthdaylog2.log2cnv	. . . . . . 7 ⊢ seq0( + , (𝑘 ∈ ℕ0 ↦ (2 / ((3 · ((2 · 𝑘) + 1)) · (9↑𝑘))))) ⇝ (log‘2)
17	16	log2ublog2 16086	. . . . . 6 ⊢ (log‘2) < (;;253 / ;;365)
18	5	nn0cni 9575	. . . . . . . . . . . 12 ⊢ 𝐾 ∈ ℂ
19	18	sqvali 11056	. . . . . . . . . . 11 ⊢ (𝐾↑2) = (𝐾 · 𝐾)
20	18	mulridi 8328	. . . . . . . . . . . 12 ⊢ (𝐾 · 1) = 𝐾
21	20	eqcomi 2242	. . . . . . . . . . 11 ⊢ 𝐾 = (𝐾 · 1)
22	19, 21	oveq12i 6097	. . . . . . . . . 10 ⊢ ((𝐾↑2) − 𝐾) = ((𝐾 · 𝐾) − (𝐾 · 1))
23		ax-1cn 8272	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
24	18, 18, 23	subdii 8734	. . . . . . . . . 10 ⊢ (𝐾 · (𝐾 − 1)) = ((𝐾 · 𝐾) − (𝐾 · 1))
25	22, 24	eqtr4i 2262	. . . . . . . . 9 ⊢ ((𝐾↑2) − 𝐾) = (𝐾 · (𝐾 − 1))
26	25	oveq1i 6095	. . . . . . . 8 ⊢ (((𝐾↑2) − 𝐾) / 2) = ((𝐾 · (𝐾 − 1)) / 2)
27	18, 23	subcli 8602	. . . . . . . . 9 ⊢ (𝐾 − 1) ∈ ℂ
28		2cn 9375	. . . . . . . . 9 ⊢ 2 ∈ ℂ
29		2ap0 9397	. . . . . . . . 9 ⊢ 2 # 0
30	18, 27, 28, 29	divassapi 9098	. . . . . . . 8 ⊢ ((𝐾 · (𝐾 − 1)) / 2) = (𝐾 · ((𝐾 − 1) / 2))
31		1nn0 9579	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
32	2, 2	deccl 9791	. . . . . . . . . . . . 13 ⊢ ;22 ∈ ℕ0
33	32	nn0cni 9575	. . . . . . . . . . . 12 ⊢ ;22 ∈ ℂ
34		2p1e3 9438	. . . . . . . . . . . . . 14 ⊢ (2 + 1) = 3
35		eqid 2238	. . . . . . . . . . . . . 14 ⊢ ;22 = ;22
36	2, 2, 34, 35	decsuc 9807	. . . . . . . . . . . . 13 ⊢ (;22 + 1) = ;23
37	1, 36	eqtr4i 2262	. . . . . . . . . . . 12 ⊢ 𝐾 = (;22 + 1)
38	33, 23, 37	mvrraddi 8543	. . . . . . . . . . 11 ⊢ (𝐾 − 1) = ;22
39	38	oveq1i 6095	. . . . . . . . . 10 ⊢ ((𝐾 − 1) / 2) = (;22 / 2)
40	2	11multnc 9844	. . . . . . . . . . 11 ⊢ (2 · ;11) = ;22
41	31, 31	deccl 9791	. . . . . . . . . . . . 13 ⊢ ;11 ∈ ℕ0
42	41	nn0cni 9575	. . . . . . . . . . . 12 ⊢ ;11 ∈ ℂ
43	33, 28, 42, 29	divmulapi 9096	. . . . . . . . . . 11 ⊢ ((;22 / 2) = ;11 ↔ (2 · ;11) = ;22)
44	40, 43	mpbir 146	. . . . . . . . . 10 ⊢ (;22 / 2) = ;11
45	39, 44	eqtri 2259	. . . . . . . . 9 ⊢ ((𝐾 − 1) / 2) = ;11
46	20, 1	eqtri 2259	. . . . . . . . . 10 ⊢ (𝐾 · 1) = ;23
47		3p2e5 9446	. . . . . . . . . 10 ⊢ (3 + 2) = 5
48	2, 3, 2, 46, 47	decaddi 9836	. . . . . . . . 9 ⊢ ((𝐾 · 1) + 2) = ;25
49	5, 31, 31, 45, 3, 2, 48, 46	decmul2c 9842	. . . . . . . 8 ⊢ (𝐾 · ((𝐾 − 1) / 2)) = ;;253
50	26, 30, 49	3eqtri 2263	. . . . . . 7 ⊢ (((𝐾↑2) − 𝐾) / 2) = ;;253
51	50, 6	oveq12i 6097	. . . . . 6 ⊢ ((((𝐾↑2) − 𝐾) / 2) / 𝑁) = (;;253 / ;;365)
52	17, 51	breqtrri 4157	. . . . 5 ⊢ (log‘2) < ((((𝐾↑2) − 𝐾) / 2) / 𝑁)
53		2rp 10059	. . . . . . 7 ⊢ 2 ∈ ℝ+
54		relogcl 15963	. . . . . . 7 ⊢ (2 ∈ ℝ+ → (log‘2) ∈ ℝ)
55	53, 54	ax-mp 5	. . . . . 6 ⊢ (log‘2) ∈ ℝ
56		5nn0 9583	. . . . . . . . . . 11 ⊢ 5 ∈ ℕ0
57	2, 56	deccl 9791	. . . . . . . . . 10 ⊢ ;25 ∈ ℕ0
58	57, 3	deccl 9791	. . . . . . . . 9 ⊢ ;;253 ∈ ℕ0
59	50, 58	eqeltri 2311	. . . . . . . 8 ⊢ (((𝐾↑2) − 𝐾) / 2) ∈ ℕ0
60	59	nn0rei 9574	. . . . . . 7 ⊢ (((𝐾↑2) − 𝐾) / 2) ∈ ℝ
61		nndivre 9340	. . . . . . 7 ⊢ (((((𝐾↑2) − 𝐾) / 2) ∈ ℝ ∧ 𝑁 ∈ ℕ) → ((((𝐾↑2) − 𝐾) / 2) / 𝑁) ∈ ℝ)
62	60, 11, 61	mp2an 430	. . . . . 6 ⊢ ((((𝐾↑2) − 𝐾) / 2) / 𝑁) ∈ ℝ
63	55, 62	ltnegi 8821	. . . . 5 ⊢ ((log‘2) < ((((𝐾↑2) − 𝐾) / 2) / 𝑁) ↔ -((((𝐾↑2) − 𝐾) / 2) / 𝑁) < -(log‘2))
64	52, 63	mpbi 145	. . . 4 ⊢ -((((𝐾↑2) − 𝐾) / 2) / 𝑁) < -(log‘2)
65	62	renegcli 8588	. . . . 5 ⊢ -((((𝐾↑2) − 𝐾) / 2) / 𝑁) ∈ ℝ
66	55	renegcli 8588	. . . . 5 ⊢ -(log‘2) ∈ ℝ
67		eflt 15876	. . . . 5 ⊢ ((-((((𝐾↑2) − 𝐾) / 2) / 𝑁) ∈ ℝ ∧ -(log‘2) ∈ ℝ) → (-((((𝐾↑2) − 𝐾) / 2) / 𝑁) < -(log‘2) ↔ (exp‘-((((𝐾↑2) − 𝐾) / 2) / 𝑁)) < (exp‘-(log‘2))))
68	65, 66, 67	mp2an 430	. . . 4 ⊢ (-((((𝐾↑2) − 𝐾) / 2) / 𝑁) < -(log‘2) ↔ (exp‘-((((𝐾↑2) − 𝐾) / 2) / 𝑁)) < (exp‘-(log‘2)))
69	64, 68	mpbi 145	. . 3 ⊢ (exp‘-((((𝐾↑2) − 𝐾) / 2) / 𝑁)) < (exp‘-(log‘2))
70	55	recni 8338	. . . . 5 ⊢ (log‘2) ∈ ℂ
71		efneg 12446	. . . . 5 ⊢ ((log‘2) ∈ ℂ → (exp‘-(log‘2)) = (1 / (exp‘(log‘2))))
72	70, 71	ax-mp 5	. . . 4 ⊢ (exp‘-(log‘2)) = (1 / (exp‘(log‘2)))
73		reeflog 15964	. . . . . 6 ⊢ (2 ∈ ℝ+ → (exp‘(log‘2)) = 2)
74	53, 73	ax-mp 5	. . . . 5 ⊢ (exp‘(log‘2)) = 2
75	74	oveq2i 6096	. . . 4 ⊢ (1 / (exp‘(log‘2))) = (1 / 2)
76	72, 75	eqtri 2259	. . 3 ⊢ (exp‘-(log‘2)) = (1 / 2)
77	69, 76	breqtri 4155	. 2 ⊢ (exp‘-((((𝐾↑2) − 𝐾) / 2) / 𝑁)) < (1 / 2)
78	31	nn0zi 9666	. . . . . . . . 9 ⊢ 1 ∈ ℤ
79	5	nn0zi 9666	. . . . . . . . 9 ⊢ 𝐾 ∈ ℤ
80		fzfig 10867	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (1...𝐾) ∈ Fin)
81	78, 79, 80	mp2an 430	. . . . . . . 8 ⊢ (1...𝐾) ∈ Fin
82	11	nnzi 9665	. . . . . . . . 9 ⊢ 𝑁 ∈ ℤ
83		fzfig 10867	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1...𝑁) ∈ Fin)
84	78, 82, 83	mp2an 430	. . . . . . . 8 ⊢ (1...𝑁) ∈ Fin
85		f1setfi 7317	. . . . . . . 8 ⊢ (((1...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)} ∈ Fin)
86	81, 84, 85	mp2an 430	. . . . . . 7 ⊢ {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)} ∈ Fin
87	13, 86	eqeltri 2311	. . . . . 6 ⊢ 𝑇 ∈ Fin
88		hashcl 11220	. . . . . 6 ⊢ (𝑇 ∈ Fin → (♯‘𝑇) ∈ ℕ0)
89	87, 88	ax-mp 5	. . . . 5 ⊢ (♯‘𝑇) ∈ ℕ0
90	89	nn0rei 9574	. . . 4 ⊢ (♯‘𝑇) ∈ ℝ
91	12, 13	birthdaylem1g 16087	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑇 ⊆ 𝑆 ∧ 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅))
92	5, 11, 91	mp2an 430	. . . . . 6 ⊢ (𝑇 ⊆ 𝑆 ∧ 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅)
93	92	simp3i 1039	. . . . 5 ⊢ 𝑆 ≠ ∅
94	92	simp2i 1038	. . . . . 6 ⊢ 𝑆 ∈ Fin
95		hashnncl 11234	. . . . . 6 ⊢ (𝑆 ∈ Fin → ((♯‘𝑆) ∈ ℕ ↔ 𝑆 ≠ ∅))
96	94, 95	ax-mp 5	. . . . 5 ⊢ ((♯‘𝑆) ∈ ℕ ↔ 𝑆 ≠ ∅)
97	93, 96	mpbir 146	. . . 4 ⊢ (♯‘𝑆) ∈ ℕ
98		nndivre 9340	. . . 4 ⊢ (((♯‘𝑇) ∈ ℝ ∧ (♯‘𝑆) ∈ ℕ) → ((♯‘𝑇) / (♯‘𝑆)) ∈ ℝ)
99	90, 97, 98	mp2an 430	. . 3 ⊢ ((♯‘𝑇) / (♯‘𝑆)) ∈ ℝ
100		reefcl 12435	. . . 4 ⊢ (-((((𝐾↑2) − 𝐾) / 2) / 𝑁) ∈ ℝ → (exp‘-((((𝐾↑2) − 𝐾) / 2) / 𝑁)) ∈ ℝ)
101	65, 100	ax-mp 5	. . 3 ⊢ (exp‘-((((𝐾↑2) − 𝐾) / 2) / 𝑁)) ∈ ℝ
102		halfre 9518	. . 3 ⊢ (1 / 2) ∈ ℝ
103	99, 101, 102	lelttri 8431	. 2 ⊢ ((((♯‘𝑇) / (♯‘𝑆)) ≤ (exp‘-((((𝐾↑2) − 𝐾) / 2) / 𝑁)) ∧ (exp‘-((((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   ∈ wcel 2209  {cab 2224   ≠ wne 2420   ⊆ wss 3220  ∅c0 3520   class class class wbr 4130   ↦ cmpt 4192  ⟶wf 5373  –1-1→wf1 5374  ‘cfv 5377  (class class class)co 6085  Fincfn 7022  ℂcc 8177  ℝcr 8178  0cc0 8179  1c1 8180   + caddc 8182   · cmul 8184   < clt 8360   ≤ cle 8361   − cmin 8497  -cneg 8498   / cdiv 9002  ℕcn 9304  2c2 9355  3c3 9356  5c5 9358  6c6 9359  9c9 9362  ℕ₀cn0 9563  ℤcz 9644  ;cdc 9777  ℝ⁺crp 10054  ...cfz 10411  seqcseq 10884  ↑cexp 10975  ♯chash 11214   ⇝ cli 12044  expce 12409  logclog 15957

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-pre-suploc 8300  ax-addf 8301  ax-mulf 8302

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-q 10020  df-rp 10055  df-xneg 10174  df-xadd 10175  df-ioo 10294  df-ico 10296  df-icc 10297  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-fac 11164  df-bc 11186  df-ihash 11215  df-shft 11580  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120  df-ef 12415  df-e 12416  df-rest 13595  df-topgen 13614  df-psmet 14880  df-xmet 14881  df-met 14882  df-bl 14883  df-mopn 14884  df-top 15099  df-topon 15112  df-bases 15144  df-ntr 15197  df-cn 15289  df-cnp 15290  df-tx 15354  df-cncf 15672  df-limced 15757  df-dvap 15758  df-relog 15959

This theorem is used by: (None)