Theorem ballotfi 13341

Description: Bertrand's ballot problem : the probability that A is ahead throughout the counting. The proof formalized here is a proof "by reflection", as opposed to other known proofs "by induction" or "by permutation". This is Metamath 100 proof #30. (Contributed by Thierry Arnoux, 7-Dec-2016.) (Revised by Jim Kingdon, 17-Jun-2026.)

Hypotheses

Ref	Expression
ballotfi.m	|- M e. NN
ballotfi.n	|- N e. NN
ballotfi.o	|- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
ballotfi.p	|- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
ballotfi.f	|- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
ballotfi.e	|- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
ballotfi.mgtn	|- N < M

Assertion

Ref	Expression
ballotfi	|- ( P ` E ) = ( ( M - N ) / ( M + N ) )

Distinct variable groups:   E, c, i, x   F, c, i, x   M, c, i, x   N, c, i, x   O, c, i, x

Allowed substitution hints:   P( x, i, c)

Proof of Theorem ballotfi

Dummy variables k q r s p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotfi.m	. 2 |- M e. NN
2		ballotfi.n	. 2 |- N e. NN
3		ballotfi.o	. 2 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
4		ballotfi.p	. 2 |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
5		ballotfi.f	. 2 |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
6		ballotfi.e	. 2 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
7		ballotfi.mgtn	. 2 |- N < M
8		fveq2 5698	. . . . . . . 8 |- ( q = c -> ( F ` q ) = ( F ` c ) )
9	8	fveq1d 5700	. . . . . . 7 |- ( q = c -> ( ( F ` q ) ` p ) = ( ( F ` c ) ` p ) )
10	9	eqeq1d 2247	. . . . . 6 |- ( q = c -> ( ( ( F ` q ) ` p ) = 0 <-> ( ( F ` c ) ` p ) = 0 ) )
11	10	rabbidv 2810	. . . . 5 |- ( q = c -> { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } = { p e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` p ) = 0 } )
12	11	infeq1d 7359	. . . 4 |- ( q = c -> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) = inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` p ) = 0 } , RR , < ) )
13	12	cbvmptv 4229	. . 3 |- ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) = ( c e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` p ) = 0 } , RR , < ) )
14		fveqeq2 5707	. . . . . 6 |- ( p = k -> ( ( ( F ` c ) ` p ) = 0 <-> ( ( F ` c ) ` k ) = 0 ) )
15	14	cbvrabv 2820	. . . . 5 |- { p e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` p ) = 0 } = { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 }
16	15	infeq1i 7360	. . . 4 |- inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` p ) = 0 } , RR , < ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < )
17	16	mpteq2i 4220	. . 3 |- ( c e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` p ) = 0 } , RR , < ) ) = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
18	13, 17	eqtri 2259	. 2 |- ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
19		fveq2 5698	. . . . . . 7 |- ( r = c -> ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) = ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) )
20	19	breq2d 4143	. . . . . 6 |- ( r = c -> ( s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) <-> s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) ) )
21	19	oveq1d 6103	. . . . . . 7 |- ( r = c -> ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) + 1 ) = ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) + 1 ) )
22	21	oveq1d 6103	. . . . . 6 |- ( r = c -> ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) + 1 ) - s ) = ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) + 1 ) - s ) )
23	20, 22	ifbieq1d 3663	. . . . 5 |- ( r = c -> if ( s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) + 1 ) - s ) , s ) = if ( s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) + 1 ) - s ) , s ) )
24	23	mpteq2dv 4224	. . . 4 |- ( r = c -> ( s e. ( 1 ... ( M + N ) ) |-> if ( s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) + 1 ) - s ) , s ) ) = ( s e. ( 1 ... ( M + N ) ) |-> if ( s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) + 1 ) - s ) , s ) ) )
25	24	cbvmptv 4229	. . 3 |- ( r e. ( O \ E ) |-> ( s e. ( 1 ... ( M + N ) ) |-> if ( s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) + 1 ) - s ) , s ) ) ) = ( c e. ( O \ E ) |-> ( s e. ( 1 ... ( M + N ) ) |-> if ( s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) + 1 ) - s ) , s ) ) )
26		breq1 4134	. . . . . 6 |- ( s = i -> ( s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) <-> i <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) ) )
27		oveq2 6096	. . . . . 6 |- ( s = i -> ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) + 1 ) - s ) = ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) + 1 ) - i ) )
28		id 19	. . . . . 6 |- ( s = i -> s = i )
29	26, 27, 28	ifbieq12d 3667	. . . . 5 |- ( s = i -> if ( s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) + 1 ) - s ) , s ) = if ( i <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) + 1 ) - i ) , i ) )
30	29	cbvmptv 4229	. . . 4 |- ( s e. ( 1 ... ( M + N ) ) |-> if ( s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) + 1 ) - s ) , s ) ) = ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) + 1 ) - i ) , i ) )
31	30	mpteq2i 4220	. . 3 |- ( c e. ( O \ E ) |-> ( s e. ( 1 ... ( M + N ) ) |-> if ( s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) + 1 ) - s ) , s ) ) ) = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) + 1 ) - i ) , i ) ) )
32	25, 31	eqtri 2259	. 2 |- ( r e. ( O \ E ) |-> ( s e. ( 1 ... ( M + N ) ) |-> if ( s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) + 1 ) - s ) , s ) ) ) = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` c ) + 1 ) - i ) , i ) ) )
33		eqid 2238	. 2 |- ( c e. ( O \ E ) |-> ( ( ( r e. ( O \ E ) |-> ( s e. ( 1 ... ( M + N ) ) |-> if ( s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) + 1 ) - s ) , s ) ) ) ` c ) " c ) ) = ( c e. ( O \ E ) |-> ( ( ( r e. ( O \ E ) |-> ( s e. ( 1 ... ( M + N ) ) |-> if ( s <_ ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) , ( ( ( ( q e. ( O \ E ) |-> inf ( { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } , RR , < ) ) ` r ) + 1 ) - s ) , s ) ) ) ` c ) " c ) )
34	1, 2, 3, 4, 5, 6, 7, 18, 32, 33	ballotfilemth 13340	1 |- ( P ` E ) = ( ( M - N ) / ( M + N ) )

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   \ cdif 3217   i^i cin 3219   ifcif 3638   ~Pcpw 3688   class class class wbr 4131   |-> cmpt 4193   "cima 4779   ` cfv 5380  (class class class)co 6088   Fincfn 7027  infcinf 7330   RRcr 8185   0cc0 8186   1c1 8187   + caddc 8189   < clt 8367   <_ cle 8368   - cmin 8505   / cdiv 9011   NNcn 9314   ZZcz 9656   ...cfz 10429  ♯chash 11237

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-13 2211  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 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304

This proof depends on definitions:  df-bi 117  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-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-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6811  df-map 6928  df-en 7028  df-dom 7029  df-fin 7030  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-fz 10430  df-fzo 10568  df-seqfrec 10907  df-fac 11187  df-bc 11209  df-ihash 11238

This theorem is used by: (None)