Theorem ballotfi 13226

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 5675	. . . . . . . 8 |- ( q = c -> ( F ` q ) = ( F ` c ) )
9	8	fveq1d 5677	. . . . . . 7 |- ( q = c -> ( ( F ` q ) ` p ) = ( ( F ` c ) ` p ) )
10	9	eqeq1d 2243	. . . . . 6 |- ( q = c -> ( ( ( F ` q ) ` p ) = 0 <-> ( ( F ` c ) ` p ) = 0 ) )
11	10	rabbidv 2804	. . . . 5 |- ( q = c -> { p e. ( 1 ... ( M + N ) ) | ( ( F ` q ) ` p ) = 0 } = { p e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` p ) = 0 } )
12	11	infeq1d 7316	. . . 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 4211	. . 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 5684	. . . . . 6 |- ( p = k -> ( ( ( F ` c ) ` p ) = 0 <-> ( ( F ` c ) ` k ) = 0 ) )
15	14	cbvrabv 2814	. . . . 5 |- { p e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` p ) = 0 } = { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 }
16	15	infeq1i 7317	. . . 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 4202	. . 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 2255	. 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 5675	. . . . . . 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 4126	. . . . . 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 6073	. . . . . . 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 6073	. . . . . 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 3649	. . . . 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 4206	. . . 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 4211	. . 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 4117	. . . . . 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 6066	. . . . . 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 3653	. . . . 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 4211	. . . 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 4202	. . 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 2255	. 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 2234	. 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 13225	1 |- ( P ` E ) = ( ( M - N ) / ( M + N ) )

Syntax hints:   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   \ cdif 3211   i^i cin 3213   ifcif 3624   ~Pcpw 3674   class class class wbr 4114   |-> cmpt 4176   "cima 4757   ` cfv 5357  (class class class)co 6058   Fincfn 6988  infcinf 7287   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   <_ cle 8325   - cmin 8460   / cdiv 8963   NNcn 9254   ZZcz 9594   ...cfz 10361  ♯chash 11163

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

This theorem depends on definitions:  df-bi 117  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-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-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-map 6897  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-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-fac 11113  df-bc 11135  df-ihash 11164

This theorem is referenced by: (None)