Theorem ballotfi 13265

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 5693	. . . . . . . 8 |- ( q = c -> ( F ` q ) = ( F ` c ) )
9	8	fveq1d 5695	. . . . . . 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 7346	. . . 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 4225	. . 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 5702	. . . . . 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 7347	. . . 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 4216	. . 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 5693	. . . . . . 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 4140	. . . . . 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 6094	. . . . . . 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 6094	. . . . . 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 4220	. . . 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 4225	. . 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 4131	. . . . . 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 6087	. . . . . 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 4225	. . . 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 4216	. . 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 13264	1 |- ( P ` E ) = ( ( M - N ) / ( M + N ) )

Syntax hints:   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   \ cdif 3217   i^i cin 3219   ifcif 3638   ~Pcpw 3688   class class class wbr 4128   |-> cmpt 4190   "cima 4775   ` cfv 5375  (class class class)co 6079   Fincfn 7016  infcinf 7317   RRcr 8172   0cc0 8173   1c1 8174   + caddc 8176   < clt 8354   <_ cle 8355   - cmin 8491   / cdiv 8996   NNcn 9287   ZZcz 9627   ...cfz 10394  ♯chash 11197

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 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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-2o 6682  df-oadd 6685  df-er 6801  df-map 6918  df-en 7017  df-dom 7018  df-fin 7019  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-seqfrec 10868  df-fac 11147  df-bc 11169  df-ihash 11198

This theorem is referenced by: (None)