Theorem ballotfilemi 13187

Description: Value of I for a given counting C. (Contributed by Thierry Arnoux, 1-Dec-2016.) (Revised by AV, 6-Oct-2020.)

Hypotheses

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

Assertion

Ref	Expression
ballotfilemi	|- ( C e. ( O \ E ) -> ( I ` C ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) )

Distinct variable groups:   M, c   N, c   O, c   i, M   i, N   i, O   k, M   k, N   k, O   i, c, F, k   C, i, k   i, E, k   C, k   k, I   k, c, E

Allowed substitution hints:   C( x, c)   P( x, i, k, c)   E( x)   F( x)   I( x, i, c)   M( x)   N( x)   O( x)

Proof of Theorem ballotfilemi

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5675	. . . . . 6 |- ( d = C -> ( F ` d ) = ( F ` C ) )
2	1	fveq1d 5677	. . . . 5 |- ( d = C -> ( ( F ` d ) ` k ) = ( ( F ` C ) ` k ) )
3	2	eqeq1d 2243	. . . 4 |- ( d = C -> ( ( ( F ` d ) ` k ) = 0 <-> ( ( F ` C ) ` k ) = 0 ) )
4	3	rabbidv 2804	. . 3 |- ( d = C -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } = { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
5	4	infeq1d 7316	. 2 |- ( d = C -> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } , RR , < ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) )
6		ballotth.i	. . 3 |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
7		fveq2 5675	. . . . . . . 8 |- ( c = d -> ( F ` c ) = ( F ` d ) )
8	7	fveq1d 5677	. . . . . . 7 |- ( c = d -> ( ( F ` c ) ` k ) = ( ( F ` d ) ` k ) )
9	8	eqeq1d 2243	. . . . . 6 |- ( c = d -> ( ( ( F ` c ) ` k ) = 0 <-> ( ( F ` d ) ` k ) = 0 ) )
10	9	rabbidv 2804	. . . . 5 |- ( c = d -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } = { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } )
11	10	infeq1d 7316	. . . 4 |- ( c = d -> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } , RR , < ) )
12	11	cbvmptv 4211	. . 3 |- ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) ) = ( d e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } , RR , < ) )
13	6, 12	eqtri 2255	. 2 |- I = ( d e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } , RR , < ) )
14		reex 8277	. . 3 |- RR e. _V
15		infex2g 7338	. . 3 |- ( RR e. _V -> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) e. _V )
16	14, 15	ax-mp 5	. 2 |- inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) e. _V
17	5, 13, 16	fvmpt 5759	1 |- ( C e. ( O \ E ) -> ( I ` C ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   _Vcvv 2815   \ cdif 3211   i^i cin 3213   ~Pcpw 3674   class class class wbr 4114   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058   Fincfn 6988  infcinf 7287   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   - 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-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-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  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-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-sup 7288  df-inf 7289

This theorem is referenced by:  ballotfilemiex  13188  ballotfilemimin  13193  ballotfilemfrcn0  13217  ballotfilemirc  13219