Theorem ballotfilemi 13226

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 5693	. . . . . 6 |- ( d = C -> ( F ` d ) = ( F ` C ) )
2	1	fveq1d 5695	. . . . 5 |- ( d = C -> ( ( F ` d ) ` k ) = ( ( F ` C ) ` k ) )
3	2	eqeq1d 2247	. . . 4 |- ( d = C -> ( ( ( F ` d ) ` k ) = 0 <-> ( ( F ` C ) ` k ) = 0 ) )
4	3	rabbidv 2810	. . 3 |- ( d = C -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } = { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
5	4	infeq1d 7346	. 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 5693	. . . . . . . 8 |- ( c = d -> ( F ` c ) = ( F ` d ) )
8	7	fveq1d 5695	. . . . . . 7 |- ( c = d -> ( ( F ` c ) ` k ) = ( ( F ` d ) ` k ) )
9	8	eqeq1d 2247	. . . . . 6 |- ( c = d -> ( ( ( F ` c ) ` k ) = 0 <-> ( ( F ` d ) ` k ) = 0 ) )
10	9	rabbidv 2810	. . . . 5 |- ( c = d -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } = { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } )
11	10	infeq1d 7346	. . . 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 4225	. . 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 2259	. 2 |- I = ( d e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } , RR , < ) )
14		reex 8307	. . 3 |- RR e. _V
15		infex2g 7368	. . 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 5779	1 |- ( C e. ( O \ E ) -> ( I ` C ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   \ cdif 3217   i^i cin 3219   ~Pcpw 3688   class class class wbr 4128   |-> cmpt 4190   ` cfv 5375  (class class class)co 6079   Fincfn 7016  infcinf 7317   RRcr 8172   0cc0 8173   1c1 8174   + caddc 8176   < clt 8354   - 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-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-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fv 5383  df-sup 7318  df-inf 7319

This theorem is referenced by:  ballotfilemiex  13227  ballotfilemimin  13232  ballotfilemfrcn0  13256  ballotfilemirc  13258