Theorem ballotlemi 30562

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
ballotth.o	|- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
ballotth.p	|- P = ( x e. ~P O |-> ( ( # ` 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
ballotlemi	|- ( 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 ballotlemi

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( d = C -> ( F ` d ) = ( F ` C ) )
2	1	fveq1d 6193	. . . . 5 |- ( d = C -> ( ( F ` d ) ` k ) = ( ( F ` C ) ` k ) )
3	2	eqeq1d 2624	. . . 4 |- ( d = C -> ( ( ( F ` d ) ` k ) = 0 <-> ( ( F ` C ) ` k ) = 0 ) )
4	3	rabbidv 3189	. . 3 |- ( d = C -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } = { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
5	4	infeq1d 8383	. 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 6191	. . . . . . . 8 |- ( c = d -> ( F ` c ) = ( F ` d ) )
8	7	fveq1d 6193	. . . . . . 7 |- ( c = d -> ( ( F ` c ) ` k ) = ( ( F ` d ) ` k ) )
9	8	eqeq1d 2624	. . . . . 6 |- ( c = d -> ( ( ( F ` c ) ` k ) = 0 <-> ( ( F ` d ) ` k ) = 0 ) )
10	9	rabbidv 3189	. . . . 5 |- ( c = d -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } = { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } )
11	10	infeq1d 8383	. . . 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 4750	. . 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 2644	. 2 |- I = ( d e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } , RR , < ) )
14		ltso 10118	. . 3 |- < Or RR
15	14	infex 8399	. 2 |- inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) e. _V
16	5, 13, 15	fvmpt 6282	1 |- ( C e. ( O \ E ) -> ( I ` C ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650  infcinf 8347   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   ZZcz 11377   ...cfz 12326   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-ltxr 10079

This theorem is referenced by:  ballotlemiex  30563  ballotlemimin  30567  ballotlemfrcn0  30591  ballotlemirc  30593