Theorem ballotlemfg 30587

Description: Express the value of ( F ` C ) in terms of .^. (Contributed by Thierry Arnoux, 21-Apr-2017.)

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 , < ) )
ballotth.s	|- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
ballotth.r	|- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
ballotlemg	|- .^ = ( u e. Fin , v e. Fin |-> ( ( # ` ( v i^i u ) ) - ( # ` ( v \ u ) ) ) )

Assertion

Ref	Expression
ballotlemfg	|- ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> ( ( F ` C ) ` J ) = ( C .^ ( 1 ... J ) ) )

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, c   E, c   i, I, c   k, J   S, k, i, c   R, i   v, u, C   u, I, v   u, J, v   u, R, v   u, S, v   i, J

Allowed substitution hints:   C( x, c)   P( x, v, u, i, k, c)   R( x, k, c)   S( x)   E( x, v, u)   .^ ( x, v, u, i, k, c)   F( x, v, u)   I( x)   J( x, c)   M( x, v, u)   N( x, v, u)   O( x, v, u)

Proof of Theorem ballotlemfg

Step	Hyp	Ref	Expression
1		ballotth.m	. . 3 |- M e. NN
2		ballotth.n	. . 3 |- N e. NN
3		ballotth.o	. . 3 |- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
4		ballotth.p	. . 3 |- P = ( x e. ~P O |-> ( ( # ` x ) / ( # ` O ) ) )
5		ballotth.f	. . 3 |- F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
6		eldifi 3732	. . . 4 |- ( C e. ( O \ E ) -> C e. O )
7	6	adantr 481	. . 3 |- ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> C e. O )
8		elfzelz 12342	. . . 4 |- ( J e. ( 0 ... ( M + N ) ) -> J e. ZZ )
9	8	adantl 482	. . 3 |- ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> J e. ZZ )
10	1, 2, 3, 4, 5, 7, 9	ballotlemfval 30551	. 2 |- ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> ( ( F ` C ) ` J ) = ( ( # ` ( ( 1 ... J ) i^i C ) ) - ( # ` ( ( 1 ... J ) \ C ) ) ) )
11		fzfi 12771	. . . . 5 |- ( 1 ... ( M + N ) ) e. Fin
12	1, 2, 3	ballotlemelo 30549	. . . . . 6 |- ( C e. O <-> ( C C_ ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) )
13	12	simplbi 476	. . . . 5 |- ( C e. O -> C C_ ( 1 ... ( M + N ) ) )
14		ssfi 8180	. . . . 5 |- ( ( ( 1 ... ( M + N ) ) e. Fin /\ C C_ ( 1 ... ( M + N ) ) ) -> C e. Fin )
15	11, 13, 14	sylancr 695	. . . 4 |- ( C e. O -> C e. Fin )
16	7, 15	syl 17	. . 3 |- ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> C e. Fin )
17		fzfid 12772	. . 3 |- ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> ( 1 ... J ) e. Fin )
18		ballotth.e	. . . 4 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
19		ballotth.mgtn	. . . 4 |- N < M
20		ballotth.i	. . . 4 |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
21		ballotth.s	. . . 4 |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
22		ballotth.r	. . . 4 |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
23		ballotlemg	. . . 4 |- .^ = ( u e. Fin , v e. Fin |-> ( ( # ` ( v i^i u ) ) - ( # ` ( v \ u ) ) ) )
24	1, 2, 3, 4, 5, 18, 19, 20, 21, 22, 23	ballotlemgval 30585	. . 3 |- ( ( C e. Fin /\ ( 1 ... J ) e. Fin ) -> ( C .^ ( 1 ... J ) ) = ( ( # ` ( ( 1 ... J ) i^i C ) ) - ( # ` ( ( 1 ... J ) \ C ) ) ) )
25	16, 17, 24	syl2anc 693	. 2 |- ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> ( C .^ ( 1 ... J ) ) = ( ( # ` ( ( 1 ... J ) i^i C ) ) - ( # ` ( ( 1 ... J ) \ C ) ) ) )
26	10, 25	eqtr4d 2659	1 |- ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> ( ( F ` C ) ` J ) = ( C .^ ( 1 ... J ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   ifcif 4086   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955  infcinf 8347   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - 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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by:  ballotlemfrci  30589  ballotlemfrceq  30590