Theorem ballotlemiex 30563

Description: Properties of ( I ` C ). (Contributed by Thierry Arnoux, 12-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
ballotlemiex	|- ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )

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 ballotlemiex

Step	Hyp	Ref	Expression
1		ballotth.m	. . . 4 |- M e. NN
2		ballotth.n	. . . 4 |- N e. NN
3		ballotth.o	. . . 4 |- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
4		ballotth.p	. . . 4 |- P = ( x e. ~P O |-> ( ( # ` x ) / ( # ` O ) ) )
5		ballotth.f	. . . 4 |- F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
6		ballotth.e	. . . 4 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
7		ballotth.mgtn	. . . 4 |- N < M
8		ballotth.i	. . . 4 |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
9	1, 2, 3, 4, 5, 6, 7, 8	ballotlemi 30562	. . 3 |- ( C e. ( O \ E ) -> ( I ` C ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) )
10		ltso 10118	. . . . 5 |- < Or RR
11	10	a1i 11	. . . 4 |- ( C e. ( O \ E ) -> < Or RR )
12		fzfi 12771	. . . . . 6 |- ( 1 ... ( M + N ) ) e. Fin
13		ssrab2 3687	. . . . . 6 |- { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ ( 1 ... ( M + N ) )
14		ssfi 8180	. . . . . 6 |- ( ( ( 1 ... ( M + N ) ) e. Fin /\ { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ ( 1 ... ( M + N ) ) ) -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } e. Fin )
15	12, 13, 14	mp2an 708	. . . . 5 |- { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } e. Fin
16	15	a1i 11	. . . 4 |- ( C e. ( O \ E ) -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } e. Fin )
17	1, 2, 3, 4, 5, 6, 7	ballotlem5 30561	. . . . 5 |- ( C e. ( O \ E ) -> E. k e. ( 1 ... ( M + N ) ) ( ( F ` C ) ` k ) = 0 )
18		rabn0 3958	. . . . 5 |- ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } =/= (/) <-> E. k e. ( 1 ... ( M + N ) ) ( ( F ` C ) ` k ) = 0 )
19	17, 18	sylibr 224	. . . 4 |- ( C e. ( O \ E ) -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } =/= (/) )
20		fzssuz 12382	. . . . . . . 8 |- ( 1 ... ( M + N ) ) C_ ( ZZ>= ` 1 )
21		uzssz 11707	. . . . . . . 8 |- ( ZZ>= ` 1 ) C_ ZZ
22	20, 21	sstri 3612	. . . . . . 7 |- ( 1 ... ( M + N ) ) C_ ZZ
23		zssre 11384	. . . . . . 7 |- ZZ C_ RR
24	22, 23	sstri 3612	. . . . . 6 |- ( 1 ... ( M + N ) ) C_ RR
25	13, 24	sstri 3612	. . . . 5 |- { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ RR
26	25	a1i 11	. . . 4 |- ( C e. ( O \ E ) -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ RR )
27		fiinfcl 8407	. . . 4 |- ( ( < Or RR /\ ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } e. Fin /\ { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } =/= (/) /\ { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ RR ) ) -> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
28	11, 16, 19, 26, 27	syl13anc 1328	. . 3 |- ( C e. ( O \ E ) -> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
29	9, 28	eqeltrd 2701	. 2 |- ( C e. ( O \ E ) -> ( I ` C ) e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
30		fveq2 6191	. . . 4 |- ( k = ( I ` C ) -> ( ( F ` C ) ` k ) = ( ( F ` C ) ` ( I ` C ) ) )
31	30	eqeq1d 2624	. . 3 |- ( k = ( I ` C ) -> ( ( ( F ` C ) ` k ) = 0 <-> ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
32	31	elrab 3363	. 2 |- ( ( I ` C ) e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } <-> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
33	29, 32	sylib 208	1 |- ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   Or wor 5034   ` cfv 5888  (class class class)co 6650   Fincfn 7955  infcinf 8347   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   ...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-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-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-int 4476  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-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  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-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by:  ballotlemi1  30564  ballotlemii  30565  ballotlemimin  30567  ballotlemic  30568  ballotlem1c  30569  ballotlemsgt1  30572  ballotlemsdom  30573  ballotlemsel1i  30574  ballotlemsf1o  30575  ballotlemsi  30576  ballotlemsima  30577  ballotlemrv2  30583  ballotlemfrc  30588  ballotlemfrci  30589  ballotlemfrceq  30590  ballotlemfrcn0  30591  ballotlemrc  30592  ballotlemirc  30593  ballotlem1ri  30596