Theorem ballotfilemiex 13293

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
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
ballotfilemiex	|- ( 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 ballotfilemiex

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotth.m	. . . 4 |- M e. NN
2		ballotth.n	. . . 4 |- N e. NN
3		ballotfilem.o	. . . 4 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
4		ballotfilem.p	. . . 4 |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` 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	ballotfilemi 13292	. . 3 |- ( C e. ( O \ E ) -> ( I ` C ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) )
10		ssrab2 3333	. . . . . 6 |- { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ ( 1 ... ( M + N ) )
11		fz1ssnn 10472	. . . . . 6 |- ( 1 ... ( M + N ) ) C_ NN
12	10, 11	sstri 3257	. . . . 5 |- { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ NN
13	12	a1i 9	. . . 4 |- ( C e. ( O \ E ) -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ NN )
14		nnz 9667	. . . . . . . . 9 |- ( z e. NN -> z e. ZZ )
15	14	adantl 277	. . . . . . . 8 |- ( ( C e. ( O \ E ) /\ z e. NN ) -> z e. ZZ )
16		1zzd 9675	. . . . . . . 8 |- ( ( C e. ( O \ E ) /\ z e. NN ) -> 1 e. ZZ )
17		nnaddcl 9326	. . . . . . . . . . 11 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
18	1, 2, 17	mp2an 430	. . . . . . . . . 10 |- ( M + N ) e. NN
19	18	nnzi 9669	. . . . . . . . 9 |- ( M + N ) e. ZZ
20	19	a1i 9	. . . . . . . 8 |- ( ( C e. ( O \ E ) /\ z e. NN ) -> ( M + N ) e. ZZ )
21		fzdcel 10454	. . . . . . . 8 |- ( ( z e. ZZ /\ 1 e. ZZ /\ ( M + N ) e. ZZ ) -> DECID z e. ( 1 ... ( M + N ) ) )
22	15, 16, 20, 21	syl3anc 1278	. . . . . . 7 |- ( ( C e. ( O \ E ) /\ z e. NN ) -> DECID z e. ( 1 ... ( M + N ) ) )
23		eldifi 3351	. . . . . . . . . 10 |- ( C e. ( O \ E ) -> C e. O )
24	23	adantr 276	. . . . . . . . 9 |- ( ( C e. ( O \ E ) /\ z e. NN ) -> C e. O )
25	1, 2, 3, 4, 5, 24, 15	ballotfilemfelz 13279	. . . . . . . 8 |- ( ( C e. ( O \ E ) /\ z e. NN ) -> ( ( F ` C ) ` z ) e. ZZ )
26		0zd 9660	. . . . . . . 8 |- ( ( C e. ( O \ E ) /\ z e. NN ) -> 0 e. ZZ )
27		zdceq 9724	. . . . . . . 8 |- ( ( ( ( F ` C ) ` z ) e. ZZ /\ 0 e. ZZ ) -> DECID ( ( F ` C ) ` z ) = 0 )
28	25, 26, 27	syl2anc 415	. . . . . . 7 |- ( ( C e. ( O \ E ) /\ z e. NN ) -> DECID ( ( F ` C ) ` z ) = 0 )
29	22, 28	dcand 945	. . . . . 6 |- ( ( C e. ( O \ E ) /\ z e. NN ) -> DECID ( z e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` z ) = 0 ) )
30		fveqeq2 5704	. . . . . . . 8 |- ( k = z -> ( ( ( F ` C ) ` k ) = 0 <-> ( ( F ` C ) ` z ) = 0 ) )
31	30	elrab 2982	. . . . . . 7 |- ( z e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } <-> ( z e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` z ) = 0 ) )
32	31	dcbii 852	. . . . . 6 |- (DECID z e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } <-> DECID ( z e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` z ) = 0 ) )
33	29, 32	sylibr 134	. . . . 5 |- ( ( C e. ( O \ E ) /\ z e. NN ) -> DECID z e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
34	33	ralrimiva 2623	. . . 4 |- ( C e. ( O \ E ) -> A. z e. NN DECID z e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
35	1, 2, 3, 4, 5, 6, 7	ballotfilem5 13291	. . . . 5 |- ( C e. ( O \ E ) -> E. k e. ( 1 ... ( M + N ) ) ( ( F ` C ) ` k ) = 0 )
36		rabn0m 3549	. . . . 5 |- ( E. y y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } <-> E. k e. ( 1 ... ( M + N ) ) ( ( F ` C ) ` k ) = 0 )
37	35, 36	sylibr 134	. . . 4 |- ( C e. ( O \ E ) -> E. y y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
38		nnmindc 12827	. . . 4 |- ( ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ NN /\ A. z e. NN DECID z e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } /\ E. y y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } ) -> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
39	13, 34, 37, 38	syl3anc 1278	. . 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 } )
40	9, 39	eqeltrd 2315	. 2 |- ( C e. ( O \ E ) -> ( I ` C ) e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
41		fveqeq2 5704	. . 3 |- ( k = ( I ` C ) -> ( ( ( F ` C ) ` k ) = 0 <-> ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
42	41	elrab 2982	. 2 |- ( ( I ` C ) e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } <-> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
43	40, 42	sylib 122	1 |- ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104  DECID wdc 846   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   \ cdif 3217   i^i cin 3219   C_ wss 3220   ~Pcpw 3688   class class class wbr 4130   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   Fincfn 7022  infcinf 7323   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   < clt 8360   - cmin 8498   / cdiv 9004   NNcn 9306   ZZcz 9648   ...cfz 10421  ♯chash 11228

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  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-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  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-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-n0 9568  df-z 9649  df-uz 9931  df-q 10029  df-rp 10065  df-fz 10422  df-fzo 10560  df-ihash 11229

This theorem is used by:  ballotfilemi1  13294  ballotfilemii  13295  ballotfilemimin  13298  ballotfilemic  13299  ballotfilem1c  13300  ballotfilemsv  13302  ballotfilemsgt1  13303  ballotfilemsdom  13304  ballotfilemsel1i  13305  ballotfilemsf1o  13306  ballotfilemsi  13307  ballotfilemsima  13308  ballotfilemrv2  13314  ballotfilemfrc  13319  ballotfilemfrci  13320  ballotfilemfrceq  13321  ballotfilemfrcn0  13322  ballotfilemrc  13323  ballotfilemirc  13324  ballotfilem1ri  13327