Theorem ballotlemimin 28084

Description: ( I ` C ) is the first tie. (Contributed by Thierry Arnoux, 1-Dec-2016.)

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 ) |-> sup ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , `' < ) )

Assertion

Ref	Expression
ballotlemimin	|- ( C e. ( O \ E ) -> -. E. k e. ( 1 ... ( ( I ` C ) - 1 ) ) ( ( F ` C ) ` k ) = 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   i, I

Allowed substitution hints:   C( x, c)   P( x, i, k, c)   E( x)   F( x)   I( x, c)   M( x)   N( x)   O( x)

Proof of Theorem ballotlemimin

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

Step	Hyp	Ref	Expression
1		elfzle2 11686	. . . . . 6 |- ( k e. ( 1 ... ( ( I ` C ) - 1 ) ) -> k <_ ( ( I ` C ) - 1 ) )
2	1	adantl 466	. . . . 5 |- ( ( C e. ( O \ E ) /\ k e. ( 1 ... ( ( I ` C ) - 1 ) ) ) -> k <_ ( ( I ` C ) - 1 ) )
3		elfzelz 11684	. . . . . 6 |- ( k e. ( 1 ... ( ( I ` C ) - 1 ) ) -> k e. ZZ )
4		ballotth.m	. . . . . . . . . 10 |- M e. NN
5		ballotth.n	. . . . . . . . . 10 |- N e. NN
6		ballotth.o	. . . . . . . . . 10 |- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
7		ballotth.p	. . . . . . . . . 10 |- P = ( x e. ~P O |-> ( ( # ` x ) / ( # ` O ) ) )
8		ballotth.f	. . . . . . . . . 10 |- F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
9		ballotth.e	. . . . . . . . . 10 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
10		ballotth.mgtn	. . . . . . . . . 10 |- N < M
11		ballotth.i	. . . . . . . . . 10 |- I = ( c e. ( O \ E ) |-> sup ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , `' < ) )
12	4, 5, 6, 7, 8, 9, 10, 11	ballotlemiex 28080	. . . . . . . . 9 |- ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
13	12	simpld 459	. . . . . . . 8 |- ( C e. ( O \ E ) -> ( I ` C ) e. ( 1 ... ( M + N ) ) )
14		elfznn 11710	. . . . . . . 8 |- ( ( I ` C ) e. ( 1 ... ( M + N ) ) -> ( I ` C ) e. NN )
15	13, 14	syl 16	. . . . . . 7 |- ( C e. ( O \ E ) -> ( I ` C ) e. NN )
16	15	nnzd 10961	. . . . . 6 |- ( C e. ( O \ E ) -> ( I ` C ) e. ZZ )
17		zltlem1 10911	. . . . . 6 |- ( ( k e. ZZ /\ ( I ` C ) e. ZZ ) -> ( k < ( I ` C ) <-> k <_ ( ( I ` C ) - 1 ) ) )
18	3, 16, 17	syl2anr 478	. . . . 5 |- ( ( C e. ( O \ E ) /\ k e. ( 1 ... ( ( I ` C ) - 1 ) ) ) -> ( k < ( I ` C ) <-> k <_ ( ( I ` C ) - 1 ) ) )
19	2, 18	mpbird 232	. . . 4 |- ( ( C e. ( O \ E ) /\ k e. ( 1 ... ( ( I ` C ) - 1 ) ) ) -> k < ( I ` C ) )
20	19	adantr 465	. . 3 |- ( ( ( C e. ( O \ E ) /\ k e. ( 1 ... ( ( I ` C ) - 1 ) ) ) /\ ( ( F ` C ) ` k ) = 0 ) -> k < ( I ` C ) )
21		1z 10890	. . . . . . . . . . . . . 14 |- 1 e. ZZ
22	21	a1i 11	. . . . . . . . . . . . 13 |- ( C e. ( O \ E ) -> 1 e. ZZ )
23	16, 22	zsubcld 10967	. . . . . . . . . . . 12 |- ( C e. ( O \ E ) -> ( ( I ` C ) - 1 ) e. ZZ )
24	23	zred 10962	. . . . . . . . . . 11 |- ( C e. ( O \ E ) -> ( ( I ` C ) - 1 ) e. RR )
25		nnaddcl 10554	. . . . . . . . . . . . . 14 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
26	4, 5, 25	mp2an 672	. . . . . . . . . . . . 13 |- ( M + N ) e. NN
27	26	a1i 11	. . . . . . . . . . . 12 |- ( C e. ( O \ E ) -> ( M + N ) e. NN )
28	27	nnred 10547	. . . . . . . . . . 11 |- ( C e. ( O \ E ) -> ( M + N ) e. RR )
29		elfzle2 11686	. . . . . . . . . . . . 13 |- ( ( I ` C ) e. ( 1 ... ( M + N ) ) -> ( I ` C ) <_ ( M + N ) )
30	13, 29	syl 16	. . . . . . . . . . . 12 |- ( C e. ( O \ E ) -> ( I ` C ) <_ ( M + N ) )
31	27	nnzd 10961	. . . . . . . . . . . . 13 |- ( C e. ( O \ E ) -> ( M + N ) e. ZZ )
32		zlem1lt 10910	. . . . . . . . . . . . 13 |- ( ( ( I ` C ) e. ZZ /\ ( M + N ) e. ZZ ) -> ( ( I ` C ) <_ ( M + N ) <-> ( ( I ` C ) - 1 ) < ( M + N ) ) )
33	16, 31, 32	syl2anc 661	. . . . . . . . . . . 12 |- ( C e. ( O \ E ) -> ( ( I ` C ) <_ ( M + N ) <-> ( ( I ` C ) - 1 ) < ( M + N ) ) )
34	30, 33	mpbid 210	. . . . . . . . . . 11 |- ( C e. ( O \ E ) -> ( ( I ` C ) - 1 ) < ( M + N ) )
35	24, 28, 34	ltled 9728	. . . . . . . . . 10 |- ( C e. ( O \ E ) -> ( ( I ` C ) - 1 ) <_ ( M + N ) )
36		eluz 11091	. . . . . . . . . . 11 |- ( ( ( ( I ` C ) - 1 ) e. ZZ /\ ( M + N ) e. ZZ ) -> ( ( M + N ) e. ( ZZ>= ` ( ( I ` C ) - 1 ) ) <-> ( ( I ` C ) - 1 ) <_ ( M + N ) ) )
37	23, 31, 36	syl2anc 661	. . . . . . . . . 10 |- ( C e. ( O \ E ) -> ( ( M + N ) e. ( ZZ>= ` ( ( I ` C ) - 1 ) ) <-> ( ( I ` C ) - 1 ) <_ ( M + N ) ) )
38	35, 37	mpbird 232	. . . . . . . . 9 |- ( C e. ( O \ E ) -> ( M + N ) e. ( ZZ>= ` ( ( I ` C ) - 1 ) ) )
39		fzss2 11719	. . . . . . . . 9 |- ( ( M + N ) e. ( ZZ>= ` ( ( I ` C ) - 1 ) ) -> ( 1 ... ( ( I ` C ) - 1 ) ) C_ ( 1 ... ( M + N ) ) )
40	38, 39	syl 16	. . . . . . . 8 |- ( C e. ( O \ E ) -> ( 1 ... ( ( I ` C ) - 1 ) ) C_ ( 1 ... ( M + N ) ) )
41	40	sseld 3503	. . . . . . 7 |- ( C e. ( O \ E ) -> ( k e. ( 1 ... ( ( I ` C ) - 1 ) ) -> k e. ( 1 ... ( M + N ) ) ) )
42		rabid 3038	. . . . . . . 8 |- ( k e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } <-> ( k e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` k ) = 0 ) )
43	4, 5, 6, 7, 8, 9, 10, 11	ballotlemsup 28083	. . . . . . . . . 10 |- ( C e. ( O \ E ) -> E. z e. RR ( A. w e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } -. z `' < w /\ A. w e. RR ( w `' < z -> E. y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } w `' < y ) ) )
44		ltso 9661	. . . . . . . . . . . . 13 |- < Or RR
45		cnvso 5544	. . . . . . . . . . . . 13 |- ( < Or RR <-> `' < Or RR )
46	44, 45	mpbi 208	. . . . . . . . . . . 12 |- `' < Or RR
47	46	a1i 11	. . . . . . . . . . 11 |- ( E. z e. RR ( A. w e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } -. z `' < w /\ A. w e. RR ( w `' < z -> E. y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } w `' < y ) ) -> `' < Or RR )
48		id 22	. . . . . . . . . . 11 |- ( E. z e. RR ( A. w e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } -. z `' < w /\ A. w e. RR ( w `' < z -> E. y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } w `' < y ) ) -> E. z e. RR ( A. w e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } -. z `' < w /\ A. w e. RR ( w `' < z -> E. y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } w `' < y ) ) )
49	47, 48	supub 7915	. . . . . . . . . 10 |- ( E. z e. RR ( A. w e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } -. z `' < w /\ A. w e. RR ( w `' < z -> E. y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } w `' < y ) ) -> ( k e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } -> -. sup ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , `' < ) `' < k ) )
50	43, 49	syl 16	. . . . . . . . 9 |- ( C e. ( O \ E ) -> ( k e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } -> -. sup ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , `' < ) `' < k ) )
51	4, 5, 6, 7, 8, 9, 10, 11	ballotlemi 28079	. . . . . . . . . . 11 |- ( C e. ( O \ E ) -> ( I ` C ) = sup ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , `' < ) )
52	51	breq1d 4457	. . . . . . . . . 10 |- ( C e. ( O \ E ) -> ( ( I ` C ) `' < k <-> sup ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , `' < ) `' < k ) )
53	52	notbid 294	. . . . . . . . 9 |- ( C e. ( O \ E ) -> ( -. ( I ` C ) `' < k <-> -. sup ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , `' < ) `' < k ) )
54	50, 53	sylibrd 234	. . . . . . . 8 |- ( C e. ( O \ E ) -> ( k e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } -> -. ( I ` C ) `' < k ) )
55	42, 54	syl5bir 218	. . . . . . 7 |- ( C e. ( O \ E ) -> ( ( k e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` k ) = 0 ) -> -. ( I ` C ) `' < k ) )
56	41, 55	syland 481	. . . . . 6 |- ( C e. ( O \ E ) -> ( ( k e. ( 1 ... ( ( I ` C ) - 1 ) ) /\ ( ( F ` C ) ` k ) = 0 ) -> -. ( I ` C ) `' < k ) )
57	56	imp 429	. . . . 5 |- ( ( C e. ( O \ E ) /\ ( k e. ( 1 ... ( ( I ` C ) - 1 ) ) /\ ( ( F ` C ) ` k ) = 0 ) ) -> -. ( I ` C ) `' < k )
58		ltrel 9645	. . . . . 6 |- Rel <
59	58	relbrcnv 5375	. . . . 5 |- ( ( I ` C ) `' < k <-> k < ( I ` C ) )
60	57, 59	sylnib 304	. . . 4 |- ( ( C e. ( O \ E ) /\ ( k e. ( 1 ... ( ( I ` C ) - 1 ) ) /\ ( ( F ` C ) ` k ) = 0 ) ) -> -. k < ( I ` C ) )
61	60	anassrs 648	. . 3 |- ( ( ( C e. ( O \ E ) /\ k e. ( 1 ... ( ( I ` C ) - 1 ) ) ) /\ ( ( F ` C ) ` k ) = 0 ) -> -. k < ( I ` C ) )
62	20, 61	pm2.65da 576	. 2 |- ( ( C e. ( O \ E ) /\ k e. ( 1 ... ( ( I ` C ) - 1 ) ) ) -> -. ( ( F ` C ) ` k ) = 0 )
63	62	nrexdv 2920	1 |- ( C e. ( O \ E ) -> -. E. k e. ( 1 ... ( ( I ` C ) - 1 ) ) ( ( F ` C ) ` k ) = 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   \ cdif 3473   i^i cin 3475   C_ wss 3476   ~Pcpw 4010   class class class wbr 4447   |-> cmpt 4505   Or wor 4799   `'ccnv 4998   ` cfv 5586  (class class class)co 6282   supcsup 7896   RRcr 9487   0cc0 9488   1c1 9489   + caddc 9491   < clt 9624   <_ cle 9625   - cmin 9801   / cdiv 10202   NNcn 10532   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668   #chash 12369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-hash 12370

This theorem is referenced by:  ballotlemic  28085  ballotlem1c  28086