Theorem ballotlemsup 27021

Description: The set of zeroes of F satisfies the conditions to have a supremum (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
ballotlemsup	|- ( 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 ) ) )

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   y, c, z, k   y, C, z   y, F, z   y, M, z   y, N, z   w, k, y, z, C   w, F   w, M   w, N

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

Proof of Theorem ballotlemsup

Step	Hyp	Ref	Expression
1		fzfi 11895	. . . . . 6 |- ( 1 ... ( M + N ) ) e. Fin
2		ssrab2 3535	. . . . . 6 |- { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ ( 1 ... ( M + N ) )
3		ssfi 7634	. . . . . 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 )
4	1, 2, 3	mp2an 672	. . . . 5 |- { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } e. Fin
5	4	a1i 11	. . . 4 |- ( C e. ( O \ E ) -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } e. Fin )
6		ballotth.m	. . . . . 6 |- M e. NN
7		ballotth.n	. . . . . 6 |- N e. NN
8		ballotth.o	. . . . . 6 |- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
9		ballotth.p	. . . . . 6 |- P = ( x e. ~P O |-> ( ( # ` x ) / ( # ` O ) ) )
10		ballotth.f	. . . . . 6 |- F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
11		ballotth.e	. . . . . 6 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
12		ballotth.mgtn	. . . . . 6 |- N < M
13	6, 7, 8, 9, 10, 11, 12	ballotlem5 27016	. . . . 5 |- ( C e. ( O \ E ) -> E. k e. ( 1 ... ( M + N ) ) ( ( F ` C ) ` k ) = 0 )
14		rabn0 3755	. . . . 5 |- ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } =/= (/) <-> E. k e. ( 1 ... ( M + N ) ) ( ( F ` C ) ` k ) = 0 )
15	13, 14	sylibr 212	. . . 4 |- ( C e. ( O \ E ) -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } =/= (/) )
16		fzssuz 11600	. . . . . . . 8 |- ( 1 ... ( M + N ) ) C_ ( ZZ>= ` 1 )
17		nnuz 10997	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
18	16, 17	sseqtr4i 3487	. . . . . . 7 |- ( 1 ... ( M + N ) ) C_ NN
19		nnssre 10427	. . . . . . 7 |- NN C_ RR
20	18, 19	sstri 3463	. . . . . 6 |- ( 1 ... ( M + N ) ) C_ RR
21	2, 20	sstri 3463	. . . . 5 |- { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ RR
22	21	a1i 11	. . . 4 |- ( C e. ( O \ E ) -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ RR )
23	5, 15, 22	3jca 1168	. . 3 |- ( C e. ( O \ E ) -> ( { 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 ) )
24		ltso 9556	. . . 4 |- < Or RR
25		cnvso 5474	. . . 4 |- ( < Or RR <-> `' < Or RR )
26	24, 25	mpbi 208	. . 3 |- `' < Or RR
27	23, 26	jctil 537	. 2 |- ( C e. ( O \ E ) -> ( `' < 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 ) ) )
28		fisup2g 7817	. 2 |- ( ( `' < 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 ) ) -> E. z e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } ( 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 ) ) )
29	21	sseli 3450	. . . 4 |- ( z e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } -> z e. RR )
30	29	anim1i 568	. . 3 |- ( ( z e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } /\ ( 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 ) ) ) -> ( 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 ) ) ) )
31	30	reximi2 2918	. 2 |- ( E. z e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } ( 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 ) ) )
32	27, 28, 31	3syl 20	1 |- ( 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 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2644   A.wral 2795   E.wrex 2796   {crab 2799   \ cdif 3423   i^i cin 3425   C_ wss 3426   (/)c0 3735   ~Pcpw 3958   class class class wbr 4390   |-> cmpt 4448   Or wor 4738   `'ccnv 4937   ` cfv 5516  (class class class)co 6190   Fincfn 7410   supcsup 7791   RRcr 9382   0cc0 9383   1c1 9384   + caddc 9386   < clt 9519   - cmin 9696   / cdiv 10094   NNcn 10423   ZZcz 10747   ZZ>=cuz 10962   ...cfz 11538   #chash 12204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-hash 12205

This theorem is referenced by:  ballotlemimin  27022  ballotlemfrcn0  27046