Theorem ballotlemsup 28071

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 12040	. . . . . 6 |- ( 1 ... ( M + N ) ) e. Fin
2		ssrab2 3580	. . . . . 6 |- { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ ( 1 ... ( M + N ) )
3		ssfi 7732	. . . . . 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 28066	. . . . 5 |- ( C e. ( O \ E ) -> E. k e. ( 1 ... ( M + N ) ) ( ( F ` C ) ` k ) = 0 )
14		rabn0 3800	. . . . 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 11715	. . . . . . . 8 |- ( 1 ... ( M + N ) ) C_ ( ZZ>= ` 1 )
17		nnuz 11108	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
18	16, 17	sseqtr4i 3532	. . . . . . 7 |- ( 1 ... ( M + N ) ) C_ NN
19		nnssre 10531	. . . . . . 7 |- NN C_ RR
20	18, 19	sstri 3508	. . . . . 6 |- ( 1 ... ( M + N ) ) C_ RR
21	2, 20	sstri 3508	. . . . 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 1171	. . 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 9656	. . . 4 |- < Or RR
25		cnvso 5539	. . . 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 7917	. 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 3495	. . . 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 2926	. 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 968   = wceq 1374   e. wcel 1762   =/= wne 2657   A.wral 2809   E.wrex 2810   {crab 2813   \ cdif 3468   i^i cin 3470   C_ wss 3471   (/)c0 3780   ~Pcpw 4005   class class class wbr 4442   |-> cmpt 4500   Or wor 4794   `'ccnv 4993   ` cfv 5581  (class class class)co 6277   Fincfn 7508   supcsup 7891   RRcr 9482   0cc0 9483   1c1 9484   + caddc 9486   < clt 9619   - cmin 9796   / cdiv 10197   NNcn 10527   ZZcz 10855   ZZ>=cuz 11073   ...cfz 11663   #chash 12362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-hash 12363

This theorem is referenced by:  ballotlemimin  28072  ballotlemfrcn0  28096