Theorem ballotlemsup 28626

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 11985	. . . . . 6 |- ( 1 ... ( M + N ) ) e. Fin
2		ssrab2 3499	. . . . . 6 |- { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ ( 1 ... ( M + N ) )
3		ssfi 7656	. . . . . 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 670	. . . . 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 28621	. . . . 5 |- ( C e. ( O \ E ) -> E. k e. ( 1 ... ( M + N ) ) ( ( F ` C ) ` k ) = 0 )
14		rabn0 3732	. . . . 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 11646	. . . . . . . 8 |- ( 1 ... ( M + N ) ) C_ ( ZZ>= ` 1 )
17		nnuz 11036	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
18	16, 17	sseqtr4i 3450	. . . . . . 7 |- ( 1 ... ( M + N ) ) C_ NN
19		nnssre 10456	. . . . . . 7 |- NN C_ RR
20	18, 19	sstri 3426	. . . . . 6 |- ( 1 ... ( M + N ) ) C_ RR
21	2, 20	sstri 3426	. . . . 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 1174	. . 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		gtso 9577	. . 3 |- `' < Or RR
25	23, 24	jctil 535	. 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 ) ) )
26		fisup2g 7841	. 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 ) ) )
27	21	sseli 3413	. . . 4 |- ( z e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } -> z e. RR )
28	27	anim1i 566	. . 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 ) ) ) )
29	28	reximi2 2849	. 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 ) ) )
30	25, 26, 29	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 367   /\ w3a 971   = wceq 1399   e. wcel 1826   =/= wne 2577   A.wral 2732   E.wrex 2733   {crab 2736   \ cdif 3386   i^i cin 3388   C_ wss 3389   (/)c0 3711   ~Pcpw 3927   class class class wbr 4367   |-> cmpt 4425   Or wor 4713   `'ccnv 4912   ` cfv 5496  (class class class)co 6196   Fincfn 7435   supcsup 7815   RRcr 9402   0cc0 9403   1c1 9404   + caddc 9406   < clt 9539   - cmin 9718   / cdiv 10123   NNcn 10452   ZZcz 10781   ZZ>=cuz 11001   ...cfz 11593   #chash 12307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-hash 12308

This theorem is referenced by:  ballotlemimin  28627  ballotlemfrcn0  28651