Theorem ballotlemroOLD 29405

Description: Range of R is included in O. (Contributed by Thierry Arnoux, 17-Apr-2017.) Obsolete version of ballotlemro 29367 as of 6-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
ballotthOLD.m	|- M e. NN
ballotthOLD.n	|- N e. NN
ballotthOLD.o	|- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
ballotthOLD.p	|- P = ( x e. ~P O |-> ( ( # ` x ) / ( # ` O ) ) )
ballotthOLD.f	|- F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
ballotthOLD.e	|- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
ballotthOLD.mgtn	|- N < M
ballotthOLD.i	|- I = ( c e. ( O \ E ) |-> sup ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , `' < ) )
ballotthOLD.s	|- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
ballotthOLD.r	|- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )

Assertion

Ref	Expression
ballotlemroOLD	|- ( C e. ( O \ E ) -> ( R ` C ) e. O )

Distinct variable groups:   M, c   N, c   O, c   i, M   i, N   i, O   k, M   k, N   k, O   C, i, k   i, c, F, k   i, E, k   k, I, c   E, c   i, I, c   S, k, i, c

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

Proof of Theorem ballotlemroOLD

Step	Hyp	Ref	Expression
1		ballotthOLD.m	. . . 4 |- M e. NN
2		ballotthOLD.n	. . . 4 |- N e. NN
3		ballotthOLD.o	. . . 4 |- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
4		ballotthOLD.p	. . . 4 |- P = ( x e. ~P O |-> ( ( # ` x ) / ( # ` O ) ) )
5		ballotthOLD.f	. . . 4 |- F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
6		ballotthOLD.e	. . . 4 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
7		ballotthOLD.mgtn	. . . 4 |- N < M
8		ballotthOLD.i	. . . 4 |- I = ( c e. ( O \ E ) |-> sup ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , `' < ) )
9		ballotthOLD.s	. . . 4 |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
10		ballotthOLD.r	. . . 4 |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotlemrvalOLD 29400	. . 3 |- ( C e. ( O \ E ) -> ( R ` C ) = ( ( S ` C ) " C ) )
12		imassrn 5182	. . . 4 |- ( ( S ` C ) " C ) C_ ran ( S ` C )
13	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsf1oOLD 29396	. . . . . 6 |- ( C e. ( O \ E ) -> ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) /\ `' ( S ` C ) = ( S ` C ) ) )
14	13	simpld 461	. . . . 5 |- ( C e. ( O \ E ) -> ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) )
15		f1ofo 5826	. . . . 5 |- ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) -> ( S ` C ) : ( 1 ... ( M + N ) ) -onto-> ( 1 ... ( M + N ) ) )
16		forn 5801	. . . . 5 |- ( ( S ` C ) : ( 1 ... ( M + N ) ) -onto-> ( 1 ... ( M + N ) ) -> ran ( S ` C ) = ( 1 ... ( M + N ) ) )
17	14, 15, 16	3syl 18	. . . 4 |- ( C e. ( O \ E ) -> ran ( S ` C ) = ( 1 ... ( M + N ) ) )
18	12, 17	syl5sseq 3482	. . 3 |- ( C e. ( O \ E ) -> ( ( S ` C ) " C ) C_ ( 1 ... ( M + N ) ) )
19	11, 18	eqsstrd 3468	. 2 |- ( C e. ( O \ E ) -> ( R ` C ) C_ ( 1 ... ( M + N ) ) )
20		f1of1 5818	. . . . . . 7 |- ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) -> ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-> ( 1 ... ( M + N ) ) )
21	14, 20	syl 17	. . . . . 6 |- ( C e. ( O \ E ) -> ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-> ( 1 ... ( M + N ) ) )
22		eldifi 3557	. . . . . . . 8 |- ( C e. ( O \ E ) -> C e. O )
23	1, 2, 3	ballotlemelo 29332	. . . . . . . 8 |- ( C e. O <-> ( C C_ ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) )
24	22, 23	sylib 200	. . . . . . 7 |- ( C e. ( O \ E ) -> ( C C_ ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) )
25	24	simpld 461	. . . . . 6 |- ( C e. ( O \ E ) -> C C_ ( 1 ... ( M + N ) ) )
26		id 22	. . . . . 6 |- ( C e. ( O \ E ) -> C e. ( O \ E ) )
27		f1imaeng 7634	. . . . . 6 |- ( ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-> ( 1 ... ( M + N ) ) /\ C C_ ( 1 ... ( M + N ) ) /\ C e. ( O \ E ) ) -> ( ( S ` C ) " C ) ~~ C )
28	21, 25, 26, 27	syl3anc 1269	. . . . 5 |- ( C e. ( O \ E ) -> ( ( S ` C ) " C ) ~~ C )
29	11, 28	eqbrtrd 4426	. . . 4 |- ( C e. ( O \ E ) -> ( R ` C ) ~~ C )
30		hasheni 12538	. . . 4 |- ( ( R ` C ) ~~ C -> ( # ` ( R ` C ) ) = ( # ` C ) )
31	29, 30	syl 17	. . 3 |- ( C e. ( O \ E ) -> ( # ` ( R ` C ) ) = ( # ` C ) )
32	24	simprd 465	. . 3 |- ( C e. ( O \ E ) -> ( # ` C ) = M )
33	31, 32	eqtrd 2487	. 2 |- ( C e. ( O \ E ) -> ( # ` ( R ` C ) ) = M )
34	1, 2, 3	ballotlemelo 29332	. 2 |- ( ( R ` C ) e. O <-> ( ( R ` C ) C_ ( 1 ... ( M + N ) ) /\ ( # ` ( R ` C ) ) = M ) )
35	19, 33, 34	sylanbrc 671	1 |- ( C e. ( O \ E ) -> ( R ` C ) e. O )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1446   e. wcel 1889   A.wral 2739   {crab 2743   \ cdif 3403   i^i cin 3405   C_ wss 3406   ifcif 3883   ~Pcpw 3953   class class class wbr 4405   |-> cmpt 4464   `'ccnv 4836   ran crn 4838   "cima 4840   -1-1->wf1 5582   -onto->wfo 5583   -1-1-onto->wf1o 5584   ` cfv 5585  (class class class)co 6295   ~~ cen 7571   supcsup 7959   RRcr 9543   0cc0 9544   1c1 9545   + caddc 9547   < clt 9680   <_ cle 9681   - cmin 9865   / cdiv 10276   NNcn 10616   ZZcz 10944   ...cfz 11791   #chash 12522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-fz 11792  df-hash 12523

This theorem is referenced by:  ballotlemfrcOLD  29409  ballotlemfrceqOLD  29411  ballotlemfrcn0OLD  29412  ballotlemrcOLD  29413