Theorem ballotlemscrOLD 29389

Description: The image of ( R ` C ) by ( S ` C ). (Contributed by Thierry Arnoux, 21-Apr-2017.) Obsolete version of ballotlemscr 29351 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
ballotlemscrOLD	|- ( C e. ( O \ E ) -> ( ( S ` C ) " ( R ` C ) ) = C )

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 ballotlemscrOLD

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 29388	. . 3 |- ( C e. ( O \ E ) -> ( R ` C ) = ( ( S ` C ) " C ) )
12	11	imaeq2d 5168	. 2 |- ( C e. ( O \ E ) -> ( ( S ` C ) " ( R ` C ) ) = ( ( S ` C ) " ( ( S ` C ) " C ) ) )
13	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsf1oOLD 29384	. . . 4 |- ( C e. ( O \ E ) -> ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) /\ `' ( S ` C ) = ( S ` C ) ) )
14	13	simprd 465	. . 3 |- ( C e. ( O \ E ) -> `' ( S ` C ) = ( S ` C ) )
15	14	imaeq1d 5167	. 2 |- ( C e. ( O \ E ) -> ( `' ( S ` C ) " ( ( S ` C ) " C ) ) = ( ( S ` C ) " ( ( S ` C ) " C ) ) )
16	13	simpld 461	. . . 4 |- ( C e. ( O \ E ) -> ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) )
17		f1of1 5813	. . . 4 |- ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) -> ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-> ( 1 ... ( M + N ) ) )
18	16, 17	syl 17	. . 3 |- ( C e. ( O \ E ) -> ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-> ( 1 ... ( M + N ) ) )
19		eldifi 3555	. . . 4 |- ( C e. ( O \ E ) -> C e. O )
20	1, 2, 3	ballotlemelo 29320	. . . . 5 |- ( C e. O <-> ( C C_ ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) )
21	20	simplbi 462	. . . 4 |- ( C e. O -> C C_ ( 1 ... ( M + N ) ) )
22	19, 21	syl 17	. . 3 |- ( C e. ( O \ E ) -> C C_ ( 1 ... ( M + N ) ) )
23		f1imacnv 5830	. . 3 |- ( ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-> ( 1 ... ( M + N ) ) /\ C C_ ( 1 ... ( M + N ) ) ) -> ( `' ( S ` C ) " ( ( S ` C ) " C ) ) = C )
24	18, 22, 23	syl2anc 667	. 2 |- ( C e. ( O \ E ) -> ( `' ( S ` C ) " ( ( S ` C ) " C ) ) = C )
25	12, 15, 24	3eqtr2d 2491	1 |- ( C e. ( O \ E ) -> ( ( S ` C ) " ( R ` C ) ) = C )

Syntax hints:   -> wi 4   = wceq 1444   e. wcel 1887   A.wral 2737   {crab 2741   \ cdif 3401   i^i cin 3403   C_ wss 3404   ifcif 3881   ~Pcpw 3951   class class class wbr 4402   |-> cmpt 4461   `'ccnv 4833   "cima 4837   -1-1->wf1 5579   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290   supcsup 7954   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   < clt 9675   <_ cle 9676   - cmin 9860   / cdiv 10269   NNcn 10609   ZZcz 10937   ...cfz 11784   #chash 12515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-hash 12516

This theorem is referenced by:  ballotlemfrcOLD  29397