Theorem ballotfilemrval 13244

Description: Value of R. (Contributed by Thierry Arnoux, 14-Apr-2017.)

Hypotheses

Ref	Expression
ballotth.m	|- M e. NN
ballotth.n	|- N e. NN
ballotfilem.o	|- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
ballotfilem.p	|- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` 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 ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
ballotth.s	|- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
ballotth.r	|- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )

Assertion

Ref	Expression
ballotfilemrval	|- ( C e. ( O \ E ) -> ( R ` C ) = ( ( S ` C ) " C ) )

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, 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 ballotfilemrval

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotth.r	. . 3 |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
2		fveq2 5693	. . . . 5 |- ( c = d -> ( S ` c ) = ( S ` d ) )
3		id 19	. . . . 5 |- ( c = d -> c = d )
4	2, 3	imaeq12d 5125	. . . 4 |- ( c = d -> ( ( S ` c ) " c ) = ( ( S ` d ) " d ) )
5	4	cbvmptv 4225	. . 3 |- ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) ) = ( d e. ( O \ E ) |-> ( ( S ` d ) " d ) )
6	1, 5	eqtri 2259	. 2 |- R = ( d e. ( O \ E ) |-> ( ( S ` d ) " d ) )
7		fveq2 5693	. . 3 |- ( d = C -> ( S ` d ) = ( S ` C ) )
8		id 19	. . 3 |- ( d = C -> d = C )
9	7, 8	imaeq12d 5125	. 2 |- ( d = C -> ( ( S ` d ) " d ) = ( ( S ` C ) " C ) )
10		id 19	. 2 |- ( C e. ( O \ E ) -> C e. ( O \ E ) )
11		ballotth.m	. . . . 5 |- M e. NN
12		ballotth.n	. . . . 5 |- N e. NN
13		ballotfilem.o	. . . . 5 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
14		ballotfilem.p	. . . . 5 |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
15		ballotth.f	. . . . 5 |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
16		ballotth.e	. . . . 5 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
17		ballotth.mgtn	. . . . 5 |- N < M
18		ballotth.i	. . . . 5 |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
19		ballotth.s	. . . . 5 |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
20	11, 12, 13, 14, 15, 16, 17, 18, 19	ballotfilemsval 13235	. . . 4 |- ( C e. ( O \ E ) -> ( S ` C ) = ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - i ) , i ) ) )
21		1z 9653	. . . . . . 7 |- 1 e. ZZ
22		nnaddcl 9307	. . . . . . . . 9 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
23	11, 12, 22	mp2an 430	. . . . . . . 8 |- ( M + N ) e. NN
24	23	nnzi 9648	. . . . . . 7 |- ( M + N ) e. ZZ
25		fzfig 10850	. . . . . . 7 |- ( ( 1 e. ZZ /\ ( M + N ) e. ZZ ) -> ( 1 ... ( M + N ) ) e. Fin )
26	21, 24, 25	mp2an 430	. . . . . 6 |- ( 1 ... ( M + N ) ) e. Fin
27	26	elexi 2834	. . . . 5 |- ( 1 ... ( M + N ) ) e. _V
28	27	mptex 5937	. . . 4 |- ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - i ) , i ) ) e. _V
29	20, 28	eqeltrdi 2329	. . 3 |- ( C e. ( O \ E ) -> ( S ` C ) e. _V )
30		imaexg 5138	. . 3 |- ( ( S ` C ) e. _V -> ( ( S ` C ) " C ) e. _V )
31	29, 30	syl 14	. 2 |- ( C e. ( O \ E ) -> ( ( S ` C ) " C ) e. _V )
32	6, 9, 10, 31	fvmptd3 5796	1 |- ( C e. ( O \ E ) -> ( R ` C ) = ( ( S ` C ) " C ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   \ cdif 3217   i^i cin 3219   ifcif 3638   ~Pcpw 3688   class class class wbr 4128   |-> cmpt 4190   "cima 4775   ` cfv 5375  (class class class)co 6079   Fincfn 7016  infcinf 7317   RRcr 8172   0cc0 8173   1c1 8174   + caddc 8176   < clt 8354   <_ cle 8355   - cmin 8491   / cdiv 8996   NNcn 9287   ZZcz 9627   ...cfz 10394  ♯chash 11197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-1o 6681  df-er 6801  df-en 7017  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-fz 10395

This theorem is referenced by:  ballotfilemscr  13245  ballotfilemrv  13246  ballotfilemro  13249  ballotfilemrinv0  13259