Theorem ballotfilemirc 13258

Description: Applying R does not change first ties. (Contributed by Thierry Arnoux, 19-Apr-2017.) (Revised by AV, 6-Oct-2020.)

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
ballotfilemirc	|- ( C e. ( O \ E ) -> ( I ` ( R ` C ) ) = ( I ` 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   R, i, k   x, k, C   x, F   x, M   x, N

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

Proof of Theorem ballotfilemirc

Dummy variables y v u p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotth.m	. . . 4 |- M e. NN
2		ballotth.n	. . . 4 |- N e. NN
3		ballotfilem.o	. . . 4 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
4		ballotfilem.p	. . . 4 |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
5		ballotth.f	. . . 4 |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
6		ballotth.e	. . . 4 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
7		ballotth.mgtn	. . . 4 |- N < M
8		ballotth.i	. . . 4 |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
9		ballotth.s	. . . 4 |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
10		ballotth.r	. . . 4 |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotfilemrc 13257	. . 3 |- ( C e. ( O \ E ) -> ( R ` C ) e. ( O \ E ) )
12	1, 2, 3, 4, 5, 6, 7, 8	ballotfilemi 13226	. . 3 |- ( ( R ` C ) e. ( O \ E ) -> ( I ` ( R ` C ) ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` ( R ` C ) ) ` k ) = 0 } , RR , < ) )
13	11, 12	syl 14	. 2 |- ( C e. ( O \ E ) -> ( I ` ( R ` C ) ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` ( R ` C ) ) ` k ) = 0 } , RR , < ) )
14		lttri3 8399	. . . 4 |- ( ( p e. RR /\ q e. RR ) -> ( p = q <-> ( -. p < q /\ -. q < p ) ) )
15	14	adantl 277	. . 3 |- ( ( C e. ( O \ E ) /\ ( p e. RR /\ q e. RR ) ) -> ( p = q <-> ( -. p < q /\ -. q < p ) ) )
16	1, 2, 3, 4, 5, 6, 7, 8	ballotfilemiex 13227	. . . . . 6 |- ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
17	16	simpld 112	. . . . 5 |- ( C e. ( O \ E ) -> ( I ` C ) e. ( 1 ... ( M + N ) ) )
18	17	elfzelzd 10412	. . . 4 |- ( C e. ( O \ E ) -> ( I ` C ) e. ZZ )
19	18	zred 9751	. . 3 |- ( C e. ( O \ E ) -> ( I ` C ) e. RR )
20		fveqeq2 5702	. . . 4 |- ( k = ( I ` C ) -> ( ( ( F ` ( R ` C ) ) ` k ) = 0 <-> ( ( F ` ( R ` C ) ) ` ( I ` C ) ) = 0 ) )
21		eqid 2238	. . . . 5 |- ( u e. O , v e. Fin |-> ( (♯ ` ( v i^i u ) ) - (♯ ` ( v \ u ) ) ) ) = ( u e. O , v e. Fin |-> ( (♯ ` ( v i^i u ) ) - (♯ ` ( v \ u ) ) ) )
22	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 21	ballotfilemfrci 13254	. . . 4 |- ( C e. ( O \ E ) -> ( ( F ` ( R ` C ) ) ` ( I ` C ) ) = 0 )
23	20, 17, 22	elrabd 2984	. . 3 |- ( C e. ( O \ E ) -> ( I ` C ) e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` ( R ` C ) ) ` k ) = 0 } )
24		elrabi 2979	. . . . 5 |- ( y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` ( R ` C ) ) ` k ) = 0 } -> y e. ( 1 ... ( M + N ) ) )
25	24	anim2i 342	. . . 4 |- ( ( C e. ( O \ E ) /\ y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` ( R ` C ) ) ` k ) = 0 } ) -> ( C e. ( O \ E ) /\ y e. ( 1 ... ( M + N ) ) ) )
26	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotfilemfrcn0 13256	. . . . . . . . 9 |- ( ( C e. ( O \ E ) /\ y e. ( 1 ... ( M + N ) ) /\ y < ( I ` C ) ) -> ( ( F ` ( R ` C ) ) ` y ) =/= 0 )
27	26	neneqd 2441	. . . . . . . 8 |- ( ( C e. ( O \ E ) /\ y e. ( 1 ... ( M + N ) ) /\ y < ( I ` C ) ) -> -. ( ( F ` ( R ` C ) ) ` y ) = 0 )
28		fveqeq2 5702	. . . . . . . . . 10 |- ( k = y -> ( ( ( F ` ( R ` C ) ) ` k ) = 0 <-> ( ( F ` ( R ` C ) ) ` y ) = 0 ) )
29	28	elrab 2982	. . . . . . . . 9 |- ( y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` ( R ` C ) ) ` k ) = 0 } <-> ( y e. ( 1 ... ( M + N ) ) /\ ( ( F ` ( R ` C ) ) ` y ) = 0 ) )
30	29	simprbi 275	. . . . . . . 8 |- ( y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` ( R ` C ) ) ` k ) = 0 } -> ( ( F ` ( R ` C ) ) ` y ) = 0 )
31	27, 30	nsyl 637	. . . . . . 7 |- ( ( C e. ( O \ E ) /\ y e. ( 1 ... ( M + N ) ) /\ y < ( I ` C ) ) -> -. y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` ( R ` C ) ) ` k ) = 0 } )
32	31	3expia 1236	. . . . . 6 |- ( ( C e. ( O \ E ) /\ y e. ( 1 ... ( M + N ) ) ) -> ( y < ( I ` C ) -> -. y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` ( R ` C ) ) ` k ) = 0 } ) )
33	32	con2d 633	. . . . 5 |- ( ( C e. ( O \ E ) /\ y e. ( 1 ... ( M + N ) ) ) -> ( y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` ( R ` C ) ) ` k ) = 0 } -> -. y < ( I ` C ) ) )
34	33	imp 124	. . . 4 |- ( ( ( C e. ( O \ E ) /\ y e. ( 1 ... ( M + N ) ) ) /\ y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` ( R ` C ) ) ` k ) = 0 } ) -> -. y < ( I ` C ) )
35	25, 34	sylancom 424	. . 3 |- ( ( C e. ( O \ E ) /\ y e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` ( R ` C ) ) ` k ) = 0 } ) -> -. y < ( I ` C ) )
36	15, 19, 23, 35	infminti 7361	. 2 |- ( C e. ( O \ E ) -> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` ( R ` C ) ) ` k ) = 0 } , RR , < ) = ( I ` C ) )
37	13, 36	eqtrd 2271	1 |- ( C e. ( O \ E ) -> ( I ` ( R ` C ) ) = ( I ` C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   \ 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   e. cmpo 6081   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-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

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-rmo 2536  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-po 4439  df-iso 4440  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-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-oadd 6685  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-ihash 11198

This theorem is referenced by:  ballotfilemrinv0  13259