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Theorem ballotlemrinv0 23107
Description: Lemma for ballotlemrinv 23108. (Contributed by Thierry Arnoux, 18-Apr-2017.)
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 ,  `'  <  ) )
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
ballotlemrinv0  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( D  e.  ( O  \  E )  /\  C  =  ( ( S `  D
) " D ) ) )
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    D, i, k    S, i, c    R, i, k    x, c    x, C    x, F    x, M    x, N, i, k
Allowed substitution hints:    C( c)    D( x, c)    P( x, i, k, c)    R( x, c)    S( x)    E( x)    I( x)    O( x)

Proof of Theorem ballotlemrinv0
StepHypRef Expression
1 ballotth.m . . . . . 6  |-  M  e.  NN
2 ballotth.n . . . . . 6  |-  N  e.  NN
3 ballotth.o . . . . . 6  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 ballotth.p . . . . . 6  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
5 ballotth.f . . . . . 6  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
6 ballotth.e . . . . . 6  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
7 ballotth.mgtn . . . . . 6  |-  N  < 
M
8 ballotth.i . . . . . 6  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
9 ballotth.s . . . . . 6  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
10 ballotth.r . . . . . 6  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemrval 23092 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  =  ( ( S `
 C ) " C ) )
1211adantr 451 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( R `  C
)  =  ( ( S `  C )
" C ) )
13 simpr 447 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  D  =  ( ( S `  C ) " C ) )
1412, 13eqtr4d 2331 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( R `  C
)  =  D )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemrc 23105 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  ( O  \  E
) )
1615adantr 451 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( R `  C
)  e.  ( O 
\  E ) )
1714, 16eqeltrrd 2371 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  D  e.  ( O  \  E ) )
18 simpl 443 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  C  e.  ( O  \  E ) )
191, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsf1o 23088 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-onto-> ( 1 ... ( M  +  N ) )  /\  `' ( S `  C )  =  ( S `  C ) ) )
2019simprd 449 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  `' ( S `  C )  =  ( S `  C ) )
2118, 20syl 15 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  `' ( S `  C )  =  ( S `  C ) )
2221eqcomd 2301 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( S `  C
)  =  `' ( S `  C ) )
2322, 13imaeq12d 5029 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( ( S `  C ) " D
)  =  ( `' ( S `  C
) " ( ( S `  C )
" C ) ) )
241, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemirc 23106 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
I `  ( R `  C ) )  =  ( I `  C
) )
2524adantr 451 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( I `  ( R `  C )
)  =  ( I `
 C ) )
2614fveq2d 5545 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( I `  ( R `  C )
)  =  ( I `
 D ) )
2725, 26eqtr3d 2330 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( I `  C
)  =  ( I `
 D ) )
281, 2, 3, 4, 5, 6, 7, 8, 9ballotlemieq 23091 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  D  e.  ( O  \  E )  /\  (
I `  C )  =  ( I `  D ) )  -> 
( S `  C
)  =  ( S `
 D ) )
2918, 17, 27, 28syl3anc 1182 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( S `  C
)  =  ( S `
 D ) )
3029imaeq1d 5027 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( ( S `  C ) " D
)  =  ( ( S `  D )
" D ) )
3119simpld 445 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
) )
3218, 31syl 15 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-onto-> ( 1 ... ( M  +  N ) ) )
33 f1of1 5487 . . . . 5  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  ( S `  C ) : ( 1 ... ( M  +  N ) )
-1-1-> ( 1 ... ( M  +  N )
) )
3432, 33syl 15 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-> ( 1 ... ( M  +  N ) ) )
35 eldifi 3311 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
361, 2, 3ballotlemelo 23062 . . . . . 6  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
3736simplbi 446 . . . . 5  |-  ( C  e.  O  ->  C  C_  ( 1 ... ( M  +  N )
) )
3818, 35, 373syl 18 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  C  C_  ( 1 ... ( M  +  N
) ) )
39 f1imacnv 5505 . . . 4  |-  ( ( ( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-> ( 1 ... ( M  +  N ) )  /\  C  C_  (
1 ... ( M  +  N ) ) )  ->  ( `' ( S `  C )
" ( ( S `
 C ) " C ) )  =  C )
4034, 38, 39syl2anc 642 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( `' ( S `
 C ) "
( ( S `  C ) " C
) )  =  C )
4123, 30, 403eqtr3rd 2337 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  C  =  ( ( S `  D ) " D ) )
4217, 41jca 518 1  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( D  e.  ( O  \  E )  /\  C  =  ( ( S `  D
) " D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162    i^i cin 3164    C_ wss 3165   ifcif 3578   ~Pcpw 3638   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   "cima 4708   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   ZZcz 10040   ...cfz 10798   #chash 11353
This theorem is referenced by:  ballotlemrinv  23108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-hash 11354
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