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Theorem ballotlemrinv0 28139
Description: Lemma for ballotlemrinv 28140. (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 28124 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  =  ( ( S `
 C ) " C ) )
1211adantr 465 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( R `  C
)  =  ( ( S `  C )
" C ) )
13 simpr 461 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  D  =  ( ( S `  C ) " C ) )
1412, 13eqtr4d 2511 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( R `  C
)  =  D )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemrc 28137 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  ( O  \  E
) )
1615adantr 465 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( R `  C
)  e.  ( O 
\  E ) )
1714, 16eqeltrrd 2556 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  D  e.  ( O  \  E ) )
181, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsf1o 28120 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-onto-> ( 1 ... ( M  +  N ) )  /\  `' ( S `  C )  =  ( S `  C ) ) )
1918simprd 463 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  `' ( S `  C )  =  ( S `  C ) )
2019adantr 465 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  `' ( S `  C )  =  ( S `  C ) )
2120eqcomd 2475 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( S `  C
)  =  `' ( S `  C ) )
2221, 13imaeq12d 5338 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( ( S `  C ) " D
)  =  ( `' ( S `  C
) " ( ( S `  C )
" C ) ) )
23 simpl 457 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  C  e.  ( O  \  E ) )
241, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemirc 28138 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
I `  ( R `  C ) )  =  ( I `  C
) )
2524adantr 465 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( I `  ( R `  C )
)  =  ( I `
 C ) )
2614fveq2d 5870 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( I `  ( R `  C )
)  =  ( I `
 D ) )
2725, 26eqtr3d 2510 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( I `  C
)  =  ( I `
 D ) )
281, 2, 3, 4, 5, 6, 7, 8, 9ballotlemieq 28123 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  D  e.  ( O  \  E )  /\  (
I `  C )  =  ( I `  D ) )  -> 
( S `  C
)  =  ( S `
 D ) )
2923, 17, 27, 28syl3anc 1228 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( S `  C
)  =  ( S `
 D ) )
3029imaeq1d 5336 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( ( S `  C ) " D
)  =  ( ( S `  D )
" D ) )
3118simpld 459 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
) )
32 f1of1 5815 . . . . 5  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  ( S `  C ) : ( 1 ... ( M  +  N ) )
-1-1-> ( 1 ... ( M  +  N )
) )
3323, 31, 323syl 20 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-> ( 1 ... ( M  +  N ) ) )
34 eldifi 3626 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
351, 2, 3ballotlemelo 28094 . . . . . 6  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
3635simplbi 460 . . . . 5  |-  ( C  e.  O  ->  C  C_  ( 1 ... ( M  +  N )
) )
3723, 34, 363syl 20 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  C  C_  ( 1 ... ( M  +  N
) ) )
38 f1imacnv 5832 . . . 4  |-  ( ( ( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-> ( 1 ... ( M  +  N ) )  /\  C  C_  (
1 ... ( M  +  N ) ) )  ->  ( `' ( S `  C )
" ( ( S `
 C ) " C ) )  =  C )
3933, 37, 38syl2anc 661 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( `' ( S `
 C ) "
( ( S `  C ) " C
) )  =  C )
4022, 30, 393eqtr3rd 2517 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  C  =  ( ( S `  D ) " D ) )
4117, 40jca 532 1  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( D  e.  ( O  \  E )  /\  C  =  ( ( S `  D
) " D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818    \ cdif 3473    i^i cin 3475    C_ wss 3476   ifcif 3939   ~Pcpw 4010   class class class wbr 4447    |-> cmpt 4505   `'ccnv 4998   "cima 5002   -1-1->wf1 5585   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284   supcsup 7900   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    < clt 9628    <_ cle 9629    - cmin 9805    / cdiv 10206   NNcn 10536   ZZcz 10864   ...cfz 11672   #chash 12373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-hash 12374
This theorem is referenced by:  ballotlemrinv  28140
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