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Theorem ballotlemrinv0 24792
Description: Lemma for ballotlemrinv 24793. (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 24777 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  =  ( ( S `
 C ) " C ) )
1211adantr 453 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( R `  C
)  =  ( ( S `  C )
" C ) )
13 simpr 449 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  D  =  ( ( S `  C ) " C ) )
1412, 13eqtr4d 2473 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( R `  C
)  =  D )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemrc 24790 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  ( O  \  E
) )
1615adantr 453 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( R `  C
)  e.  ( O 
\  E ) )
1714, 16eqeltrrd 2513 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  D  e.  ( O  \  E ) )
181, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsf1o 24773 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-onto-> ( 1 ... ( M  +  N ) )  /\  `' ( S `  C )  =  ( S `  C ) ) )
1918simprd 451 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  `' ( S `  C )  =  ( S `  C ) )
2019adantr 453 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  `' ( S `  C )  =  ( S `  C ) )
2120eqcomd 2443 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( S `  C
)  =  `' ( S `  C ) )
2221, 13imaeq12d 5206 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( ( S `  C ) " D
)  =  ( `' ( S `  C
) " ( ( S `  C )
" C ) ) )
23 simpl 445 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  C  e.  ( O  \  E ) )
241, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemirc 24791 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
I `  ( R `  C ) )  =  ( I `  C
) )
2524adantr 453 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( I `  ( R `  C )
)  =  ( I `
 C ) )
2614fveq2d 5734 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( I `  ( R `  C )
)  =  ( I `
 D ) )
2725, 26eqtr3d 2472 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( I `  C
)  =  ( I `
 D ) )
281, 2, 3, 4, 5, 6, 7, 8, 9ballotlemieq 24776 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  D  e.  ( O  \  E )  /\  (
I `  C )  =  ( I `  D ) )  -> 
( S `  C
)  =  ( S `
 D ) )
2923, 17, 27, 28syl3anc 1185 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( S `  C
)  =  ( S `
 D ) )
3029imaeq1d 5204 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( ( S `  C ) " D
)  =  ( ( S `  D )
" D ) )
3118simpld 447 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
) )
32 f1of1 5675 . . . . 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 19 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-> ( 1 ... ( M  +  N ) ) )
34 eldifi 3471 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
351, 2, 3ballotlemelo 24747 . . . . . 6  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
3635simplbi 448 . . . . 5  |-  ( C  e.  O  ->  C  C_  ( 1 ... ( M  +  N )
) )
3723, 34, 363syl 19 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  C  C_  ( 1 ... ( M  +  N
) ) )
38 f1imacnv 5693 . . . 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 644 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  -> 
( `' ( S `
 C ) "
( ( S `  C ) " C
) )  =  C )
4022, 30, 393eqtr3rd 2479 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  D  =  ( ( S `  C ) " C ) )  ->  C  =  ( ( S `  D ) " D ) )
4117, 40jca 520 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 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711    \ cdif 3319    i^i cin 3321    C_ wss 3322   ifcif 3741   ~Pcpw 3801   class class class wbr 4214    e. cmpt 4268   `'ccnv 4879   "cima 4883   -1-1->wf1 5453   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083   supcsup 7447   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    < clt 9122    <_ cle 9123    - cmin 9293    / cdiv 9679   NNcn 10002   ZZcz 10284   ...cfz 11045   #chash 11620
This theorem is referenced by:  ballotlemrinv  24793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-hash 11621
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