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Theorem ballotlemgun 23083
Description: A property of the defined  .^ operator (Contributed by Thierry Arnoux, 26-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
) )
ballotlemg  |-  .^  =  ( u  e.  Fin ,  v  e.  Fin  |->  ( ( # `  (
v  i^i  u )
)  -  ( # `  ( v  \  u
) ) ) )
ballotlemgun.1  |-  ( ph  ->  U  e.  Fin )
ballotlemgun.2  |-  ( ph  ->  V  e.  Fin )
ballotlemgun.3  |-  ( ph  ->  W  e.  Fin )
ballotlemgun.4  |-  ( ph  ->  ( V  i^i  W
)  =  (/) )
Assertion
Ref Expression
ballotlemgun  |-  ( ph  ->  ( U  .^  ( V  u.  W )
)  =  ( ( U  .^  V )  +  ( U  .^  W ) ) )
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   
i, E, k    k, I, c    E, c    i, I, c    S, k, i, c    R, i    v, u, I    u, R, v   
u, S, v    u, U, v    u, V, v   
u, W, v
Allowed substitution hints:    ph( x, v, u, i, k, c)    P( x, v, u, i, k, c)    R( x, k, c)    S( x)    U( x, i, k, c)    E( x, v, u)    .^ ( x, v, u, i, k, c)    F( x, v, u)    I( x)    M( x, v, u)    N( x, v, u)    O( x, v, u)    V( x, i, k, c)    W( x, i, k, c)

Proof of Theorem ballotlemgun
StepHypRef Expression
1 indir 3417 . . . . . 6  |-  ( ( V  u.  W )  i^i  U )  =  ( ( V  i^i  U )  u.  ( W  i^i  U ) )
21fveq2i 5528 . . . . 5  |-  ( # `  ( ( V  u.  W )  i^i  U
) )  =  (
# `  ( ( V  i^i  U )  u.  ( W  i^i  U
) ) )
3 ballotlemgun.2 . . . . . . 7  |-  ( ph  ->  V  e.  Fin )
4 infi 23029 . . . . . . 7  |-  ( V  e.  Fin  ->  ( V  i^i  U )  e. 
Fin )
53, 4syl 15 . . . . . 6  |-  ( ph  ->  ( V  i^i  U
)  e.  Fin )
6 ballotlemgun.3 . . . . . . 7  |-  ( ph  ->  W  e.  Fin )
7 infi 23029 . . . . . . 7  |-  ( W  e.  Fin  ->  ( W  i^i  U )  e. 
Fin )
86, 7syl 15 . . . . . 6  |-  ( ph  ->  ( W  i^i  U
)  e.  Fin )
9 ballotlemgun.4 . . . . . . . 8  |-  ( ph  ->  ( V  i^i  W
)  =  (/) )
109ineq1d 3369 . . . . . . 7  |-  ( ph  ->  ( ( V  i^i  W )  i^i  U )  =  ( (/)  i^i  U
) )
11 inindir 3387 . . . . . . 7  |-  ( ( V  i^i  W )  i^i  U )  =  ( ( V  i^i  U )  i^i  ( W  i^i  U ) )
12 incom 3361 . . . . . . . 8  |-  ( U  i^i  (/) )  =  (
(/)  i^i  U )
13 in0 3480 . . . . . . . 8  |-  ( U  i^i  (/) )  =  (/)
1412, 13eqtr3i 2305 . . . . . . 7  |-  ( (/)  i^i 
U )  =  (/)
1510, 11, 143eqtr3g 2338 . . . . . 6  |-  ( ph  ->  ( ( V  i^i  U )  i^i  ( W  i^i  U ) )  =  (/) )
16 hashun 11364 . . . . . 6  |-  ( ( ( V  i^i  U
)  e.  Fin  /\  ( W  i^i  U )  e.  Fin  /\  (
( V  i^i  U
)  i^i  ( W  i^i  U ) )  =  (/) )  ->  ( # `  ( ( V  i^i  U )  u.  ( W  i^i  U ) ) )  =  ( (
# `  ( V  i^i  U ) )  +  ( # `  ( W  i^i  U ) ) ) )
175, 8, 15, 16syl3anc 1182 . . . . 5  |-  ( ph  ->  ( # `  (
( V  i^i  U
)  u.  ( W  i^i  U ) ) )  =  ( (
# `  ( V  i^i  U ) )  +  ( # `  ( W  i^i  U ) ) ) )
182, 17syl5eq 2327 . . . 4  |-  ( ph  ->  ( # `  (
( V  u.  W
)  i^i  U )
)  =  ( (
# `  ( V  i^i  U ) )  +  ( # `  ( W  i^i  U ) ) ) )
19 difundir 3422 . . . . . 6  |-  ( ( V  u.  W ) 
\  U )  =  ( ( V  \  U )  u.  ( W  \  U ) )
2019fveq2i 5528 . . . . 5  |-  ( # `  ( ( V  u.  W )  \  U
) )  =  (
# `  ( ( V  \  U )  u.  ( W  \  U
) ) )
21 diffi 7089 . . . . . . 7  |-  ( V  e.  Fin  ->  ( V  \  U )  e. 
Fin )
223, 21syl 15 . . . . . 6  |-  ( ph  ->  ( V  \  U
)  e.  Fin )
23 diffi 7089 . . . . . . 7  |-  ( W  e.  Fin  ->  ( W  \  U )  e. 
Fin )
246, 23syl 15 . . . . . 6  |-  ( ph  ->  ( W  \  U
)  e.  Fin )
259difeq1d 3293 . . . . . . 7  |-  ( ph  ->  ( ( V  i^i  W )  \  U )  =  ( (/)  \  U
) )
26 difindir 3424 . . . . . . 7  |-  ( ( V  i^i  W ) 
\  U )  =  ( ( V  \  U )  i^i  ( W  \  U ) )
27 0dif 3525 . . . . . . 7  |-  ( (/)  \  U )  =  (/)
2825, 26, 273eqtr3g 2338 . . . . . 6  |-  ( ph  ->  ( ( V  \  U )  i^i  ( W  \  U ) )  =  (/) )
29 hashun 11364 . . . . . 6  |-  ( ( ( V  \  U
)  e.  Fin  /\  ( W  \  U )  e.  Fin  /\  (
( V  \  U
)  i^i  ( W  \  U ) )  =  (/) )  ->  ( # `  ( ( V  \  U )  u.  ( W  \  U ) ) )  =  ( (
# `  ( V  \  U ) )  +  ( # `  ( W  \  U ) ) ) )
3022, 24, 28, 29syl3anc 1182 . . . . 5  |-  ( ph  ->  ( # `  (
( V  \  U
)  u.  ( W 
\  U ) ) )  =  ( (
# `  ( V  \  U ) )  +  ( # `  ( W  \  U ) ) ) )
3120, 30syl5eq 2327 . . . 4  |-  ( ph  ->  ( # `  (
( V  u.  W
)  \  U )
)  =  ( (
# `  ( V  \  U ) )  +  ( # `  ( W  \  U ) ) ) )
3218, 31oveq12d 5876 . . 3  |-  ( ph  ->  ( ( # `  (
( V  u.  W
)  i^i  U )
)  -  ( # `  ( ( V  u.  W )  \  U
) ) )  =  ( ( ( # `  ( V  i^i  U
) )  +  (
# `  ( W  i^i  U ) ) )  -  ( ( # `  ( V  \  U
) )  +  (
# `  ( W  \  U ) ) ) ) )
33 hashcl 11350 . . . . . 6  |-  ( ( V  i^i  U )  e.  Fin  ->  ( # `
 ( V  i^i  U ) )  e.  NN0 )
343, 4, 333syl 18 . . . . 5  |-  ( ph  ->  ( # `  ( V  i^i  U ) )  e.  NN0 )
3534nn0cnd 10020 . . . 4  |-  ( ph  ->  ( # `  ( V  i^i  U ) )  e.  CC )
36 hashcl 11350 . . . . . 6  |-  ( ( W  i^i  U )  e.  Fin  ->  ( # `
 ( W  i^i  U ) )  e.  NN0 )
376, 7, 363syl 18 . . . . 5  |-  ( ph  ->  ( # `  ( W  i^i  U ) )  e.  NN0 )
3837nn0cnd 10020 . . . 4  |-  ( ph  ->  ( # `  ( W  i^i  U ) )  e.  CC )
39 hashcl 11350 . . . . . 6  |-  ( ( V  \  U )  e.  Fin  ->  ( # `
 ( V  \  U ) )  e. 
NN0 )
403, 21, 393syl 18 . . . . 5  |-  ( ph  ->  ( # `  ( V  \  U ) )  e.  NN0 )
4140nn0cnd 10020 . . . 4  |-  ( ph  ->  ( # `  ( V  \  U ) )  e.  CC )
42 hashcl 11350 . . . . . 6  |-  ( ( W  \  U )  e.  Fin  ->  ( # `
 ( W  \  U ) )  e. 
NN0 )
436, 23, 423syl 18 . . . . 5  |-  ( ph  ->  ( # `  ( W  \  U ) )  e.  NN0 )
4443nn0cnd 10020 . . . 4  |-  ( ph  ->  ( # `  ( W  \  U ) )  e.  CC )
4535, 38, 41, 44addsub4d 9204 . . 3  |-  ( ph  ->  ( ( ( # `  ( V  i^i  U
) )  +  (
# `  ( W  i^i  U ) ) )  -  ( ( # `  ( V  \  U
) )  +  (
# `  ( W  \  U ) ) ) )  =  ( ( ( # `  ( V  i^i  U ) )  -  ( # `  ( V  \  U ) ) )  +  ( (
# `  ( W  i^i  U ) )  -  ( # `  ( W 
\  U ) ) ) ) )
4632, 45eqtrd 2315 . 2  |-  ( ph  ->  ( ( # `  (
( V  u.  W
)  i^i  U )
)  -  ( # `  ( ( V  u.  W )  \  U
) ) )  =  ( ( ( # `  ( V  i^i  U
) )  -  ( # `
 ( V  \  U ) ) )  +  ( ( # `  ( W  i^i  U
) )  -  ( # `
 ( W  \  U ) ) ) ) )
47 ballotlemgun.1 . . 3  |-  ( ph  ->  U  e.  Fin )
48 unfi 7124 . . . 4  |-  ( ( V  e.  Fin  /\  W  e.  Fin )  ->  ( V  u.  W
)  e.  Fin )
493, 6, 48syl2anc 642 . . 3  |-  ( ph  ->  ( V  u.  W
)  e.  Fin )
50 ballotth.m . . . 4  |-  M  e.  NN
51 ballotth.n . . . 4  |-  N  e.  NN
52 ballotth.o . . . 4  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
53 ballotth.p . . . 4  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
54 ballotth.f . . . 4  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
55 ballotth.e . . . 4  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
56 ballotth.mgtn . . . 4  |-  N  < 
M
57 ballotth.i . . . 4  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
58 ballotth.s . . . 4  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
59 ballotth.r . . . 4  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
60 ballotlemg . . . 4  |-  .^  =  ( u  e.  Fin ,  v  e.  Fin  |->  ( ( # `  (
v  i^i  u )
)  -  ( # `  ( v  \  u
) ) ) )
6150, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60ballotlemgval 23082 . . 3  |-  ( ( U  e.  Fin  /\  ( V  u.  W
)  e.  Fin )  ->  ( U  .^  ( V  u.  W )
)  =  ( (
# `  ( ( V  u.  W )  i^i  U ) )  -  ( # `  ( ( V  u.  W ) 
\  U ) ) ) )
6247, 49, 61syl2anc 642 . 2  |-  ( ph  ->  ( U  .^  ( V  u.  W )
)  =  ( (
# `  ( ( V  u.  W )  i^i  U ) )  -  ( # `  ( ( V  u.  W ) 
\  U ) ) ) )
6350, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60ballotlemgval 23082 . . . 4  |-  ( ( U  e.  Fin  /\  V  e.  Fin )  ->  ( U  .^  V
)  =  ( (
# `  ( V  i^i  U ) )  -  ( # `  ( V 
\  U ) ) ) )
6447, 3, 63syl2anc 642 . . 3  |-  ( ph  ->  ( U  .^  V
)  =  ( (
# `  ( V  i^i  U ) )  -  ( # `  ( V 
\  U ) ) ) )
6550, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60ballotlemgval 23082 . . . 4  |-  ( ( U  e.  Fin  /\  W  e.  Fin )  ->  ( U  .^  W
)  =  ( (
# `  ( W  i^i  U ) )  -  ( # `  ( W 
\  U ) ) ) )
6647, 6, 65syl2anc 642 . . 3  |-  ( ph  ->  ( U  .^  W
)  =  ( (
# `  ( W  i^i  U ) )  -  ( # `  ( W 
\  U ) ) ) )
6764, 66oveq12d 5876 . 2  |-  ( ph  ->  ( ( U  .^  V )  +  ( U  .^  W )
)  =  ( ( ( # `  ( V  i^i  U ) )  -  ( # `  ( V  \  U ) ) )  +  ( (
# `  ( W  i^i  U ) )  -  ( # `  ( W 
\  U ) ) ) ) )
6846, 62, 673eqtr4d 2325 1  |-  ( ph  ->  ( U  .^  ( V  u.  W )
)  =  ( ( U  .^  V )  +  ( U  .^  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    \ cdif 3149    u. cun 3150    i^i cin 3151   (/)c0 3455   ifcif 3565   ~Pcpw 3625   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Fincfn 6863   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   ...cfz 10782   #chash 11337
This theorem is referenced by:  ballotlemfrceq  23087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-hash 11338
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