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Theorem ballotlemgun 26912
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 3603 . . . . . 6  |-  ( ( V  u.  W )  i^i  U )  =  ( ( V  i^i  U )  u.  ( W  i^i  U ) )
21fveq2i 5699 . . . . 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 7541 . . . . . . 7  |-  ( V  e.  Fin  ->  ( V  i^i  U )  e. 
Fin )
53, 4syl 16 . . . . . 6  |-  ( ph  ->  ( V  i^i  U
)  e.  Fin )
6 ballotlemgun.3 . . . . . . 7  |-  ( ph  ->  W  e.  Fin )
7 infi 7541 . . . . . . 7  |-  ( W  e.  Fin  ->  ( W  i^i  U )  e. 
Fin )
86, 7syl 16 . . . . . 6  |-  ( ph  ->  ( W  i^i  U
)  e.  Fin )
9 ballotlemgun.4 . . . . . . . 8  |-  ( ph  ->  ( V  i^i  W
)  =  (/) )
109ineq1d 3556 . . . . . . 7  |-  ( ph  ->  ( ( V  i^i  W )  i^i  U )  =  ( (/)  i^i  U
) )
11 inindir 3573 . . . . . . 7  |-  ( ( V  i^i  W )  i^i  U )  =  ( ( V  i^i  U )  i^i  ( W  i^i  U ) )
12 incom 3548 . . . . . . . 8  |-  ( U  i^i  (/) )  =  (
(/)  i^i  U )
13 in0 3668 . . . . . . . 8  |-  ( U  i^i  (/) )  =  (/)
1412, 13eqtr3i 2465 . . . . . . 7  |-  ( (/)  i^i 
U )  =  (/)
1510, 11, 143eqtr3g 2498 . . . . . 6  |-  ( ph  ->  ( ( V  i^i  U )  i^i  ( W  i^i  U ) )  =  (/) )
16 hashun 12150 . . . . . 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 1218 . . . . 5  |-  ( ph  ->  ( # `  (
( V  i^i  U
)  u.  ( W  i^i  U ) ) )  =  ( (
# `  ( V  i^i  U ) )  +  ( # `  ( W  i^i  U ) ) ) )
182, 17syl5eq 2487 . . . 4  |-  ( ph  ->  ( # `  (
( V  u.  W
)  i^i  U )
)  =  ( (
# `  ( V  i^i  U ) )  +  ( # `  ( W  i^i  U ) ) ) )
19 difundir 3608 . . . . . 6  |-  ( ( V  u.  W ) 
\  U )  =  ( ( V  \  U )  u.  ( W  \  U ) )
2019fveq2i 5699 . . . . 5  |-  ( # `  ( ( V  u.  W )  \  U
) )  =  (
# `  ( ( V  \  U )  u.  ( W  \  U
) ) )
21 diffi 7548 . . . . . . 7  |-  ( V  e.  Fin  ->  ( V  \  U )  e. 
Fin )
223, 21syl 16 . . . . . 6  |-  ( ph  ->  ( V  \  U
)  e.  Fin )
23 diffi 7548 . . . . . . 7  |-  ( W  e.  Fin  ->  ( W  \  U )  e. 
Fin )
246, 23syl 16 . . . . . 6  |-  ( ph  ->  ( W  \  U
)  e.  Fin )
259difeq1d 3478 . . . . . . 7  |-  ( ph  ->  ( ( V  i^i  W )  \  U )  =  ( (/)  \  U
) )
26 difindir 3610 . . . . . . 7  |-  ( ( V  i^i  W ) 
\  U )  =  ( ( V  \  U )  i^i  ( W  \  U ) )
27 0dif 3755 . . . . . . 7  |-  ( (/)  \  U )  =  (/)
2825, 26, 273eqtr3g 2498 . . . . . 6  |-  ( ph  ->  ( ( V  \  U )  i^i  ( W  \  U ) )  =  (/) )
29 hashun 12150 . . . . . 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 1218 . . . . 5  |-  ( ph  ->  ( # `  (
( V  \  U
)  u.  ( W 
\  U ) ) )  =  ( (
# `  ( V  \  U ) )  +  ( # `  ( W  \  U ) ) ) )
3120, 30syl5eq 2487 . . . 4  |-  ( ph  ->  ( # `  (
( V  u.  W
)  \  U )
)  =  ( (
# `  ( V  \  U ) )  +  ( # `  ( W  \  U ) ) ) )
3218, 31oveq12d 6114 . . 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 12131 . . . . . 6  |-  ( ( V  i^i  U )  e.  Fin  ->  ( # `
 ( V  i^i  U ) )  e.  NN0 )
343, 4, 333syl 20 . . . . 5  |-  ( ph  ->  ( # `  ( V  i^i  U ) )  e.  NN0 )
3534nn0cnd 10643 . . . 4  |-  ( ph  ->  ( # `  ( V  i^i  U ) )  e.  CC )
36 hashcl 12131 . . . . . 6  |-  ( ( W  i^i  U )  e.  Fin  ->  ( # `
 ( W  i^i  U ) )  e.  NN0 )
376, 7, 363syl 20 . . . . 5  |-  ( ph  ->  ( # `  ( W  i^i  U ) )  e.  NN0 )
3837nn0cnd 10643 . . . 4  |-  ( ph  ->  ( # `  ( W  i^i  U ) )  e.  CC )
39 hashcl 12131 . . . . . 6  |-  ( ( V  \  U )  e.  Fin  ->  ( # `
 ( V  \  U ) )  e. 
NN0 )
403, 21, 393syl 20 . . . . 5  |-  ( ph  ->  ( # `  ( V  \  U ) )  e.  NN0 )
4140nn0cnd 10643 . . . 4  |-  ( ph  ->  ( # `  ( V  \  U ) )  e.  CC )
42 hashcl 12131 . . . . . 6  |-  ( ( W  \  U )  e.  Fin  ->  ( # `
 ( W  \  U ) )  e. 
NN0 )
436, 23, 423syl 20 . . . . 5  |-  ( ph  ->  ( # `  ( W  \  U ) )  e.  NN0 )
4443nn0cnd 10643 . . . 4  |-  ( ph  ->  ( # `  ( W  \  U ) )  e.  CC )
4535, 38, 41, 44addsub4d 9771 . . 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 2475 . 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 7584 . . . 4  |-  ( ( V  e.  Fin  /\  W  e.  Fin )  ->  ( V  u.  W
)  e.  Fin )
493, 6, 48syl2anc 661 . . 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 26911 . . 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 661 . 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 26911 . . . 4  |-  ( ( U  e.  Fin  /\  V  e.  Fin )  ->  ( U  .^  V
)  =  ( (
# `  ( V  i^i  U ) )  -  ( # `  ( V 
\  U ) ) ) )
6447, 3, 63syl2anc 661 . . 3  |-  ( ph  ->  ( U  .^  V
)  =  ( (
# `  ( V  i^i  U ) )  -  ( # `  ( V 
\  U ) ) ) )
6550, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60ballotlemgval 26911 . . . 4  |-  ( ( U  e.  Fin  /\  W  e.  Fin )  ->  ( U  .^  W
)  =  ( (
# `  ( W  i^i  U ) )  -  ( # `  ( W 
\  U ) ) ) )
6647, 6, 65syl2anc 661 . . 3  |-  ( ph  ->  ( U  .^  W
)  =  ( (
# `  ( W  i^i  U ) )  -  ( # `  ( W 
\  U ) ) ) )
6764, 66oveq12d 6114 . 2  |-  ( ph  ->  ( ( U  .^  V )  +  ( U  .^  W )
)  =  ( ( ( # `  ( V  i^i  U ) )  -  ( # `  ( V  \  U ) ) )  +  ( (
# `  ( W  i^i  U ) )  -  ( # `  ( W 
\  U ) ) ) ) )
6846, 62, 673eqtr4d 2485 1  |-  ( ph  ->  ( U  .^  ( V  u.  W )
)  =  ( ( U  .^  V )  +  ( U  .^  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2720   {crab 2724    \ cdif 3330    u. cun 3331    i^i cin 3332   (/)c0 3642   ifcif 3796   ~Pcpw 3865   class class class wbr 4297    e. cmpt 4355   `'ccnv 4844   "cima 4848   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   Fincfn 7315   supcsup 7695   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290    < clt 9423    <_ cle 9424    - cmin 9600    / cdiv 9998   NNcn 10327   NN0cn0 10584   ZZcz 10651   ...cfz 11442   #chash 12108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-hash 12109
This theorem is referenced by:  ballotlemfrceq  26916
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