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Theorem ballotlemgun 23099
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 3430 . . . . . 6  |-  ( ( V  u.  W )  i^i  U )  =  ( ( V  i^i  U )  u.  ( W  i^i  U ) )
21fveq2i 5544 . . . . 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 23045 . . . . . . 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 23045 . . . . . . 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 3382 . . . . . . 7  |-  ( ph  ->  ( ( V  i^i  W )  i^i  U )  =  ( (/)  i^i  U
) )
11 inindir 3400 . . . . . . 7  |-  ( ( V  i^i  W )  i^i  U )  =  ( ( V  i^i  U )  i^i  ( W  i^i  U ) )
12 incom 3374 . . . . . . . 8  |-  ( U  i^i  (/) )  =  (
(/)  i^i  U )
13 in0 3493 . . . . . . . 8  |-  ( U  i^i  (/) )  =  (/)
1412, 13eqtr3i 2318 . . . . . . 7  |-  ( (/)  i^i 
U )  =  (/)
1510, 11, 143eqtr3g 2351 . . . . . 6  |-  ( ph  ->  ( ( V  i^i  U )  i^i  ( W  i^i  U ) )  =  (/) )
16 hashun 11380 . . . . . 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 2340 . . . 4  |-  ( ph  ->  ( # `  (
( V  u.  W
)  i^i  U )
)  =  ( (
# `  ( V  i^i  U ) )  +  ( # `  ( W  i^i  U ) ) ) )
19 difundir 3435 . . . . . 6  |-  ( ( V  u.  W ) 
\  U )  =  ( ( V  \  U )  u.  ( W  \  U ) )
2019fveq2i 5544 . . . . 5  |-  ( # `  ( ( V  u.  W )  \  U
) )  =  (
# `  ( ( V  \  U )  u.  ( W  \  U
) ) )
21 diffi 7105 . . . . . . 7  |-  ( V  e.  Fin  ->  ( V  \  U )  e. 
Fin )
223, 21syl 15 . . . . . 6  |-  ( ph  ->  ( V  \  U
)  e.  Fin )
23 diffi 7105 . . . . . . 7  |-  ( W  e.  Fin  ->  ( W  \  U )  e. 
Fin )
246, 23syl 15 . . . . . 6  |-  ( ph  ->  ( W  \  U
)  e.  Fin )
259difeq1d 3306 . . . . . . 7  |-  ( ph  ->  ( ( V  i^i  W )  \  U )  =  ( (/)  \  U
) )
26 difindir 3437 . . . . . . 7  |-  ( ( V  i^i  W ) 
\  U )  =  ( ( V  \  U )  i^i  ( W  \  U ) )
27 0dif 3538 . . . . . . 7  |-  ( (/)  \  U )  =  (/)
2825, 26, 273eqtr3g 2351 . . . . . 6  |-  ( ph  ->  ( ( V  \  U )  i^i  ( W  \  U ) )  =  (/) )
29 hashun 11380 . . . . . 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 2340 . . . 4  |-  ( ph  ->  ( # `  (
( V  u.  W
)  \  U )
)  =  ( (
# `  ( V  \  U ) )  +  ( # `  ( W  \  U ) ) ) )
3218, 31oveq12d 5892 . . 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 11366 . . . . . 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 10036 . . . 4  |-  ( ph  ->  ( # `  ( V  i^i  U ) )  e.  CC )
36 hashcl 11366 . . . . . 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 10036 . . . 4  |-  ( ph  ->  ( # `  ( W  i^i  U ) )  e.  CC )
39 hashcl 11366 . . . . . 6  |-  ( ( V  \  U )  e.  Fin  ->  ( # `
 ( V  \  U ) )  e. 
NN0 )
403, 21, 393syl 18 . . . . 5  |-  ( ph  ->  ( # `  ( V  \  U ) )  e.  NN0 )
4140nn0cnd 10036 . . . 4  |-  ( ph  ->  ( # `  ( V  \  U ) )  e.  CC )
42 hashcl 11366 . . . . . 6  |-  ( ( W  \  U )  e.  Fin  ->  ( # `
 ( W  \  U ) )  e. 
NN0 )
436, 23, 423syl 18 . . . . 5  |-  ( ph  ->  ( # `  ( W  \  U ) )  e.  NN0 )
4443nn0cnd 10036 . . . 4  |-  ( ph  ->  ( # `  ( W  \  U ) )  e.  CC )
4535, 38, 41, 44addsub4d 9220 . . 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 2328 . 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 7140 . . . 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 23098 . . 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 23098 . . . 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 23098 . . . 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 5892 . 2  |-  ( ph  ->  ( ( U  .^  V )  +  ( U  .^  W )
)  =  ( ( ( # `  ( V  i^i  U ) )  -  ( # `  ( V  \  U ) ) )  +  ( (
# `  ( W  i^i  U ) )  -  ( # `  ( W 
\  U ) ) ) ) )
6846, 62, 673eqtr4d 2338 1  |-  ( ph  ->  ( U  .^  ( V  u.  W )
)  =  ( ( U  .^  V )  +  ( U  .^  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162    u. cun 3163    i^i cin 3164   (/)c0 3468   ifcif 3578   ~Pcpw 3638   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Fincfn 6879   supcsup 7209   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ...cfz 10798   #chash 11353
This theorem is referenced by:  ballotlemfrceq  23103
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-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-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-n0 9982  df-z 10041  df-uz 10247  df-hash 11354
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