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Theorem ballotlemgun 28441
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 3731 . . . . . 6  |-  ( ( V  u.  W )  i^i  U )  =  ( ( V  i^i  U )  u.  ( W  i^i  U ) )
21fveq2i 5859 . . . . 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 7745 . . . . . . 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 7745 . . . . . . 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 3684 . . . . . . 7  |-  ( ph  ->  ( ( V  i^i  W )  i^i  U )  =  ( (/)  i^i  U
) )
11 inindir 3701 . . . . . . 7  |-  ( ( V  i^i  W )  i^i  U )  =  ( ( V  i^i  U )  i^i  ( W  i^i  U ) )
12 incom 3676 . . . . . . . 8  |-  ( U  i^i  (/) )  =  (
(/)  i^i  U )
13 in0 3797 . . . . . . . 8  |-  ( U  i^i  (/) )  =  (/)
1412, 13eqtr3i 2474 . . . . . . 7  |-  ( (/)  i^i 
U )  =  (/)
1510, 11, 143eqtr3g 2507 . . . . . 6  |-  ( ph  ->  ( ( V  i^i  U )  i^i  ( W  i^i  U ) )  =  (/) )
16 hashun 12432 . . . . . 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 1229 . . . . 5  |-  ( ph  ->  ( # `  (
( V  i^i  U
)  u.  ( W  i^i  U ) ) )  =  ( (
# `  ( V  i^i  U ) )  +  ( # `  ( W  i^i  U ) ) ) )
182, 17syl5eq 2496 . . . 4  |-  ( ph  ->  ( # `  (
( V  u.  W
)  i^i  U )
)  =  ( (
# `  ( V  i^i  U ) )  +  ( # `  ( W  i^i  U ) ) ) )
19 difundir 3736 . . . . . 6  |-  ( ( V  u.  W ) 
\  U )  =  ( ( V  \  U )  u.  ( W  \  U ) )
2019fveq2i 5859 . . . . 5  |-  ( # `  ( ( V  u.  W )  \  U
) )  =  (
# `  ( ( V  \  U )  u.  ( W  \  U
) ) )
21 diffi 7753 . . . . . . 7  |-  ( V  e.  Fin  ->  ( V  \  U )  e. 
Fin )
223, 21syl 16 . . . . . 6  |-  ( ph  ->  ( V  \  U
)  e.  Fin )
23 diffi 7753 . . . . . . 7  |-  ( W  e.  Fin  ->  ( W  \  U )  e. 
Fin )
246, 23syl 16 . . . . . 6  |-  ( ph  ->  ( W  \  U
)  e.  Fin )
259difeq1d 3606 . . . . . . 7  |-  ( ph  ->  ( ( V  i^i  W )  \  U )  =  ( (/)  \  U
) )
26 difindir 3738 . . . . . . 7  |-  ( ( V  i^i  W ) 
\  U )  =  ( ( V  \  U )  i^i  ( W  \  U ) )
27 0dif 3885 . . . . . . 7  |-  ( (/)  \  U )  =  (/)
2825, 26, 273eqtr3g 2507 . . . . . 6  |-  ( ph  ->  ( ( V  \  U )  i^i  ( W  \  U ) )  =  (/) )
29 hashun 12432 . . . . . 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 1229 . . . . 5  |-  ( ph  ->  ( # `  (
( V  \  U
)  u.  ( W 
\  U ) ) )  =  ( (
# `  ( V  \  U ) )  +  ( # `  ( W  \  U ) ) ) )
3120, 30syl5eq 2496 . . . 4  |-  ( ph  ->  ( # `  (
( V  u.  W
)  \  U )
)  =  ( (
# `  ( V  \  U ) )  +  ( # `  ( W  \  U ) ) ) )
3218, 31oveq12d 6299 . . 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 12410 . . . . . 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 10861 . . . 4  |-  ( ph  ->  ( # `  ( V  i^i  U ) )  e.  CC )
36 hashcl 12410 . . . . . 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 10861 . . . 4  |-  ( ph  ->  ( # `  ( W  i^i  U ) )  e.  CC )
39 hashcl 12410 . . . . . 6  |-  ( ( V  \  U )  e.  Fin  ->  ( # `
 ( V  \  U ) )  e. 
NN0 )
403, 21, 393syl 20 . . . . 5  |-  ( ph  ->  ( # `  ( V  \  U ) )  e.  NN0 )
4140nn0cnd 10861 . . . 4  |-  ( ph  ->  ( # `  ( V  \  U ) )  e.  CC )
42 hashcl 12410 . . . . . 6  |-  ( ( W  \  U )  e.  Fin  ->  ( # `
 ( W  \  U ) )  e. 
NN0 )
436, 23, 423syl 20 . . . . 5  |-  ( ph  ->  ( # `  ( W  \  U ) )  e.  NN0 )
4443nn0cnd 10861 . . . 4  |-  ( ph  ->  ( # `  ( W  \  U ) )  e.  CC )
4535, 38, 41, 44addsub4d 9983 . . 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 2484 . 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 7789 . . . 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 28440 . . 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 28440 . . . 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 28440 . . . 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 6299 . 2  |-  ( ph  ->  ( ( U  .^  V )  +  ( U  .^  W )
)  =  ( ( ( # `  ( V  i^i  U ) )  -  ( # `  ( V  \  U ) ) )  +  ( (
# `  ( W  i^i  U ) )  -  ( # `  ( W 
\  U ) ) ) ) )
6846, 62, 673eqtr4d 2494 1  |-  ( ph  ->  ( U  .^  ( V  u.  W )
)  =  ( ( U  .^  V )  +  ( U  .^  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   A.wral 2793   {crab 2797    \ cdif 3458    u. cun 3459    i^i cin 3460   (/)c0 3770   ifcif 3926   ~Pcpw 3997   class class class wbr 4437    |-> cmpt 4495   `'ccnv 4988   "cima 4992   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   Fincfn 7518   supcsup 7902   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    < clt 9631    <_ cle 9632    - cmin 9810    / cdiv 10213   NNcn 10543   NN0cn0 10802   ZZcz 10871   ...cfz 11683   #chash 12387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-n0 10803  df-z 10872  df-uz 11093  df-hash 12388
This theorem is referenced by:  ballotlemfrceq  28445
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