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Theorem hashbc 12613
Description: The binomial coefficient counts the number of subsets of a finite set of a given size. This is Metamath 100 proof #58 (formula for the number of combinations). (Contributed by Mario Carneiro, 13-Jul-2014.)
Assertion
Ref Expression
hashbc  |-  ( ( A  e.  Fin  /\  K  e.  ZZ )  ->  ( ( # `  A
)  _C  K )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  K }
) )
Distinct variable groups:    x, A    x, K

Proof of Theorem hashbc
Dummy variables  j 
k  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5877 . . . . . 6  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
21oveq1d 6316 . . . . 5  |-  ( w  =  (/)  ->  ( (
# `  w )  _C  k )  =  ( ( # `  (/) )  _C  k ) )
3 pweq 3982 . . . . . . 7  |-  ( w  =  (/)  ->  ~P w  =  ~P (/) )
4 rabeq 3074 . . . . . . 7  |-  ( ~P w  =  ~P (/)  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } )
53, 4syl 17 . . . . . 6  |-  ( w  =  (/)  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } )
65fveq2d 5881 . . . . 5  |-  ( w  =  (/)  ->  ( # `  { x  e.  ~P w  |  ( # `  x
)  =  k } )  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) )
72, 6eqeq12d 2444 . . . 4  |-  ( w  =  (/)  ->  ( ( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) ) )
87ralbidv 2864 . . 3  |-  ( w  =  (/)  ->  ( A. k  e.  ZZ  (
( # `  w )  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  A. k  e.  ZZ  ( ( # `  (/) )  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) ) )
9 fveq2 5877 . . . . . 6  |-  ( w  =  y  ->  ( # `
 w )  =  ( # `  y
) )
109oveq1d 6316 . . . . 5  |-  ( w  =  y  ->  (
( # `  w )  _C  k )  =  ( ( # `  y
)  _C  k ) )
11 pweq 3982 . . . . . . 7  |-  ( w  =  y  ->  ~P w  =  ~P y
)
12 rabeq 3074 . . . . . . 7  |-  ( ~P w  =  ~P y  ->  { x  e.  ~P w  |  ( # `  x
)  =  k }  =  { x  e. 
~P y  |  (
# `  x )  =  k } )
1311, 12syl 17 . . . . . 6  |-  ( w  =  y  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P y  |  ( # `  x
)  =  k } )
1413fveq2d 5881 . . . . 5  |-  ( w  =  y  ->  ( # `
 { x  e. 
~P w  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } ) )
1510, 14eqeq12d 2444 . . . 4  |-  ( w  =  y  ->  (
( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  y )  _C  k
)  =  ( # `  { x  e.  ~P y  |  ( # `  x
)  =  k } ) ) )
1615ralbidv 2864 . . 3  |-  ( w  =  y  ->  ( A. k  e.  ZZ  ( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  A. k  e.  ZZ  ( ( # `  y
)  _C  k )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } ) ) )
17 fveq2 5877 . . . . . 6  |-  ( w  =  ( y  u. 
{ z } )  ->  ( # `  w
)  =  ( # `  ( y  u.  {
z } ) ) )
1817oveq1d 6316 . . . . 5  |-  ( w  =  ( y  u. 
{ z } )  ->  ( ( # `  w )  _C  k
)  =  ( (
# `  ( y  u.  { z } ) )  _C  k ) )
19 pweq 3982 . . . . . . 7  |-  ( w  =  ( y  u. 
{ z } )  ->  ~P w  =  ~P ( y  u. 
{ z } ) )
20 rabeq 3074 . . . . . . 7  |-  ( ~P w  =  ~P (
y  u.  { z } )  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P ( y  u.  {
z } )  |  ( # `  x
)  =  k } )
2119, 20syl 17 . . . . . 6  |-  ( w  =  ( y  u. 
{ z } )  ->  { x  e. 
~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P ( y  u.  {
z } )  |  ( # `  x
)  =  k } )
2221fveq2d 5881 . . . . 5  |-  ( w  =  ( y  u. 
{ z } )  ->  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  =  ( # `  { x  e.  ~P ( y  u.  {
z } )  |  ( # `  x
)  =  k } ) )
2318, 22eqeq12d 2444 . . . 4  |-  ( w  =  ( y  u. 
{ z } )  ->  ( ( (
# `  w )  _C  k )  =  (
# `  { x  e.  ~P w  |  (
# `  x )  =  k } )  <-> 
( ( # `  (
y  u.  { z } ) )  _C  k )  =  (
# `  { x  e.  ~P ( y  u. 
{ z } )  |  ( # `  x
)  =  k } ) ) )
2423ralbidv 2864 . . 3  |-  ( w  =  ( y  u. 
{ z } )  ->  ( A. k  e.  ZZ  ( ( # `  w )  _C  k
)  =  ( # `  { x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  A. k  e.  ZZ  ( ( # `  (
y  u.  { z } ) )  _C  k )  =  (
# `  { x  e.  ~P ( y  u. 
{ z } )  |  ( # `  x
)  =  k } ) ) )
25 fveq2 5877 . . . . . 6  |-  ( w  =  A  ->  ( # `
 w )  =  ( # `  A
) )
2625oveq1d 6316 . . . . 5  |-  ( w  =  A  ->  (
( # `  w )  _C  k )  =  ( ( # `  A
)  _C  k ) )
27 pweq 3982 . . . . . . 7  |-  ( w  =  A  ->  ~P w  =  ~P A
)
28 rabeq 3074 . . . . . . 7  |-  ( ~P w  =  ~P A  ->  { x  e.  ~P w  |  ( # `  x
)  =  k }  =  { x  e. 
~P A  |  (
# `  x )  =  k } )
2927, 28syl 17 . . . . . 6  |-  ( w  =  A  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P A  |  ( # `  x
)  =  k } )
3029fveq2d 5881 . . . . 5  |-  ( w  =  A  ->  ( # `
 { x  e. 
~P w  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  k } ) )
3126, 30eqeq12d 2444 . . . 4  |-  ( w  =  A  ->  (
( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  A )  _C  k
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  k } ) ) )
3231ralbidv 2864 . . 3  |-  ( w  =  A  ->  ( A. k  e.  ZZ  ( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  A. k  e.  ZZ  ( ( # `  A
)  _C  k )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  k } ) ) )
33 hash0 12547 . . . . . . . . . 10  |-  ( # `  (/) )  =  0
3433a1i 11 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 (/) )  =  0 )
35 elfz1eq 11810 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  k  =  0 )
3634, 35oveq12d 6319 . . . . . . . 8  |-  ( k  e.  ( 0 ... 0 )  ->  (
( # `  (/) )  _C  k )  =  ( 0  _C  0 ) )
37 0nn0 10884 . . . . . . . . 9  |-  0  e.  NN0
38 bcn0 12494 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( 0  _C  0 )  =  1 )
3937, 38ax-mp 5 . . . . . . . 8  |-  ( 0  _C  0 )  =  1
4036, 39syl6eq 2479 . . . . . . 7  |-  ( k  e.  ( 0 ... 0 )  ->  (
( # `  (/) )  _C  k )  =  1 )
41 pw0 4144 . . . . . . . . . 10  |-  ~P (/)  =  { (/)
}
4235eqcomd 2430 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... 0 )  ->  0  =  k )
4341raleqi 3029 . . . . . . . . . . . . 13  |-  ( A. x  e.  ~P  (/) ( # `  x )  =  k  <->  A. x  e.  { (/) }  ( # `  x
)  =  k )
44 0ex 4552 . . . . . . . . . . . . . 14  |-  (/)  e.  _V
45 fveq2 5877 . . . . . . . . . . . . . . . 16  |-  ( x  =  (/)  ->  ( # `  x )  =  (
# `  (/) ) )
4645, 33syl6eq 2479 . . . . . . . . . . . . . . 15  |-  ( x  =  (/)  ->  ( # `  x )  =  0 )
4746eqeq1d 2424 . . . . . . . . . . . . . 14  |-  ( x  =  (/)  ->  ( (
# `  x )  =  k  <->  0  =  k ) )
4844, 47ralsn 4035 . . . . . . . . . . . . 13  |-  ( A. x  e.  { (/) }  ( # `
 x )  =  k  <->  0  =  k )
4943, 48bitri 252 . . . . . . . . . . . 12  |-  ( A. x  e.  ~P  (/) ( # `  x )  =  k  <->  0  =  k )
5042, 49sylibr 215 . . . . . . . . . . 11  |-  ( k  e.  ( 0 ... 0 )  ->  A. x  e.  ~P  (/) ( # `  x
)  =  k )
51 rabid2 3006 . . . . . . . . . . 11  |-  ( ~P (/)  =  { x  e. 
~P (/)  |  ( # `  x )  =  k }  <->  A. x  e.  ~P  (/) ( # `  x
)  =  k )
5250, 51sylibr 215 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... 0 )  ->  ~P (/)  =  { x  e. 
~P (/)  |  ( # `  x )  =  k } )
5341, 52syl5reqr 2478 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  { x  e.  ~P (/)  |  ( # `
 x )  =  k }  =  { (/)
} )
5453fveq2d 5881 . . . . . . . 8  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 { x  e. 
~P (/)  |  ( # `  x )  =  k } )  =  (
# `  { (/) } ) )
55 hashsng 12548 . . . . . . . . 9  |-  ( (/)  e.  _V  ->  ( # `  { (/)
} )  =  1 )
5644, 55ax-mp 5 . . . . . . . 8  |-  ( # `  { (/) } )  =  1
5754, 56syl6eq 2479 . . . . . . 7  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 { x  e. 
~P (/)  |  ( # `  x )  =  k } )  =  1 )
5840, 57eqtr4d 2466 . . . . . 6  |-  ( k  e.  ( 0 ... 0 )  ->  (
( # `  (/) )  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) )
5958adantl 467 . . . . 5  |-  ( ( k  e.  ZZ  /\  k  e.  ( 0 ... 0 ) )  ->  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) )
6033oveq1i 6311 . . . . . 6  |-  ( (
# `  (/) )  _C  k )  =  ( 0  _C  k )
61 bcval3 12490 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  k  e.  ZZ  /\  -.  k  e.  ( 0 ... 0 ) )  ->  ( 0  _C  k )  =  0 )
6237, 61mp3an1 1347 . . . . . . 7  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( 0  _C  k )  =  0 )
63 id 23 . . . . . . . . . . . . . 14  |-  ( 0  =  k  ->  0  =  k )
64 0z 10948 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
65 elfz3 11809 . . . . . . . . . . . . . . 15  |-  ( 0  e.  ZZ  ->  0  e.  ( 0 ... 0
) )
6664, 65ax-mp 5 . . . . . . . . . . . . . 14  |-  0  e.  ( 0 ... 0
)
6763, 66syl6eqelr 2519 . . . . . . . . . . . . 13  |-  ( 0  =  k  ->  k  e.  ( 0 ... 0
) )
6867con3i 140 . . . . . . . . . . . 12  |-  ( -.  k  e.  ( 0 ... 0 )  ->  -.  0  =  k
)
6968adantl 467 . . . . . . . . . . 11  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  -.  0  =  k )
7041raleqi 3029 . . . . . . . . . . . 12  |-  ( A. x  e.  ~P  (/)  -.  ( # `
 x )  =  k  <->  A. x  e.  { (/)
}  -.  ( # `  x )  =  k )
7147notbid 295 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  ( -.  ( # `  x
)  =  k  <->  -.  0  =  k ) )
7244, 71ralsn 4035 . . . . . . . . . . . 12  |-  ( A. x  e.  { (/) }  -.  ( # `  x )  =  k  <->  -.  0  =  k )
7370, 72bitri 252 . . . . . . . . . . 11  |-  ( A. x  e.  ~P  (/)  -.  ( # `
 x )  =  k  <->  -.  0  =  k )
7469, 73sylibr 215 . . . . . . . . . 10  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  A. x  e.  ~P  (/) 
-.  ( # `  x
)  =  k )
75 rabeq0 3784 . . . . . . . . . 10  |-  ( { x  e.  ~P (/)  |  (
# `  x )  =  k }  =  (/)  <->  A. x  e.  ~P  (/)  -.  ( # `
 x )  =  k )
7674, 75sylibr 215 . . . . . . . . 9  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  { x  e. 
~P (/)  |  ( # `  x )  =  k }  =  (/) )
7776fveq2d 5881 . . . . . . . 8  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( # `  {
x  e.  ~P (/)  |  (
# `  x )  =  k } )  =  ( # `  (/) ) )
7877, 33syl6eq 2479 . . . . . . 7  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( # `  {
x  e.  ~P (/)  |  (
# `  x )  =  k } )  =  0 )
7962, 78eqtr4d 2466 . . . . . 6  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( 0  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) )
8060, 79syl5eq 2475 . . . . 5  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) )
8159, 80pm2.61dan 798 . . . 4  |-  ( k  e.  ZZ  ->  (
( # `  (/) )  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) )
8281rgen 2785 . . 3  |-  A. k  e.  ZZ  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } )
83 oveq2 6309 . . . . . 6  |-  ( k  =  j  ->  (
( # `  y )  _C  k )  =  ( ( # `  y
)  _C  j ) )
84 eqeq2 2437 . . . . . . . . 9  |-  ( k  =  j  ->  (
( # `  x )  =  k  <->  ( # `  x
)  =  j ) )
8584rabbidv 3072 . . . . . . . 8  |-  ( k  =  j  ->  { x  e.  ~P y  |  (
# `  x )  =  k }  =  { x  e.  ~P y  |  ( # `  x
)  =  j } )
86 fveq2 5877 . . . . . . . . . 10  |-  ( x  =  z  ->  ( # `
 x )  =  ( # `  z
) )
8786eqeq1d 2424 . . . . . . . . 9  |-  ( x  =  z  ->  (
( # `  x )  =  j  <->  ( # `  z
)  =  j ) )
8887cbvrabv 3080 . . . . . . . 8  |-  { x  e.  ~P y  |  (
# `  x )  =  j }  =  { z  e.  ~P y  |  ( # `  z
)  =  j }
8985, 88syl6eq 2479 . . . . . . 7  |-  ( k  =  j  ->  { x  e.  ~P y  |  (
# `  x )  =  k }  =  { z  e.  ~P y  |  ( # `  z
)  =  j } )
9089fveq2d 5881 . . . . . 6  |-  ( k  =  j  ->  ( # `
 { x  e. 
~P y  |  (
# `  x )  =  k } )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) )
9183, 90eqeq12d 2444 . . . . 5  |-  ( k  =  j  ->  (
( ( # `  y
)  _C  k )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } )  <->  ( ( # `  y )  _C  j
)  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )
9291cbvralv 3055 . . . 4  |-  ( A. k  e.  ZZ  (
( # `  y )  _C  k )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } )  <->  A. j  e.  ZZ  ( ( # `  y
)  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) )
93 simpll 758 . . . . . . 7  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( k  e.  ZZ  /\  A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )  -> 
y  e.  Fin )
94 simplr 760 . . . . . . 7  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( k  e.  ZZ  /\  A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )  ->  -.  z  e.  y
)
95 simprr 764 . . . . . . . 8  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( k  e.  ZZ  /\  A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )  ->  A. j  e.  ZZ  ( ( # `  y
)  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) )
9688fveq2i 5880 . . . . . . . . . 10  |-  ( # `  { x  e.  ~P y  |  ( # `  x
)  =  j } )  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } )
9796eqeq2i 2440 . . . . . . . . 9  |-  ( ( ( # `  y
)  _C  j )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  j } )  <->  ( ( # `  y )  _C  j
)  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } ) )
9897ralbii 2856 . . . . . . . 8  |-  ( A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  j } )  <->  A. j  e.  ZZ  ( ( # `  y
)  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) )
9995, 98sylibr 215 . . . . . . 7  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( k  e.  ZZ  /\  A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )  ->  A. j  e.  ZZ  ( ( # `  y
)  _C  j )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  j } ) )
100 simprl 762 . . . . . . 7  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( k  e.  ZZ  /\  A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )  -> 
k  e.  ZZ )
10193, 94, 99, 100hashbclem 12612 . . . . . 6  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( k  e.  ZZ  /\  A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )  -> 
( ( # `  (
y  u.  { z } ) )  _C  k )  =  (
# `  { x  e.  ~P ( y  u. 
{ z } )  |  ( # `  x
)  =  k } ) )
102101expr 618 . . . . 5  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  k  e.  ZZ )  ->  ( A. j  e.  ZZ  ( ( # `  y
)  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } )  ->  ( ( # `
 ( y  u. 
{ z } ) )  _C  k )  =  ( # `  {
x  e.  ~P (
y  u.  { z } )  |  (
# `  x )  =  k } ) ) )
103102ralrimdva 2843 . . . 4  |-  ( ( y  e.  Fin  /\  -.  z  e.  y
)  ->  ( A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } )  ->  A. k  e.  ZZ  ( ( # `  ( y  u.  {
z } ) )  _C  k )  =  ( # `  {
x  e.  ~P (
y  u.  { z } )  |  (
# `  x )  =  k } ) ) )
10492, 103syl5bi 220 . . 3  |-  ( ( y  e.  Fin  /\  -.  z  e.  y
)  ->  ( A. k  e.  ZZ  (
( # `  y )  _C  k )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } )  ->  A. k  e.  ZZ  ( ( # `  ( y  u.  {
z } ) )  _C  k )  =  ( # `  {
x  e.  ~P (
y  u.  { z } )  |  (
# `  x )  =  k } ) ) )
1058, 16, 24, 32, 82, 104findcard2s 7814 . 2  |-  ( A  e.  Fin  ->  A. k  e.  ZZ  ( ( # `  A )  _C  k
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  k } ) )
106 oveq2 6309 . . . 4  |-  ( k  =  K  ->  (
( # `  A )  _C  k )  =  ( ( # `  A
)  _C  K ) )
107 eqeq2 2437 . . . . . 6  |-  ( k  =  K  ->  (
( # `  x )  =  k  <->  ( # `  x
)  =  K ) )
108107rabbidv 3072 . . . . 5  |-  ( k  =  K  ->  { x  e.  ~P A  |  (
# `  x )  =  k }  =  { x  e.  ~P A  |  ( # `  x
)  =  K }
)
109108fveq2d 5881 . . . 4  |-  ( k  =  K  ->  ( # `
 { x  e. 
~P A  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  K }
) )
110106, 109eqeq12d 2444 . . 3  |-  ( k  =  K  ->  (
( ( # `  A
)  _C  k )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  k } )  <->  ( ( # `  A )  _C  K
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  K }
) ) )
111110rspccva 3181 . 2  |-  ( ( A. k  e.  ZZ  ( ( # `  A
)  _C  k )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  k } )  /\  K  e.  ZZ )  ->  (
( # `  A )  _C  K )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  K }
) )
112105, 111sylan 473 1  |-  ( ( A  e.  Fin  /\  K  e.  ZZ )  ->  ( ( # `  A
)  _C  K )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  K }
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775   {crab 2779   _Vcvv 3081    u. cun 3434   (/)c0 3761   ~Pcpw 3979   {csn 3996   ` cfv 5597  (class class class)co 6301   Fincfn 7573   0cc0 9539   1c1 9540   NN0cn0 10869   ZZcz 10937   ...cfz 11784    _C cbc 12486   #chash 12514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-seq 12213  df-fac 12459  df-bc 12487  df-hash 12515
This theorem is referenced by:  hashbc2  14945  sylow1lem1  17237  musum  24106  ballotlem1  29314  ballotlem2  29316
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