MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hashbc Structured version   Unicode version

Theorem hashbc 12202
Description: The binomial coefficient counts the number of subsets of a finite set of a given size. (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 5688 . . . . . 6  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
21oveq1d 6105 . . . . 5  |-  ( w  =  (/)  ->  ( (
# `  w )  _C  k )  =  ( ( # `  (/) )  _C  k ) )
3 pweq 3860 . . . . . . 7  |-  ( w  =  (/)  ->  ~P w  =  ~P (/) )
4 rabeq 2964 . . . . . . 7  |-  ( ~P w  =  ~P (/)  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } )
53, 4syl 16 . . . . . 6  |-  ( w  =  (/)  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } )
65fveq2d 5692 . . . . 5  |-  ( w  =  (/)  ->  ( # `  { x  e.  ~P w  |  ( # `  x
)  =  k } )  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) )
72, 6eqeq12d 2455 . . . 4  |-  ( w  =  (/)  ->  ( ( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) ) )
87ralbidv 2733 . . 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 5688 . . . . . 6  |-  ( w  =  y  ->  ( # `
 w )  =  ( # `  y
) )
109oveq1d 6105 . . . . 5  |-  ( w  =  y  ->  (
( # `  w )  _C  k )  =  ( ( # `  y
)  _C  k ) )
11 pweq 3860 . . . . . . 7  |-  ( w  =  y  ->  ~P w  =  ~P y
)
12 rabeq 2964 . . . . . . 7  |-  ( ~P w  =  ~P y  ->  { x  e.  ~P w  |  ( # `  x
)  =  k }  =  { x  e. 
~P y  |  (
# `  x )  =  k } )
1311, 12syl 16 . . . . . 6  |-  ( w  =  y  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P y  |  ( # `  x
)  =  k } )
1413fveq2d 5692 . . . . 5  |-  ( w  =  y  ->  ( # `
 { x  e. 
~P w  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } ) )
1510, 14eqeq12d 2455 . . . 4  |-  ( w  =  y  ->  (
( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  y )  _C  k
)  =  ( # `  { x  e.  ~P y  |  ( # `  x
)  =  k } ) ) )
1615ralbidv 2733 . . 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 5688 . . . . . 6  |-  ( w  =  ( y  u. 
{ z } )  ->  ( # `  w
)  =  ( # `  ( y  u.  {
z } ) ) )
1817oveq1d 6105 . . . . 5  |-  ( w  =  ( y  u. 
{ z } )  ->  ( ( # `  w )  _C  k
)  =  ( (
# `  ( y  u.  { z } ) )  _C  k ) )
19 pweq 3860 . . . . . . 7  |-  ( w  =  ( y  u. 
{ z } )  ->  ~P w  =  ~P ( y  u. 
{ z } ) )
20 rabeq 2964 . . . . . . 7  |-  ( ~P w  =  ~P (
y  u.  { z } )  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P ( y  u.  {
z } )  |  ( # `  x
)  =  k } )
2119, 20syl 16 . . . . . 6  |-  ( w  =  ( y  u. 
{ z } )  ->  { x  e. 
~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P ( y  u.  {
z } )  |  ( # `  x
)  =  k } )
2221fveq2d 5692 . . . . 5  |-  ( w  =  ( y  u. 
{ z } )  ->  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  =  ( # `  { x  e.  ~P ( y  u.  {
z } )  |  ( # `  x
)  =  k } ) )
2318, 22eqeq12d 2455 . . . 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 2733 . . 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 5688 . . . . . 6  |-  ( w  =  A  ->  ( # `
 w )  =  ( # `  A
) )
2625oveq1d 6105 . . . . 5  |-  ( w  =  A  ->  (
( # `  w )  _C  k )  =  ( ( # `  A
)  _C  k ) )
27 pweq 3860 . . . . . . 7  |-  ( w  =  A  ->  ~P w  =  ~P A
)
28 rabeq 2964 . . . . . . 7  |-  ( ~P w  =  ~P A  ->  { x  e.  ~P w  |  ( # `  x
)  =  k }  =  { x  e. 
~P A  |  (
# `  x )  =  k } )
2927, 28syl 16 . . . . . 6  |-  ( w  =  A  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P A  |  ( # `  x
)  =  k } )
3029fveq2d 5692 . . . . 5  |-  ( w  =  A  ->  ( # `
 { x  e. 
~P w  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  k } ) )
3126, 30eqeq12d 2455 . . . 4  |-  ( w  =  A  ->  (
( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  A )  _C  k
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  k } ) ) )
3231ralbidv 2733 . . 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 12131 . . . . . . . . . 10  |-  ( # `  (/) )  =  0
3433a1i 11 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 (/) )  =  0 )
35 elfz1eq 11458 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  k  =  0 )
3634, 35oveq12d 6108 . . . . . . . 8  |-  ( k  e.  ( 0 ... 0 )  ->  (
( # `  (/) )  _C  k )  =  ( 0  _C  0 ) )
37 0nn0 10590 . . . . . . . . 9  |-  0  e.  NN0
38 bcn0 12082 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( 0  _C  0 )  =  1 )
3937, 38ax-mp 5 . . . . . . . 8  |-  ( 0  _C  0 )  =  1
4036, 39syl6eq 2489 . . . . . . 7  |-  ( k  e.  ( 0 ... 0 )  ->  (
( # `  (/) )  _C  k )  =  1 )
41 pw0 4017 . . . . . . . . . 10  |-  ~P (/)  =  { (/)
}
4235eqcomd 2446 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... 0 )  ->  0  =  k )
4341raleqi 2919 . . . . . . . . . . . . 13  |-  ( A. x  e.  ~P  (/) ( # `  x )  =  k  <->  A. x  e.  { (/) }  ( # `  x
)  =  k )
44 0ex 4419 . . . . . . . . . . . . . 14  |-  (/)  e.  _V
45 fveq2 5688 . . . . . . . . . . . . . . . 16  |-  ( x  =  (/)  ->  ( # `  x )  =  (
# `  (/) ) )
4645, 33syl6eq 2489 . . . . . . . . . . . . . . 15  |-  ( x  =  (/)  ->  ( # `  x )  =  0 )
4746eqeq1d 2449 . . . . . . . . . . . . . 14  |-  ( x  =  (/)  ->  ( (
# `  x )  =  k  <->  0  =  k ) )
4844, 47ralsn 3912 . . . . . . . . . . . . 13  |-  ( A. x  e.  { (/) }  ( # `
 x )  =  k  <->  0  =  k )
4943, 48bitri 249 . . . . . . . . . . . 12  |-  ( A. x  e.  ~P  (/) ( # `  x )  =  k  <->  0  =  k )
5042, 49sylibr 212 . . . . . . . . . . 11  |-  ( k  e.  ( 0 ... 0 )  ->  A. x  e.  ~P  (/) ( # `  x
)  =  k )
51 rabid2 2896 . . . . . . . . . . 11  |-  ( ~P (/)  =  { x  e. 
~P (/)  |  ( # `  x )  =  k }  <->  A. x  e.  ~P  (/) ( # `  x
)  =  k )
5250, 51sylibr 212 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... 0 )  ->  ~P (/)  =  { x  e. 
~P (/)  |  ( # `  x )  =  k } )
5341, 52syl5reqr 2488 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  { x  e.  ~P (/)  |  ( # `
 x )  =  k }  =  { (/)
} )
5453fveq2d 5692 . . . . . . . 8  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 { x  e. 
~P (/)  |  ( # `  x )  =  k } )  =  (
# `  { (/) } ) )
55 hashsng 12132 . . . . . . . . 9  |-  ( (/)  e.  _V  ->  ( # `  { (/)
} )  =  1 )
5644, 55ax-mp 5 . . . . . . . 8  |-  ( # `  { (/) } )  =  1
5754, 56syl6eq 2489 . . . . . . 7  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 { x  e. 
~P (/)  |  ( # `  x )  =  k } )  =  1 )
5840, 57eqtr4d 2476 . . . . . 6  |-  ( k  e.  ( 0 ... 0 )  ->  (
( # `  (/) )  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) )
5958adantl 463 . . . . 5  |-  ( ( k  e.  ZZ  /\  k  e.  ( 0 ... 0 ) )  ->  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) )
6033oveq1i 6100 . . . . . 6  |-  ( (
# `  (/) )  _C  k )  =  ( 0  _C  k )
61 bcval3 12078 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  k  e.  ZZ  /\  -.  k  e.  ( 0 ... 0 ) )  ->  ( 0  _C  k )  =  0 )
6237, 61mp3an1 1296 . . . . . . 7  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( 0  _C  k )  =  0 )
63 id 22 . . . . . . . . . . . . . 14  |-  ( 0  =  k  ->  0  =  k )
64 0z 10653 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
65 elfz3 11457 . . . . . . . . . . . . . . 15  |-  ( 0  e.  ZZ  ->  0  e.  ( 0 ... 0
) )
6664, 65ax-mp 5 . . . . . . . . . . . . . 14  |-  0  e.  ( 0 ... 0
)
6763, 66syl6eqelr 2530 . . . . . . . . . . . . 13  |-  ( 0  =  k  ->  k  e.  ( 0 ... 0
) )
6867con3i 135 . . . . . . . . . . . 12  |-  ( -.  k  e.  ( 0 ... 0 )  ->  -.  0  =  k
)
6968adantl 463 . . . . . . . . . . 11  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  -.  0  =  k )
7041raleqi 2919 . . . . . . . . . . . 12  |-  ( A. x  e.  ~P  (/)  -.  ( # `
 x )  =  k  <->  A. x  e.  { (/)
}  -.  ( # `  x )  =  k )
7147notbid 294 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  ( -.  ( # `  x
)  =  k  <->  -.  0  =  k ) )
7244, 71ralsn 3912 . . . . . . . . . . . 12  |-  ( A. x  e.  { (/) }  -.  ( # `  x )  =  k  <->  -.  0  =  k )
7370, 72bitri 249 . . . . . . . . . . 11  |-  ( A. x  e.  ~P  (/)  -.  ( # `
 x )  =  k  <->  -.  0  =  k )
7469, 73sylibr 212 . . . . . . . . . 10  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  A. x  e.  ~P  (/) 
-.  ( # `  x
)  =  k )
75 rabeq0 3656 . . . . . . . . . 10  |-  ( { x  e.  ~P (/)  |  (
# `  x )  =  k }  =  (/)  <->  A. x  e.  ~P  (/)  -.  ( # `
 x )  =  k )
7674, 75sylibr 212 . . . . . . . . 9  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  { x  e. 
~P (/)  |  ( # `  x )  =  k }  =  (/) )
7776fveq2d 5692 . . . . . . . 8  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( # `  {
x  e.  ~P (/)  |  (
# `  x )  =  k } )  =  ( # `  (/) ) )
7877, 33syl6eq 2489 . . . . . . 7  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( # `  {
x  e.  ~P (/)  |  (
# `  x )  =  k } )  =  0 )
7962, 78eqtr4d 2476 . . . . . 6  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( 0  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) )
8060, 79syl5eq 2485 . . . . 5  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) )
8159, 80pm2.61dan 784 . . . 4  |-  ( k  e.  ZZ  ->  (
( # `  (/) )  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) )
8281rgen 2779 . . 3  |-  A. k  e.  ZZ  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } )
83 oveq2 6098 . . . . . 6  |-  ( k  =  j  ->  (
( # `  y )  _C  k )  =  ( ( # `  y
)  _C  j ) )
84 eqeq2 2450 . . . . . . . . 9  |-  ( k  =  j  ->  (
( # `  x )  =  k  <->  ( # `  x
)  =  j ) )
8584rabbidv 2962 . . . . . . . 8  |-  ( k  =  j  ->  { x  e.  ~P y  |  (
# `  x )  =  k }  =  { x  e.  ~P y  |  ( # `  x
)  =  j } )
86 fveq2 5688 . . . . . . . . . 10  |-  ( x  =  z  ->  ( # `
 x )  =  ( # `  z
) )
8786eqeq1d 2449 . . . . . . . . 9  |-  ( x  =  z  ->  (
( # `  x )  =  j  <->  ( # `  z
)  =  j ) )
8887cbvrabv 2969 . . . . . . . 8  |-  { x  e.  ~P y  |  (
# `  x )  =  j }  =  { z  e.  ~P y  |  ( # `  z
)  =  j }
8985, 88syl6eq 2489 . . . . . . 7  |-  ( k  =  j  ->  { x  e.  ~P y  |  (
# `  x )  =  k }  =  { z  e.  ~P y  |  ( # `  z
)  =  j } )
9089fveq2d 5692 . . . . . 6  |-  ( k  =  j  ->  ( # `
 { x  e. 
~P y  |  (
# `  x )  =  k } )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) )
9183, 90eqeq12d 2455 . . . . 5  |-  ( k  =  j  ->  (
( ( # `  y
)  _C  k )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } )  <->  ( ( # `  y )  _C  j
)  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )
9291cbvralv 2945 . . . 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 748 . . . . . . 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 749 . . . . . . 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 751 . . . . . . . 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 5691 . . . . . . . . . 10  |-  ( # `  { x  e.  ~P y  |  ( # `  x
)  =  j } )  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } )
9796eqeq2i 2451 . . . . . . . . 9  |-  ( ( ( # `  y
)  _C  j )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  j } )  <->  ( ( # `  y )  _C  j
)  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } ) )
9897ralbii 2737 . . . . . . . 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 212 . . . . . . 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 750 . . . . . . 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 12201 . . . . . 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 612 . . . . 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 2804 . . . 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 217 . . 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 7549 . 2  |-  ( A  e.  Fin  ->  A. k  e.  ZZ  ( ( # `  A )  _C  k
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  k } ) )
106 oveq2 6098 . . . 4  |-  ( k  =  K  ->  (
( # `  A )  _C  k )  =  ( ( # `  A
)  _C  K ) )
107 eqeq2 2450 . . . . . 6  |-  ( k  =  K  ->  (
( # `  x )  =  k  <->  ( # `  x
)  =  K ) )
108107rabbidv 2962 . . . . 5  |-  ( k  =  K  ->  { x  e.  ~P A  |  (
# `  x )  =  k }  =  { x  e.  ~P A  |  ( # `  x
)  =  K }
)
109108fveq2d 5692 . . . 4  |-  ( k  =  K  ->  ( # `
 { x  e. 
~P A  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  K }
) )
110106, 109eqeq12d 2455 . . 3  |-  ( k  =  K  ->  (
( ( # `  A
)  _C  k )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  k } )  <->  ( ( # `  A )  _C  K
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  K }
) ) )
111110rspccva 3069 . 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 468 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 369    = wceq 1364    e. wcel 1761   A.wral 2713   {crab 2717   _Vcvv 2970    u. cun 3323   (/)c0 3634   ~Pcpw 3857   {csn 3874   ` cfv 5415  (class class class)co 6090   Fincfn 7306   0cc0 9278   1c1 9279   NN0cn0 10575   ZZcz 10642   ...cfz 11433    _C cbc 12074   #chash 12099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-seq 11803  df-fac 12048  df-bc 12075  df-hash 12100
This theorem is referenced by:  hashbc2  14063  sylow1lem1  16090  musum  22490  ballotlem1  26799  ballotlem2  26801
  Copyright terms: Public domain W3C validator