Step |
Hyp |
Ref |
Expression |
1 |
|
ballotth.m |
|- M e. NN |
2 |
|
ballotth.n |
|- N e. NN |
3 |
|
ballotth.o |
|- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M } |
4 |
|
ballotth.p |
|- P = ( x e. ~P O |-> ( ( # ` x ) / ( # ` O ) ) ) |
5 |
|
ballotth.f |
|- F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) ) |
6 |
|
ballotth.e |
|- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) } |
7 |
|
ballotth.mgtn |
|- N < M |
8 |
|
ballotth.i |
|- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) ) |
9 |
|
ballotth.s |
|- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) ) |
10 |
|
ballotth.r |
|- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) ) |
11 |
|
fveq2 |
|- ( d = C -> ( S ` d ) = ( S ` C ) ) |
12 |
|
id |
|- ( d = C -> d = C ) |
13 |
11 12
|
imaeq12d |
|- ( d = C -> ( ( S ` d ) " d ) = ( ( S ` C ) " C ) ) |
14 |
|
fveq2 |
|- ( c = d -> ( S ` c ) = ( S ` d ) ) |
15 |
|
id |
|- ( c = d -> c = d ) |
16 |
14 15
|
imaeq12d |
|- ( c = d -> ( ( S ` c ) " c ) = ( ( S ` d ) " d ) ) |
17 |
16
|
cbvmptv |
|- ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) ) = ( d e. ( O \ E ) |-> ( ( S ` d ) " d ) ) |
18 |
10 17
|
eqtri |
|- R = ( d e. ( O \ E ) |-> ( ( S ` d ) " d ) ) |
19 |
|
fvex |
|- ( S ` C ) e. _V |
20 |
|
imaexg |
|- ( ( S ` C ) e. _V -> ( ( S ` C ) " C ) e. _V ) |
21 |
19 20
|
ax-mp |
|- ( ( S ` C ) " C ) e. _V |
22 |
13 18 21
|
fvmpt |
|- ( C e. ( O \ E ) -> ( R ` C ) = ( ( S ` C ) " C ) ) |