Theorem ballotlemgval

Description: Expand the value of .^ . (Contributed by Thierry Arnoux, 21-Apr-2017)

	Ref	Expression
Hypotheses	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 ) |-> inf ( { 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 ) ) ) )
Assertion	ballotlemgval	|- ( ( U e. Fin /\ V e. Fin ) -> ( U .^ V ) = ( ( # ` ( V i^i U ) ) - ( # ` ( V \ U ) ) ) )

Proof

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		ballotlemg	|- .^ = ( u e. Fin , v e. Fin |-> ( ( # ` ( v i^i u ) ) - ( # ` ( v \ u ) ) ) )
12		ineq2	|- ( u = U -> ( v i^i u ) = ( v i^i U ) )
13	12	fveq2d	|- ( u = U -> ( # ` ( v i^i u ) ) = ( # ` ( v i^i U ) ) )
14		difeq2	|- ( u = U -> ( v \ u ) = ( v \ U ) )
15	14	fveq2d	|- ( u = U -> ( # ` ( v \ u ) ) = ( # ` ( v \ U ) ) )
16	13 15	oveq12d	|- ( u = U -> ( ( # ` ( v i^i u ) ) - ( # ` ( v \ u ) ) ) = ( ( # ` ( v i^i U ) ) - ( # ` ( v \ U ) ) ) )
17		ineq1	|- ( v = V -> ( v i^i U ) = ( V i^i U ) )
18	17	fveq2d	|- ( v = V -> ( # ` ( v i^i U ) ) = ( # ` ( V i^i U ) ) )
19		difeq1	|- ( v = V -> ( v \ U ) = ( V \ U ) )
20	19	fveq2d	|- ( v = V -> ( # ` ( v \ U ) ) = ( # ` ( V \ U ) ) )
21	18 20	oveq12d	|- ( v = V -> ( ( # ` ( v i^i U ) ) - ( # ` ( v \ U ) ) ) = ( ( # ` ( V i^i U ) ) - ( # ` ( V \ U ) ) ) )
22		ovex	|- ( ( # ` ( V i^i U ) ) - ( # ` ( V \ U ) ) ) e. _V
23	16 21 11 22	ovmpo	|- ( ( U e. Fin /\ V e. Fin ) -> ( U .^ V ) = ( ( # ` ( V i^i U ) ) - ( # ` ( V \ U ) ) ) )