Theorem ballotlemgval 30585

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

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

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

Allowed substitution hints:   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)

Proof of Theorem ballotlemgval

Step	Hyp	Ref	Expression
1		ineq2 3808	. . . 4 |- ( u = U -> ( v i^i u ) = ( v i^i U ) )
2	1	fveq2d 6195	. . 3 |- ( u = U -> ( # ` ( v i^i u ) ) = ( # ` ( v i^i U ) ) )
3		difeq2 3722	. . . 4 |- ( u = U -> ( v \ u ) = ( v \ U ) )
4	3	fveq2d 6195	. . 3 |- ( u = U -> ( # ` ( v \ u ) ) = ( # ` ( v \ U ) ) )
5	2, 4	oveq12d 6668	. 2 |- ( u = U -> ( ( # ` ( v i^i u ) ) - ( # ` ( v \ u ) ) ) = ( ( # ` ( v i^i U ) ) - ( # ` ( v \ U ) ) ) )
6		ineq1 3807	. . . 4 |- ( v = V -> ( v i^i U ) = ( V i^i U ) )
7	6	fveq2d 6195	. . 3 |- ( v = V -> ( # ` ( v i^i U ) ) = ( # ` ( V i^i U ) ) )
8		difeq1 3721	. . . 4 |- ( v = V -> ( v \ U ) = ( V \ U ) )
9	8	fveq2d 6195	. . 3 |- ( v = V -> ( # ` ( v \ U ) ) = ( # ` ( V \ U ) ) )
10	7, 9	oveq12d 6668	. 2 |- ( v = V -> ( ( # ` ( v i^i U ) ) - ( # ` ( v \ U ) ) ) = ( ( # ` ( V i^i U ) ) - ( # ` ( V \ U ) ) ) )
11		ballotlemg	. 2 |- .^ = ( u e. Fin , v e. Fin |-> ( ( # ` ( v i^i u ) ) - ( # ` ( v \ u ) ) ) )
12		ovex 6678	. 2 |- ( ( # ` ( V i^i U ) ) - ( # ` ( V \ U ) ) ) e. _V
13	5, 10, 11, 12	ovmpt2 6796	1 |- ( ( U e. Fin /\ V e. Fin ) -> ( U .^ V ) = ( ( # ` ( V i^i U ) ) - ( # ` ( V \ U ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   ifcif 4086   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955  infcinf 8347   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   ZZcz 11377   ...cfz 12326   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  ballotlemgun  30586  ballotlemfg  30587  ballotlemfrc  30588