Theorem ballotlemfval 30551

Description: The value of F. (Contributed by Thierry Arnoux, 23-Nov-2016.)

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 ) ) ) ) )
ballotlemfval.c	|- ( ph -> C e. O )
ballotlemfval.j	|- ( ph -> J e. ZZ )

Assertion

Ref	Expression
ballotlemfval	|- ( ph -> ( ( F ` C ) ` J ) = ( ( # ` ( ( 1 ... J ) i^i C ) ) - ( # ` ( ( 1 ... J ) \ C ) ) ) )

Distinct variable groups:   M, c   N, c   O, c   i, M   i, N   i, O, c   F, c, i   C, i   i, J   ph, i

Allowed substitution hints:   ph( x, c)   C( x, c)   P( x, i, c)   F( x)   J( x, c)   M( x)   N( x)   O( x)

Proof of Theorem ballotlemfval

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotlemfval.c	. . 3 |- ( ph -> C e. O )
2		simpl 473	. . . . . . . 8 |- ( ( b = C /\ i e. ZZ ) -> b = C )
3	2	ineq2d 3814	. . . . . . 7 |- ( ( b = C /\ i e. ZZ ) -> ( ( 1 ... i ) i^i b ) = ( ( 1 ... i ) i^i C ) )
4	3	fveq2d 6195	. . . . . 6 |- ( ( b = C /\ i e. ZZ ) -> ( # ` ( ( 1 ... i ) i^i b ) ) = ( # ` ( ( 1 ... i ) i^i C ) ) )
5	2	difeq2d 3728	. . . . . . 7 |- ( ( b = C /\ i e. ZZ ) -> ( ( 1 ... i ) \ b ) = ( ( 1 ... i ) \ C ) )
6	5	fveq2d 6195	. . . . . 6 |- ( ( b = C /\ i e. ZZ ) -> ( # ` ( ( 1 ... i ) \ b ) ) = ( # ` ( ( 1 ... i ) \ C ) ) )
7	4, 6	oveq12d 6668	. . . . 5 |- ( ( b = C /\ i e. ZZ ) -> ( ( # ` ( ( 1 ... i ) i^i b ) ) - ( # ` ( ( 1 ... i ) \ b ) ) ) = ( ( # ` ( ( 1 ... i ) i^i C ) ) - ( # ` ( ( 1 ... i ) \ C ) ) ) )
8	7	mpteq2dva 4744	. . . 4 |- ( b = C -> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i b ) ) - ( # ` ( ( 1 ... i ) \ b ) ) ) ) = ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i C ) ) - ( # ` ( ( 1 ... i ) \ C ) ) ) ) )
9		ballotth.f	. . . . 5 |- F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
10		ineq2 3808	. . . . . . . . 9 |- ( b = c -> ( ( 1 ... i ) i^i b ) = ( ( 1 ... i ) i^i c ) )
11	10	fveq2d 6195	. . . . . . . 8 |- ( b = c -> ( # ` ( ( 1 ... i ) i^i b ) ) = ( # ` ( ( 1 ... i ) i^i c ) ) )
12		difeq2 3722	. . . . . . . . 9 |- ( b = c -> ( ( 1 ... i ) \ b ) = ( ( 1 ... i ) \ c ) )
13	12	fveq2d 6195	. . . . . . . 8 |- ( b = c -> ( # ` ( ( 1 ... i ) \ b ) ) = ( # ` ( ( 1 ... i ) \ c ) ) )
14	11, 13	oveq12d 6668	. . . . . . 7 |- ( b = c -> ( ( # ` ( ( 1 ... i ) i^i b ) ) - ( # ` ( ( 1 ... i ) \ b ) ) ) = ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) )
15	14	mpteq2dv 4745	. . . . . 6 |- ( b = c -> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i b ) ) - ( # ` ( ( 1 ... i ) \ b ) ) ) ) = ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
16	15	cbvmptv 4750	. . . . 5 |- ( b e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i b ) ) - ( # ` ( ( 1 ... i ) \ b ) ) ) ) ) = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
17	9, 16	eqtr4i 2647	. . . 4 |- F = ( b e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i b ) ) - ( # ` ( ( 1 ... i ) \ b ) ) ) ) )
18		zex 11386	. . . . 5 |- ZZ e. _V
19	18	mptex 6486	. . . 4 |- ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i C ) ) - ( # ` ( ( 1 ... i ) \ C ) ) ) ) e. _V
20	8, 17, 19	fvmpt 6282	. . 3 |- ( C e. O -> ( F ` C ) = ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i C ) ) - ( # ` ( ( 1 ... i ) \ C ) ) ) ) )
21	1, 20	syl 17	. 2 |- ( ph -> ( F ` C ) = ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i C ) ) - ( # ` ( ( 1 ... i ) \ C ) ) ) ) )
22		oveq2 6658	. . . . . 6 |- ( i = J -> ( 1 ... i ) = ( 1 ... J ) )
23	22	ineq1d 3813	. . . . 5 |- ( i = J -> ( ( 1 ... i ) i^i C ) = ( ( 1 ... J ) i^i C ) )
24	23	fveq2d 6195	. . . 4 |- ( i = J -> ( # ` ( ( 1 ... i ) i^i C ) ) = ( # ` ( ( 1 ... J ) i^i C ) ) )
25	22	difeq1d 3727	. . . . 5 |- ( i = J -> ( ( 1 ... i ) \ C ) = ( ( 1 ... J ) \ C ) )
26	25	fveq2d 6195	. . . 4 |- ( i = J -> ( # ` ( ( 1 ... i ) \ C ) ) = ( # ` ( ( 1 ... J ) \ C ) ) )
27	24, 26	oveq12d 6668	. . 3 |- ( i = J -> ( ( # ` ( ( 1 ... i ) i^i C ) ) - ( # ` ( ( 1 ... i ) \ C ) ) ) = ( ( # ` ( ( 1 ... J ) i^i C ) ) - ( # ` ( ( 1 ... J ) \ C ) ) ) )
28	27	adantl 482	. 2 |- ( ( ph /\ i = J ) -> ( ( # ` ( ( 1 ... i ) i^i C ) ) - ( # ` ( ( 1 ... i ) \ C ) ) ) = ( ( # ` ( ( 1 ... J ) i^i C ) ) - ( # ` ( ( 1 ... J ) \ C ) ) ) )
29		ballotlemfval.j	. 2 |- ( ph -> J e. ZZ )
30		ovexd 6680	. 2 |- ( ph -> ( ( # ` ( ( 1 ... J ) i^i C ) ) - ( # ` ( ( 1 ... J ) \ C ) ) ) e. _V )
31	21, 28, 29, 30	fvmptd 6288	1 |- ( ph -> ( ( F ` C ) ` J ) = ( ( # ` ( ( 1 ... J ) i^i C ) ) - ( # ` ( ( 1 ... J ) \ C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   ~Pcpw 4158   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   - 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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-neg 10269  df-z 11378

This theorem is referenced by:  ballotlemfelz  30552  ballotlemfp1  30553  ballotlemfmpn  30556  ballotlemfval0  30557  ballotlemfg  30587  ballotlemfrc  30588