Theorem ballotfilemfval 13207

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
ballotfilem.o	|- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
ballotfilem.p	|- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` 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
ballotfilemfval	|- ( 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 ballotfilemfval

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotlemfval.c	. . 3 |- ( ph -> C e. O )
2		simpl 109	. . . . . . . 8 |- ( ( b = C /\ i e. ZZ ) -> b = C )
3	2	ineq2d 3432	. . . . . . 7 |- ( ( b = C /\ i e. ZZ ) -> ( ( 1 ... i ) i^i b ) = ( ( 1 ... i ) i^i C ) )
4	3	fveq2d 5694	. . . . . 6 |- ( ( b = C /\ i e. ZZ ) -> (♯ ` ( ( 1 ... i ) i^i b ) ) = (♯ ` ( ( 1 ... i ) i^i C ) ) )
5	2	difeq2d 3347	. . . . . . 7 |- ( ( b = C /\ i e. ZZ ) -> ( ( 1 ... i ) \ b ) = ( ( 1 ... i ) \ C ) )
6	5	fveq2d 5694	. . . . . 6 |- ( ( b = C /\ i e. ZZ ) -> (♯ ` ( ( 1 ... i ) \ b ) ) = (♯ ` ( ( 1 ... i ) \ C ) ) )
7	4, 6	oveq12d 6093	. . . . 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 4216	. . . 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 3426	. . . . . . . . 9 |- ( b = c -> ( ( 1 ... i ) i^i b ) = ( ( 1 ... i ) i^i c ) )
11	10	fveq2d 5694	. . . . . . . 8 |- ( b = c -> (♯ ` ( ( 1 ... i ) i^i b ) ) = (♯ ` ( ( 1 ... i ) i^i c ) ) )
12		difeq2 3341	. . . . . . . . 9 |- ( b = c -> ( ( 1 ... i ) \ b ) = ( ( 1 ... i ) \ c ) )
13	12	fveq2d 5694	. . . . . . . 8 |- ( b = c -> (♯ ` ( ( 1 ... i ) \ b ) ) = (♯ ` ( ( 1 ... i ) \ c ) ) )
14	11, 13	oveq12d 6093	. . . . . . 7 |- ( b = c -> ( (♯ ` ( ( 1 ... i ) i^i b ) ) - (♯ ` ( ( 1 ... i ) \ b ) ) ) = ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) )
15	14	mpteq2dv 4217	. . . . . 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 4222	. . . . 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 2262	. . . 4 |- F = ( b e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i b ) ) - (♯ ` ( ( 1 ... i ) \ b ) ) ) ) )
18		zex 9632	. . . . 5 |- ZZ e. _V
19	18	mptex 5934	. . . 4 |- ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i C ) ) - (♯ ` ( ( 1 ... i ) \ C ) ) ) ) e. _V
20	8, 17, 19	fvmpt 5776	. . 3 |- ( C e. O -> ( F ` C ) = ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i C ) ) - (♯ ` ( ( 1 ... i ) \ C ) ) ) ) )
21	1, 20	syl 14	. 2 |- ( ph -> ( F ` C ) = ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i C ) ) - (♯ ` ( ( 1 ... i ) \ C ) ) ) ) )
22		oveq2 6083	. . . . . 6 |- ( i = J -> ( 1 ... i ) = ( 1 ... J ) )
23	22	ineq1d 3431	. . . . 5 |- ( i = J -> ( ( 1 ... i ) i^i C ) = ( ( 1 ... J ) i^i C ) )
24	23	fveq2d 5694	. . . 4 |- ( i = J -> (♯ ` ( ( 1 ... i ) i^i C ) ) = (♯ ` ( ( 1 ... J ) i^i C ) ) )
25	22	difeq1d 3346	. . . . 5 |- ( i = J -> ( ( 1 ... i ) \ C ) = ( ( 1 ... J ) \ C ) )
26	25	fveq2d 5694	. . . 4 |- ( i = J -> (♯ ` ( ( 1 ... i ) \ C ) ) = (♯ ` ( ( 1 ... J ) \ C ) ) )
27	24, 26	oveq12d 6093	. . 3 |- ( i = J -> ( (♯ ` ( ( 1 ... i ) i^i C ) ) - (♯ ` ( ( 1 ... i ) \ C ) ) ) = ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) )
28	27	adantl 277	. 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		ballotth.m	. . . . . 6 |- M e. NN
31		ballotth.n	. . . . . 6 |- N e. NN
32		ballotfilem.o	. . . . . 6 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
33	30, 31, 32, 1, 29	ballotfilemcinfi 13202	. . . . 5 |- ( ph -> ( ( 1 ... J ) i^i C ) e. Fin )
34		hashcl 11198	. . . . 5 |- ( ( ( 1 ... J ) i^i C ) e. Fin -> (♯ ` ( ( 1 ... J ) i^i C ) ) e. NN0 )
35	33, 34	syl 14	. . . 4 |- ( ph -> (♯ ` ( ( 1 ... J ) i^i C ) ) e. NN0 )
36	35	nn0zd 9745	. . 3 |- ( ph -> (♯ ` ( ( 1 ... J ) i^i C ) ) e. ZZ )
37	30, 31, 32, 1, 29	ballotfilemdifcfi 13203	. . . . 5 |- ( ph -> ( ( 1 ... J ) \ C ) e. Fin )
38		hashcl 11198	. . . . 5 |- ( ( ( 1 ... J ) \ C ) e. Fin -> (♯ ` ( ( 1 ... J ) \ C ) ) e. NN0 )
39	37, 38	syl 14	. . . 4 |- ( ph -> (♯ ` ( ( 1 ... J ) \ C ) ) e. NN0 )
40	39	nn0zd 9745	. . 3 |- ( ph -> (♯ ` ( ( 1 ... J ) \ C ) ) e. ZZ )
41	36, 40	zsubcld 9752	. 2 |- ( ph -> ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) e. ZZ )
42	21, 28, 29, 41	fvmptd 5780	1 |- ( ph -> ( ( F ` C ) ` J ) = ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {crab 2532   \ cdif 3217   i^i cin 3219   ~Pcpw 3685   |-> cmpt 4187   ` cfv 5372  (class class class)co 6075   Fincfn 7012   1c1 8170   + caddc 8172   - cmin 8487   / cdiv 8992   NNcn 9283   NN0cn0 9542   ZZcz 9623   ...cfz 10390  ♯chash 11192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-ihash 11193

This theorem is referenced by:  ballotfilemfelz  13208  ballotfilemfp1  13209  ballotfilemfmpn  13212  ballotfilemfval0  13213  ballotfilemfg  13247  ballotfilemfrc  13248