Theorem ballotfilemfval 13229

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 5699	. . . . . 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 5699	. . . . . 6 |- ( ( b = C /\ i e. ZZ ) -> (♯ ` ( ( 1 ... i ) \ b ) ) = (♯ ` ( ( 1 ... i ) \ C ) ) )
7	4, 6	oveq12d 6103	. . . . 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 4221	. . . 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 5699	. . . . . . . 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 5699	. . . . . . . 8 |- ( b = c -> (♯ ` ( ( 1 ... i ) \ b ) ) = (♯ ` ( ( 1 ... i ) \ c ) ) )
14	11, 13	oveq12d 6103	. . . . . . 7 |- ( b = c -> ( (♯ ` ( ( 1 ... i ) i^i b ) ) - (♯ ` ( ( 1 ... i ) \ b ) ) ) = ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) )
15	14	mpteq2dv 4222	. . . . . 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 4227	. . . . 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 9653	. . . . 5 |- ZZ e. _V
19	18	mptex 5943	. . . 4 |- ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i C ) ) - (♯ ` ( ( 1 ... i ) \ C ) ) ) ) e. _V
20	8, 17, 19	fvmpt 5782	. . 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 6093	. . . . . 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 5699	. . . 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 5699	. . . 4 |- ( i = J -> (♯ ` ( ( 1 ... i ) \ C ) ) = (♯ ` ( ( 1 ... J ) \ C ) ) )
27	24, 26	oveq12d 6103	. . 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 13224	. . . . 5 |- ( ph -> ( ( 1 ... J ) i^i C ) e. Fin )
34		hashcl 11220	. . . . 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 9766	. . 3 |- ( ph -> (♯ ` ( ( 1 ... J ) i^i C ) ) e. ZZ )
37	30, 31, 32, 1, 29	ballotfilemdifcfi 13225	. . . . 5 |- ( ph -> ( ( 1 ... J ) \ C ) e. Fin )
38		hashcl 11220	. . . . 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 9766	. . 3 |- ( ph -> (♯ ` ( ( 1 ... J ) \ C ) ) e. ZZ )
41	36, 40	zsubcld 9773	. 2 |- ( ph -> ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) e. ZZ )
42	21, 28, 29, 41	fvmptd 5786	1 |- ( ph -> ( ( F ` C ) ` J ) = ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {crab 2532   \ cdif 3217   i^i cin 3219   ~Pcpw 3688   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   Fincfn 7022   1c1 8180   + caddc 8182   - cmin 8497   / cdiv 9002   NNcn 9304   NN0cn0 9563   ZZcz 9644   ...cfz 10411  ♯chash 11214

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-ihash 11215

This theorem is used by:  ballotfilemfelz  13230  ballotfilemfp1  13231  ballotfilemfmpn  13234  ballotfilemfval0  13235  ballotfilemfg  13269  ballotfilemfrc  13270