Theorem ballotfilemfval 13150

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
ballotfi.o	|- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
ballotfi.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 3424	. . . . . . 7 |- ( ( b = C /\ i e. ZZ ) -> ( ( 1 ... i ) i^i b ) = ( ( 1 ... i ) i^i C ) )
4	3	fveq2d 5676	. . . . . 6 |- ( ( b = C /\ i e. ZZ ) -> (♯ ` ( ( 1 ... i ) i^i b ) ) = (♯ ` ( ( 1 ... i ) i^i C ) ) )
5	2	difeq2d 3339	. . . . . . 7 |- ( ( b = C /\ i e. ZZ ) -> ( ( 1 ... i ) \ b ) = ( ( 1 ... i ) \ C ) )
6	5	fveq2d 5676	. . . . . 6 |- ( ( b = C /\ i e. ZZ ) -> (♯ ` ( ( 1 ... i ) \ b ) ) = (♯ ` ( ( 1 ... i ) \ C ) ) )
7	4, 6	oveq12d 6070	. . . . 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 4202	. . . 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 3418	. . . . . . . . 9 |- ( b = c -> ( ( 1 ... i ) i^i b ) = ( ( 1 ... i ) i^i c ) )
11	10	fveq2d 5676	. . . . . . . 8 |- ( b = c -> (♯ ` ( ( 1 ... i ) i^i b ) ) = (♯ ` ( ( 1 ... i ) i^i c ) ) )
12		difeq2 3333	. . . . . . . . 9 |- ( b = c -> ( ( 1 ... i ) \ b ) = ( ( 1 ... i ) \ c ) )
13	12	fveq2d 5676	. . . . . . . 8 |- ( b = c -> (♯ ` ( ( 1 ... i ) \ b ) ) = (♯ ` ( ( 1 ... i ) \ c ) ) )
14	11, 13	oveq12d 6070	. . . . . . 7 |- ( b = c -> ( (♯ ` ( ( 1 ... i ) i^i b ) ) - (♯ ` ( ( 1 ... i ) \ b ) ) ) = ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) )
15	14	mpteq2dv 4203	. . . . . 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 4208	. . . . 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 2258	. . . 4 |- F = ( b e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i b ) ) - (♯ ` ( ( 1 ... i ) \ b ) ) ) ) )
18		zex 9588	. . . . 5 |- ZZ e. _V
19	18	mptex 5914	. . . 4 |- ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i C ) ) - (♯ ` ( ( 1 ... i ) \ C ) ) ) ) e. _V
20	8, 17, 19	fvmpt 5756	. . 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 6060	. . . . . 6 |- ( i = J -> ( 1 ... i ) = ( 1 ... J ) )
23	22	ineq1d 3423	. . . . 5 |- ( i = J -> ( ( 1 ... i ) i^i C ) = ( ( 1 ... J ) i^i C ) )
24	23	fveq2d 5676	. . . 4 |- ( i = J -> (♯ ` ( ( 1 ... i ) i^i C ) ) = (♯ ` ( ( 1 ... J ) i^i C ) ) )
25	22	difeq1d 3338	. . . . 5 |- ( i = J -> ( ( 1 ... i ) \ C ) = ( ( 1 ... J ) \ C ) )
26	25	fveq2d 5676	. . . 4 |- ( i = J -> (♯ ` ( ( 1 ... i ) \ C ) ) = (♯ ` ( ( 1 ... J ) \ C ) ) )
27	24, 26	oveq12d 6070	. . 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		ballotfi.o	. . . . . 6 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
33	30, 31, 32, 1, 29	ballotfilemcinfi 13147	. . . . 5 |- ( ph -> ( ( 1 ... J ) i^i C ) e. Fin )
34		hashcl 11148	. . . . 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 9701	. . 3 |- ( ph -> (♯ ` ( ( 1 ... J ) i^i C ) ) e. ZZ )
37	30, 31, 32, 1, 29	ballotfilemdifcfi 13148	. . . . 5 |- ( ph -> ( ( 1 ... J ) \ C ) e. Fin )
38		hashcl 11148	. . . . 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 9701	. . 3 |- ( ph -> (♯ ` ( ( 1 ... J ) \ C ) ) e. ZZ )
41	36, 40	zsubcld 9708	. 2 |- ( ph -> ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) e. ZZ )
42	21, 28, 29, 41	fvmptd 5760	1 |- ( ph -> ( ( F ` C ) ` J ) = ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   {crab 2526   \ cdif 3210   i^i cin 3212   ~Pcpw 3671   |-> cmpt 4173   ` cfv 5354  (class class class)co 6052   Fincfn 6977   1c1 8130   + caddc 8132   - cmin 8446   / cdiv 8948   NNcn 9239   NN0cn0 9498   ZZcz 9579   ...cfz 10345  ♯chash 11142

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-fz 10346  df-ihash 11143

This theorem is referenced by:  ballotfilemfelz  13151  ballotfilemfp1  13152  ballotfilemfmpn  13155  ballotfilemfval0  13156