Theorem ballotlemfmpn 30556

Description: ( F ` C ) finishes counting at ( M - N ). (Contributed by Thierry Arnoux, 25-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 ) ) ) ) )

Assertion

Ref	Expression
ballotlemfmpn	|- ( C e. O -> ( ( F ` C ) ` ( M + N ) ) = ( M - N ) )

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

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

Proof of Theorem ballotlemfmpn

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotth.m	. . 3 |- M e. NN
2		ballotth.n	. . 3 |- N e. NN
3		ballotth.o	. . 3 |- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
4		ballotth.p	. . 3 |- P = ( x e. ~P O |-> ( ( # ` x ) / ( # ` O ) ) )
5		ballotth.f	. . 3 |- F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
6		id 22	. . 3 |- ( C e. O -> C e. O )
7		nnaddcl 11042	. . . . . 6 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
8	1, 2, 7	mp2an 708	. . . . 5 |- ( M + N ) e. NN
9	8	nnzi 11401	. . . 4 |- ( M + N ) e. ZZ
10	9	a1i 11	. . 3 |- ( C e. O -> ( M + N ) e. ZZ )
11	1, 2, 3, 4, 5, 6, 10	ballotlemfval 30551	. 2 |- ( C e. O -> ( ( F ` C ) ` ( M + N ) ) = ( ( # ` ( ( 1 ... ( M + N ) ) i^i C ) ) - ( # ` ( ( 1 ... ( M + N ) ) \ C ) ) ) )
12		ssrab2 3687	. . . . . . . . 9 |- { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M } C_ ~P ( 1 ... ( M + N ) )
13	3, 12	eqsstri 3635	. . . . . . . 8 |- O C_ ~P ( 1 ... ( M + N ) )
14	13	sseli 3599	. . . . . . 7 |- ( C e. O -> C e. ~P ( 1 ... ( M + N ) ) )
15	14	elpwid 4170	. . . . . 6 |- ( C e. O -> C C_ ( 1 ... ( M + N ) ) )
16		sseqin2 3817	. . . . . 6 |- ( C C_ ( 1 ... ( M + N ) ) <-> ( ( 1 ... ( M + N ) ) i^i C ) = C )
17	15, 16	sylib 208	. . . . 5 |- ( C e. O -> ( ( 1 ... ( M + N ) ) i^i C ) = C )
18	17	fveq2d 6195	. . . 4 |- ( C e. O -> ( # ` ( ( 1 ... ( M + N ) ) i^i C ) ) = ( # ` C ) )
19		rabssab 3690	. . . . . . 7 |- { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M } C_ { c | ( # ` c ) = M }
20	19	sseli 3599	. . . . . 6 |- ( C e. { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M } -> C e. { c | ( # ` c ) = M } )
21	20, 3	eleq2s 2719	. . . . 5 |- ( C e. O -> C e. { c | ( # ` c ) = M } )
22		fveq2 6191	. . . . . . 7 |- ( b = C -> ( # ` b ) = ( # ` C ) )
23	22	eqeq1d 2624	. . . . . 6 |- ( b = C -> ( ( # ` b ) = M <-> ( # ` C ) = M ) )
24		fveq2 6191	. . . . . . . 8 |- ( c = b -> ( # ` c ) = ( # ` b ) )
25	24	eqeq1d 2624	. . . . . . 7 |- ( c = b -> ( ( # ` c ) = M <-> ( # ` b ) = M ) )
26	25	cbvabv 2747	. . . . . 6 |- { c | ( # ` c ) = M } = { b | ( # ` b ) = M }
27	23, 26	elab2g 3353	. . . . 5 |- ( C e. O -> ( C e. { c | ( # ` c ) = M } <-> ( # ` C ) = M ) )
28	21, 27	mpbid 222	. . . 4 |- ( C e. O -> ( # ` C ) = M )
29	18, 28	eqtrd 2656	. . 3 |- ( C e. O -> ( # ` ( ( 1 ... ( M + N ) ) i^i C ) ) = M )
30		fzfi 12771	. . . . 5 |- ( 1 ... ( M + N ) ) e. Fin
31		hashssdif 13200	. . . . 5 |- ( ( ( 1 ... ( M + N ) ) e. Fin /\ C C_ ( 1 ... ( M + N ) ) ) -> ( # ` ( ( 1 ... ( M + N ) ) \ C ) ) = ( ( # ` ( 1 ... ( M + N ) ) ) - ( # ` C ) ) )
32	30, 15, 31	sylancr 695	. . . 4 |- ( C e. O -> ( # ` ( ( 1 ... ( M + N ) ) \ C ) ) = ( ( # ` ( 1 ... ( M + N ) ) ) - ( # ` C ) ) )
33	8	nnnn0i 11300	. . . . . 6 |- ( M + N ) e. NN0
34		hashfz1 13134	. . . . . 6 |- ( ( M + N ) e. NN0 -> ( # ` ( 1 ... ( M + N ) ) ) = ( M + N ) )
35	33, 34	mp1i 13	. . . . 5 |- ( C e. O -> ( # ` ( 1 ... ( M + N ) ) ) = ( M + N ) )
36	35, 28	oveq12d 6668	. . . 4 |- ( C e. O -> ( ( # ` ( 1 ... ( M + N ) ) ) - ( # ` C ) ) = ( ( M + N ) - M ) )
37	1	nncni 11030	. . . . . 6 |- M e. CC
38	2	nncni 11030	. . . . . 6 |- N e. CC
39		pncan2 10288	. . . . . 6 |- ( ( M e. CC /\ N e. CC ) -> ( ( M + N ) - M ) = N )
40	37, 38, 39	mp2an 708	. . . . 5 |- ( ( M + N ) - M ) = N
41	40	a1i 11	. . . 4 |- ( C e. O -> ( ( M + N ) - M ) = N )
42	32, 36, 41	3eqtrd 2660	. . 3 |- ( C e. O -> ( # ` ( ( 1 ... ( M + N ) ) \ C ) ) = N )
43	29, 42	oveq12d 6668	. 2 |- ( C e. O -> ( ( # ` ( ( 1 ... ( M + N ) ) i^i C ) ) - ( # ` ( ( 1 ... ( M + N ) ) \ C ) ) ) = ( M - N ) )
44	11, 43	eqtrd 2656	1 |- ( C e. O -> ( ( F ` C ) ` ( M + N ) ) = ( M - N ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   1c1 9937   + caddc 9939   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by:  ballotlem5  30561