Theorem ballotfilemdifcfz 13210

Description: Lemma for ballotfi . The portion of a counting representing votes for B within a specified integer range is finite. (Contributed by Jim Kingdon, 15-Jun-2026.)

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 }
ballotfilemc.c	|- ( ph -> C e. O )
ballotfilemc.j	|- ( ph -> J e. ZZ )
ballotfilemc.k	|- ( ph -> K e. ZZ )

Assertion

Ref	Expression
ballotfilemdifcfz	|- ( ph -> ( ( J ... K ) \ C ) e. Fin )

Distinct variable groups:   M, c   N, c   O, c

Allowed substitution hints:   ph( c)   C( c)   J( c)   K( c)

Proof of Theorem ballotfilemdifcfz

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotfilemc.j	. . 3 |- ( ph -> J e. ZZ )
2		ballotfilemc.k	. . 3 |- ( ph -> K e. ZZ )
3	1, 2	fzfigd 10851	. 2 |- ( ph -> ( J ... K ) e. Fin )
4		difssd 3356	. 2 |- ( ph -> ( ( J ... K ) \ C ) C_ ( J ... K ) )
5		ballotth.m	. . . . . . 7 |- M e. NN
6		ballotth.n	. . . . . . 7 |- N e. NN
7		ballotfilem.o	. . . . . . 7 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
8		ballotfilemc.c	. . . . . . . 8 |- ( ph -> C e. O )
9	8	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. ( J ... K ) ) -> C e. O )
10		elfzelz 10411	. . . . . . . 8 |- ( x e. ( J ... K ) -> x e. ZZ )
11	10	adantl 277	. . . . . . 7 |- ( ( ph /\ x e. ( J ... K ) ) -> x e. ZZ )
12	5, 6, 7, 9, 11	ballotfilemcdc 13206	. . . . . 6 |- ( ( ph /\ x e. ( J ... K ) ) -> DECID x e. C )
13		dcn 854	. . . . . 6 |- (DECID x e. C -> DECID -. x e. C )
14	12, 13	syl 14	. . . . 5 |- ( ( ph /\ x e. ( J ... K ) ) -> DECID -. x e. C )
15		ibar 301	. . . . . . 7 |- ( x e. ( J ... K ) -> ( -. x e. C <-> ( x e. ( J ... K ) /\ -. x e. C ) ) )
16	15	dcbid 850	. . . . . 6 |- ( x e. ( J ... K ) -> (DECID -. x e. C <-> DECID ( x e. ( J ... K ) /\ -. x e. C ) ) )
17	16	adantl 277	. . . . 5 |- ( ( ph /\ x e. ( J ... K ) ) -> (DECID -. x e. C <-> DECID ( x e. ( J ... K ) /\ -. x e. C ) ) )
18	14, 17	mpbid 147	. . . 4 |- ( ( ph /\ x e. ( J ... K ) ) -> DECID ( x e. ( J ... K ) /\ -. x e. C ) )
19		eldif 3229	. . . . 5 |- ( x e. ( ( J ... K ) \ C ) <-> ( x e. ( J ... K ) /\ -. x e. C ) )
20	19	dcbii 852	. . . 4 |- (DECID x e. ( ( J ... K ) \ C ) <-> DECID ( x e. ( J ... K ) /\ -. x e. C ) )
21	18, 20	sylibr 134	. . 3 |- ( ( ph /\ x e. ( J ... K ) ) -> DECID x e. ( ( J ... K ) \ C ) )
22	21	ralrimiva 2623	. 2 |- ( ph -> A. x e. ( J ... K )DECID x e. ( ( J ... K ) \ C ) )
23		ssfidc 7239	. 2 |- ( ( ( J ... K ) e. Fin /\ ( ( J ... K ) \ C ) C_ ( J ... K ) /\ A. x e. ( J ... K )DECID x e. ( ( J ... K ) \ C ) ) -> ( ( J ... K ) \ C ) e. Fin )
24	3, 4, 22, 23	syl3anc 1278	1 |- ( ph -> ( ( J ... K ) \ C ) e. Fin )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   \ cdif 3217   i^i cin 3219   C_ wss 3220   ~Pcpw 3688   ` cfv 5375  (class class class)co 6079   Fincfn 7016   1c1 8174   + caddc 8176   NNcn 9287   ZZcz 9627   ...cfz 10394  ♯chash 11197

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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289

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 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-1o 6681  df-er 6801  df-en 7017  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-fz 10395

This theorem is referenced by:  ballotfilemgval  13250  ballotfilemgun  13251