Theorem ballotfilemcdc 13223

Description: Lemma for ballotfi . It is decidable whether a given integer is an element of a particular element of O. (Contributed by Jim Kingdon, 7-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 )
ballotfilemcdc.dc	|- ( ph -> K e. ZZ )

Assertion

Ref	Expression
ballotfilemcdc	|- ( ph -> DECID K e. C )

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

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

Proof of Theorem ballotfilemcdc

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2302	. . 3 |- ( w = (/) -> ( K e. w <-> K e. (/) ) )
2	1	dcbid 850	. 2 |- ( w = (/) -> (DECID K e. w <-> DECID K e. (/) ) )
3		eleq2 2302	. . 3 |- ( w = y -> ( K e. w <-> K e. y ) )
4	3	dcbid 850	. 2 |- ( w = y -> (DECID K e. w <-> DECID K e. y ) )
5		eleq2 2302	. . 3 |- ( w = ( y u. { z } ) -> ( K e. w <-> K e. ( y u. { z } ) ) )
6	5	dcbid 850	. 2 |- ( w = ( y u. { z } ) -> (DECID K e. w <-> DECID K e. ( y u. { z } ) ) )
7		eleq2 2302	. . 3 |- ( w = C -> ( K e. w <-> K e. C ) )
8	7	dcbid 850	. 2 |- ( w = C -> (DECID K e. w <-> DECID K e. C ) )
9		noel 3525	. . . . 5 |- -. K e. (/)
10	9	olci 744	. . . 4 |- ( K e. (/) \/ -. K e. (/) )
11		df-dc 847	. . . 4 |- (DECID K e. (/) <-> ( K e. (/) \/ -. K e. (/) ) )
12	10, 11	mpbir 146	. . 3 |- DECID K e. (/)
13	12	a1i 9	. 2 |- ( ph -> DECID K e. (/) )
14		simpr 110	. . . 4 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> DECID K e. y )
15		ballotfilemcdc.dc	. . . . . . 7 |- ( ph -> K e. ZZ )
16	15	ad3antrrr 496	. . . . . 6 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> K e. ZZ )
17		ballotfilemc.c	. . . . . . . . . . 11 |- ( ph -> C e. O )
18		ballotth.m	. . . . . . . . . . . 12 |- M e. NN
19		ballotth.n	. . . . . . . . . . . 12 |- N e. NN
20		ballotfilem.o	. . . . . . . . . . . 12 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
21	18, 19, 20	ballotfilemelo 13222	. . . . . . . . . . 11 |- ( C e. O <-> ( C C_ ( 1 ... ( M + N ) ) /\ C e. Fin /\ (♯ ` C ) = M ) )
22	17, 21	sylib 122	. . . . . . . . . 10 |- ( ph -> ( C C_ ( 1 ... ( M + N ) ) /\ C e. Fin /\ (♯ ` C ) = M ) )
23	22	simp1d 1040	. . . . . . . . 9 |- ( ph -> C C_ ( 1 ... ( M + N ) ) )
24	23	ad3antrrr 496	. . . . . . . 8 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> C C_ ( 1 ... ( M + N ) ) )
25		simplrr 542	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> z e. ( C \ y ) )
26	25	eldifad 3231	. . . . . . . 8 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> z e. C )
27	24, 26	sseldd 3249	. . . . . . 7 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> z e. ( 1 ... ( M + N ) ) )
28	27	elfzelzd 10429	. . . . . 6 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> z e. ZZ )
29		zdceq 9720	. . . . . 6 |- ( ( K e. ZZ /\ z e. ZZ ) -> DECID K = z )
30	16, 28, 29	syl2anc 415	. . . . 5 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> DECID K = z )
31		vex 2824	. . . . . . 7 |- z e. _V
32	31	elsn2 3743	. . . . . 6 |- ( K e. { z } <-> K = z )
33	32	dcbii 852	. . . . 5 |- (DECID K e. { z } <-> DECID K = z )
34	30, 33	sylibr 134	. . . 4 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> DECID K e. { z } )
35	14, 34	dcun 3637	. . 3 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> DECID K e. ( y u. { z } ) )
36	35	ex 115	. 2 |- ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) -> (DECID K e. y -> DECID K e. ( y u. { z } ) ) )
37	22	simp2d 1041	. 2 |- ( ph -> C e. Fin )
38	2, 4, 6, 8, 13, 36, 37	findcard2sd 7196	1 |- ( ph -> DECID K e. C )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   {crab 2532   \ cdif 3217   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520   ~Pcpw 3688   {csn 3709   ` cfv 5377  (class class class)co 6085   Fincfn 7022   1c1 8180   + caddc 8182   NNcn 9304   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-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-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-er 6807  df-en 7023  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

This theorem is used by:  ballotfilemcinfi  13224  ballotfilemdifcfi  13225  ballotfilemcinfz  13226  ballotfilemdifcfz  13227  ballotfilemafi  13238  ballotfilembfi  13239  ballotfilemic  13250  ballotfilemth  13281