Theorem ballotfilemimin 13251

Description: ( I ` C ) is the first tie. (Contributed by Thierry Arnoux, 1-Dec-2016.) (Revised by AV, 6-Oct-2020.)

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 ) ) ) ) )
ballotth.e	|- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
ballotth.mgtn	|- N < M
ballotth.i	|- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )

Assertion

Ref	Expression
ballotfilemimin	|- ( C e. ( O \ E ) -> -. E. k e. ( 1 ... ( ( I ` C ) - 1 ) ) ( ( F ` C ) ` k ) = 0 )

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

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

Proof of Theorem ballotfilemimin

Dummy variables j l are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfzle2 10434	. . . . . . 7 |- ( j e. ( 1 ... ( ( I ` C ) - 1 ) ) -> j <_ ( ( I ` C ) - 1 ) )
2	1	adantl 277	. . . . . 6 |- ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) -> j <_ ( ( I ` C ) - 1 ) )
3		elfzelz 10430	. . . . . . 7 |- ( j e. ( 1 ... ( ( I ` C ) - 1 ) ) -> j e. ZZ )
4		ballotth.m	. . . . . . . . . 10 |- M e. NN
5		ballotth.n	. . . . . . . . . 10 |- N e. NN
6		ballotfilem.o	. . . . . . . . . 10 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
7		ballotfilem.p	. . . . . . . . . 10 |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
8		ballotth.f	. . . . . . . . . 10 |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
9		ballotth.e	. . . . . . . . . 10 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
10		ballotth.mgtn	. . . . . . . . . 10 |- N < M
11		ballotth.i	. . . . . . . . . 10 |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
12	4, 5, 6, 7, 8, 9, 10, 11	ballotfilemiex 13246	. . . . . . . . 9 |- ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
13	12	simpld 112	. . . . . . . 8 |- ( C e. ( O \ E ) -> ( I ` C ) e. ( 1 ... ( M + N ) ) )
14	13	elfzelzd 10431	. . . . . . 7 |- ( C e. ( O \ E ) -> ( I ` C ) e. ZZ )
15		zltlem1 9704	. . . . . . 7 |- ( ( j e. ZZ /\ ( I ` C ) e. ZZ ) -> ( j < ( I ` C ) <-> j <_ ( ( I ` C ) - 1 ) ) )
16	3, 14, 15	syl2anr 290	. . . . . 6 |- ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) -> ( j < ( I ` C ) <-> j <_ ( ( I ` C ) - 1 ) ) )
17	2, 16	mpbird 167	. . . . 5 |- ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) -> j < ( I ` C ) )
18	17	adantr 276	. . . 4 |- ( ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) /\ ( ( F ` C ) ` j ) = 0 ) -> j < ( I ` C ) )
19	14	ad2antrr 492	. . . . . 6 |- ( ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) /\ ( ( F ` C ) ` j ) = 0 ) -> ( I ` C ) e. ZZ )
20	19	zred 9770	. . . . 5 |- ( ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) /\ ( ( F ` C ) ` j ) = 0 ) -> ( I ` C ) e. RR )
21	3	adantl 277	. . . . . . 7 |- ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) -> j e. ZZ )
22	21	adantr 276	. . . . . 6 |- ( ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) /\ ( ( F ` C ) ` j ) = 0 ) -> j e. ZZ )
23	22	zred 9770	. . . . 5 |- ( ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) /\ ( ( F ` C ) ` j ) = 0 ) -> j e. RR )
24		1zzd 9673	. . . . . . . . . . . . 13 |- ( C e. ( O \ E ) -> 1 e. ZZ )
25	14, 24	zsubcld 9775	. . . . . . . . . . . 12 |- ( C e. ( O \ E ) -> ( ( I ` C ) - 1 ) e. ZZ )
26	25	zred 9770	. . . . . . . . . . 11 |- ( C e. ( O \ E ) -> ( ( I ` C ) - 1 ) e. RR )
27		nnaddcl 9325	. . . . . . . . . . . . . 14 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
28	4, 5, 27	mp2an 430	. . . . . . . . . . . . 13 |- ( M + N ) e. NN
29	28	a1i 9	. . . . . . . . . . . 12 |- ( C e. ( O \ E ) -> ( M + N ) e. NN )
30	29	nnred 9318	. . . . . . . . . . 11 |- ( C e. ( O \ E ) -> ( M + N ) e. RR )
31		elfzle2 10434	. . . . . . . . . . . . 13 |- ( ( I ` C ) e. ( 1 ... ( M + N ) ) -> ( I ` C ) <_ ( M + N ) )
32	13, 31	syl 14	. . . . . . . . . . . 12 |- ( C e. ( O \ E ) -> ( I ` C ) <_ ( M + N ) )
33	29	nnzd 9769	. . . . . . . . . . . . 13 |- ( C e. ( O \ E ) -> ( M + N ) e. ZZ )
34		zlem1lt 9703	. . . . . . . . . . . . 13 |- ( ( ( I ` C ) e. ZZ /\ ( M + N ) e. ZZ ) -> ( ( I ` C ) <_ ( M + N ) <-> ( ( I ` C ) - 1 ) < ( M + N ) ) )
35	14, 33, 34	syl2anc 415	. . . . . . . . . . . 12 |- ( C e. ( O \ E ) -> ( ( I ` C ) <_ ( M + N ) <-> ( ( I ` C ) - 1 ) < ( M + N ) ) )
36	32, 35	mpbid 147	. . . . . . . . . . 11 |- ( C e. ( O \ E ) -> ( ( I ` C ) - 1 ) < ( M + N ) )
37	26, 30, 36	ltled 8445	. . . . . . . . . 10 |- ( C e. ( O \ E ) -> ( ( I ` C ) - 1 ) <_ ( M + N ) )
38		eluz 9937	. . . . . . . . . . 11 |- ( ( ( ( I ` C ) - 1 ) e. ZZ /\ ( M + N ) e. ZZ ) -> ( ( M + N ) e. ( ZZ>= ` ( ( I ` C ) - 1 ) ) <-> ( ( I ` C ) - 1 ) <_ ( M + N ) ) )
39	25, 33, 38	syl2anc 415	. . . . . . . . . 10 |- ( C e. ( O \ E ) -> ( ( M + N ) e. ( ZZ>= ` ( ( I ` C ) - 1 ) ) <-> ( ( I ` C ) - 1 ) <_ ( M + N ) ) )
40	37, 39	mpbird 167	. . . . . . . . 9 |- ( C e. ( O \ E ) -> ( M + N ) e. ( ZZ>= ` ( ( I ` C ) - 1 ) ) )
41		fzss2 10472	. . . . . . . . 9 |- ( ( M + N ) e. ( ZZ>= ` ( ( I ` C ) - 1 ) ) -> ( 1 ... ( ( I ` C ) - 1 ) ) C_ ( 1 ... ( M + N ) ) )
42	40, 41	syl 14	. . . . . . . 8 |- ( C e. ( O \ E ) -> ( 1 ... ( ( I ` C ) - 1 ) ) C_ ( 1 ... ( M + N ) ) )
43	42	sseld 3247	. . . . . . 7 |- ( C e. ( O \ E ) -> ( j e. ( 1 ... ( ( I ` C ) - 1 ) ) -> j e. ( 1 ... ( M + N ) ) ) )
44		fveqeq2 5704	. . . . . . . . . 10 |- ( l = j -> ( ( ( F ` C ) ` l ) = 0 <-> ( ( F ` C ) ` j ) = 0 ) )
45	44	elrab 2982	. . . . . . . . 9 |- ( j e. { l e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` l ) = 0 } <-> ( j e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` j ) = 0 ) )
46	4, 5, 6, 7, 8, 9, 10, 11	ballotfilemi 13245	. . . . . . . . . . . 12 |- ( C e. ( O \ E ) -> ( I ` C ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) )
47		fveqeq2 5704	. . . . . . . . . . . . . 14 |- ( l = k -> ( ( ( F ` C ) ` l ) = 0 <-> ( ( F ` C ) ` k ) = 0 ) )
48	47	cbvrabv 2820	. . . . . . . . . . . . 13 |- { l e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` l ) = 0 } = { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 }
49	48	infeq1i 7353	. . . . . . . . . . . 12 |- inf ( { l e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` l ) = 0 } , RR , < ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < )
50	46, 49	eqtr4di 2289	. . . . . . . . . . 11 |- ( C e. ( O \ E ) -> ( I ` C ) = inf ( { l e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` l ) = 0 } , RR , < ) )
51	50	adantr 276	. . . . . . . . . 10 |- ( ( C e. ( O \ E ) /\ j e. { l e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` l ) = 0 } ) -> ( I ` C ) = inf ( { l e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` l ) = 0 } , RR , < ) )
52	4, 5, 6, 7, 8, 9, 10, 11, 48	ballotfilemsle 13250	. . . . . . . . . 10 |- ( ( C e. ( O \ E ) /\ j e. { l e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` l ) = 0 } ) -> inf ( { l e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` l ) = 0 } , RR , < ) <_ j )
53	51, 52	eqbrtrd 4152	. . . . . . . . 9 |- ( ( C e. ( O \ E ) /\ j e. { l e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` l ) = 0 } ) -> ( I ` C ) <_ j )
54	45, 53	sylan2br 288	. . . . . . . 8 |- ( ( C e. ( O \ E ) /\ ( j e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` j ) = 0 ) ) -> ( I ` C ) <_ j )
55	54	ex 115	. . . . . . 7 |- ( C e. ( O \ E ) -> ( ( j e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` j ) = 0 ) -> ( I ` C ) <_ j ) )
56	43, 55	syland 293	. . . . . 6 |- ( C e. ( O \ E ) -> ( ( j e. ( 1 ... ( ( I ` C ) - 1 ) ) /\ ( ( F ` C ) ` j ) = 0 ) -> ( I ` C ) <_ j ) )
57	56	impl 380	. . . . 5 |- ( ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) /\ ( ( F ` C ) ` j ) = 0 ) -> ( I ` C ) <_ j )
58	20, 23, 57	lensymd 8448	. . . 4 |- ( ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) /\ ( ( F ` C ) ` j ) = 0 ) -> -. j < ( I ` C ) )
59	18, 58	pm2.65da 671	. . 3 |- ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) -> -. ( ( F ` C ) ` j ) = 0 )
60	59	nrexdv 2643	. 2 |- ( C e. ( O \ E ) -> -. E. j e. ( 1 ... ( ( I ` C ) - 1 ) ) ( ( F ` C ) ` j ) = 0 )
61		fveqeq2 5704	. . 3 |- ( j = k -> ( ( ( F ` C ) ` j ) = 0 <-> ( ( F ` C ) ` k ) = 0 ) )
62	61	cbvrexv 2787	. 2 |- ( E. j e. ( 1 ... ( ( I ` C ) - 1 ) ) ( ( F ` C ) ` j ) = 0 <-> E. k e. ( 1 ... ( ( I ` C ) - 1 ) ) ( ( F ` C ) ` k ) = 0 )
63	60, 62	sylnib 687	1 |- ( C e. ( O \ E ) -> -. E. k e. ( 1 ... ( ( I ` C ) - 1 ) ) ( ( F ` C ) ` k ) = 0 )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   \ cdif 3217   i^i cin 3219   C_ wss 3220   ~Pcpw 3688   class class class wbr 4130   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   Fincfn 7022  infcinf 7323   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   < clt 8360   <_ cle 8361   - cmin 8497   / cdiv 9003   NNcn 9305   ZZcz 9646   ZZ>=cuz 9923   ...cfz 10413  ♯chash 11216

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-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

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-rmo 2536  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-po 4441  df-iso 4442  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-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-fz 10414  df-fzo 10552  df-ihash 11217

This theorem is used by:  ballotfilemic  13252  ballotfilem1c  13253