Theorem ballotfilemimin 13193

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 10382	. . . . . . 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 10378	. . . . . . 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 13188	. . . . . . . . 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 10379	. . . . . . 7 |- ( C e. ( O \ E ) -> ( I ` C ) e. ZZ )
15		zltlem1 9652	. . . . . . 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 488	. . . . . 6 |- ( ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) /\ ( ( F ` C ) ` j ) = 0 ) -> ( I ` C ) e. ZZ )
20	19	zred 9718	. . . . 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 9718	. . . . 5 |- ( ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) /\ ( ( F ` C ) ` j ) = 0 ) -> j e. RR )
24		1zzd 9621	. . . . . . . . . . . . 13 |- ( C e. ( O \ E ) -> 1 e. ZZ )
25	14, 24	zsubcld 9723	. . . . . . . . . . . 12 |- ( C e. ( O \ E ) -> ( ( I ` C ) - 1 ) e. ZZ )
26	25	zred 9718	. . . . . . . . . . 11 |- ( C e. ( O \ E ) -> ( ( I ` C ) - 1 ) e. RR )
27		nnaddcl 9274	. . . . . . . . . . . . . 14 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
28	4, 5, 27	mp2an 426	. . . . . . . . . . . . 13 |- ( M + N ) e. NN
29	28	a1i 9	. . . . . . . . . . . 12 |- ( C e. ( O \ E ) -> ( M + N ) e. NN )
30	29	nnred 9267	. . . . . . . . . . 11 |- ( C e. ( O \ E ) -> ( M + N ) e. RR )
31		elfzle2 10382	. . . . . . . . . . . . 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 9717	. . . . . . . . . . . . 13 |- ( C e. ( O \ E ) -> ( M + N ) e. ZZ )
34		zlem1lt 9651	. . . . . . . . . . . . 13 |- ( ( ( I ` C ) e. ZZ /\ ( M + N ) e. ZZ ) -> ( ( I ` C ) <_ ( M + N ) <-> ( ( I ` C ) - 1 ) < ( M + N ) ) )
35	14, 33, 34	syl2anc 411	. . . . . . . . . . . 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 8408	. . . . . . . . . 10 |- ( C e. ( O \ E ) -> ( ( I ` C ) - 1 ) <_ ( M + N ) )
38		eluz 9885	. . . . . . . . . . 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 411	. . . . . . . . . 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 10419	. . . . . . . . 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 3241	. . . . . . 7 |- ( C e. ( O \ E ) -> ( j e. ( 1 ... ( ( I ` C ) - 1 ) ) -> j e. ( 1 ... ( M + N ) ) ) )
44		fveqeq2 5684	. . . . . . . . . 10 |- ( l = j -> ( ( ( F ` C ) ` l ) = 0 <-> ( ( F ` C ) ` j ) = 0 ) )
45	44	elrab 2976	. . . . . . . . 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 13187	. . . . . . . . . . . 12 |- ( C e. ( O \ E ) -> ( I ` C ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) )
47		fveqeq2 5684	. . . . . . . . . . . . . 14 |- ( l = k -> ( ( ( F ` C ) ` l ) = 0 <-> ( ( F ` C ) ` k ) = 0 ) )
48	47	cbvrabv 2814	. . . . . . . . . . . . 13 |- { l e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` l ) = 0 } = { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 }
49	48	infeq1i 7317	. . . . . . . . . . . 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 2285	. . . . . . . . . . 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 13192	. . . . . . . . . 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 4136	. . . . . . . . 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 8411	. . . 4 |- ( ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) /\ ( ( F ` C ) ` j ) = 0 ) -> -. j < ( I ` C ) )
59	18, 58	pm2.65da 667	. . 3 |- ( ( C e. ( O \ E ) /\ j e. ( 1 ... ( ( I ` C ) - 1 ) ) ) -> -. ( ( F ` C ) ` j ) = 0 )
60	59	nrexdv 2637	. 2 |- ( C e. ( O \ E ) -> -. E. j e. ( 1 ... ( ( I ` C ) - 1 ) ) ( ( F ` C ) ` j ) = 0 )
61		fveqeq2 5684	. . 3 |- ( j = k -> ( ( ( F ` C ) ` j ) = 0 <-> ( ( F ` C ) ` k ) = 0 ) )
62	61	cbvrexv 2781	. 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 683	1 |- ( C e. ( O \ E ) -> -. E. k e. ( 1 ... ( ( I ` C ) - 1 ) ) ( ( F ` C ) ` k ) = 0 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   {crab 2526   \ cdif 3211   i^i cin 3213   C_ wss 3214   ~Pcpw 3674   class class class wbr 4114   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058   Fincfn 6988  infcinf 7287   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   <_ cle 8325   - cmin 8460   / cdiv 8963   NNcn 9254   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361  ♯chash 11163

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 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

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-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-ihash 11164

This theorem is referenced by:  ballotfilemic  13194  ballotfilem1c  13195