Theorem ballotfilem4 13190

Description: If the first pick is a vote for B, A is not ahead throughout the count. (Contributed by Thierry Arnoux, 25-Nov-2016.)

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 ) }

Assertion

Ref	Expression
ballotfilem4	|- ( C e. O -> ( -. 1 e. C -> -. C e. E ) )

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)   E( x, i, c)   F( x)   M( x)   N( x)   O( x)

Proof of Theorem ballotfilem4

Step	Hyp	Ref	Expression
1		ballotth.m	. . . . . . . 8 |- M e. NN
2		ballotth.n	. . . . . . . 8 |- N e. NN
3		nnaddcl 9278	. . . . . . . 8 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
4	1, 2, 3	mp2an 426	. . . . . . 7 |- ( M + N ) e. NN
5		elnnuz 9913	. . . . . . 7 |- ( ( M + N ) e. NN <-> ( M + N ) e. ( ZZ>= ` 1 ) )
6	4, 5	mpbi 145	. . . . . 6 |- ( M + N ) e. ( ZZ>= ` 1 )
7		eluzfz1 10389	. . . . . 6 |- ( ( M + N ) e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... ( M + N ) ) )
8	6, 7	ax-mp 5	. . . . 5 |- 1 e. ( 1 ... ( M + N ) )
9		0le1 8774	. . . . . . . . . 10 |- 0 <_ 1
10		0re 8291	. . . . . . . . . . 11 |- 0 e. RR
11		1re 8290	. . . . . . . . . . 11 |- 1 e. RR
12	10, 11	lenlti 8391	. . . . . . . . . 10 |- ( 0 <_ 1 <-> -. 1 < 0 )
13	9, 12	mpbi 145	. . . . . . . . 9 |- -. 1 < 0
14		ltsub13 8736	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 0 e. RR /\ 1 e. RR ) -> ( 0 < ( 0 - 1 ) <-> 1 < ( 0 - 0 ) ) )
15	10, 10, 11, 14	mp3an 1374	. . . . . . . . . 10 |- ( 0 < ( 0 - 1 ) <-> 1 < ( 0 - 0 ) )
16		0m0e0 9370	. . . . . . . . . . 11 |- ( 0 - 0 ) = 0
17	16	breq2i 4123	. . . . . . . . . 10 |- ( 1 < ( 0 - 0 ) <-> 1 < 0 )
18	15, 17	bitri 184	. . . . . . . . 9 |- ( 0 < ( 0 - 1 ) <-> 1 < 0 )
19	13, 18	mtbir 678	. . . . . . . 8 |- -. 0 < ( 0 - 1 )
20		1m1e0 9327	. . . . . . . . . . . 12 |- ( 1 - 1 ) = 0
21	20	fveq2i 5679	. . . . . . . . . . 11 |- ( ( F ` C ) ` ( 1 - 1 ) ) = ( ( F ` C ) ` 0 )
22		ballotfilem.o	. . . . . . . . . . . 12 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
23		ballotfilem.p	. . . . . . . . . . . 12 |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
24		ballotth.f	. . . . . . . . . . . 12 |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
25	1, 2, 22, 23, 24	ballotfilemfval0 13184	. . . . . . . . . . 11 |- ( C e. O -> ( ( F ` C ) ` 0 ) = 0 )
26	21, 25	eqtrid 2279	. . . . . . . . . 10 |- ( C e. O -> ( ( F ` C ) ` ( 1 - 1 ) ) = 0 )
27	26	oveq1d 6074	. . . . . . . . 9 |- ( C e. O -> ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) = ( 0 - 1 ) )
28	27	breq2d 4127	. . . . . . . 8 |- ( C e. O -> ( 0 < ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) <-> 0 < ( 0 - 1 ) ) )
29	19, 28	mtbiri 682	. . . . . . 7 |- ( C e. O -> -. 0 < ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) )
30	29	adantr 276	. . . . . 6 |- ( ( C e. O /\ -. 1 e. C ) -> -. 0 < ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) )
31		simpl 109	. . . . . . . . . . 11 |- ( ( C e. O /\ 1 e. ( 1 ... ( M + N ) ) ) -> C e. O )
32		1nn 9269	. . . . . . . . . . . 12 |- 1 e. NN
33	32	a1i 9	. . . . . . . . . . 11 |- ( ( C e. O /\ 1 e. ( 1 ... ( M + N ) ) ) -> 1 e. NN )
34	1, 2, 22, 23, 24, 31, 33	ballotfilemfp1 13180	. . . . . . . . . 10 |- ( ( C e. O /\ 1 e. ( 1 ... ( M + N ) ) ) -> ( ( -. 1 e. C -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) ) /\ ( 1 e. C -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) + 1 ) ) ) )
35	34	simpld 112	. . . . . . . . 9 |- ( ( C e. O /\ 1 e. ( 1 ... ( M + N ) ) ) -> ( -. 1 e. C -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) ) )
36	8, 35	mpan2 425	. . . . . . . 8 |- ( C e. O -> ( -. 1 e. C -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) ) )
37	36	imp 124	. . . . . . 7 |- ( ( C e. O /\ -. 1 e. C ) -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) )
38	37	breq2d 4127	. . . . . 6 |- ( ( C e. O /\ -. 1 e. C ) -> ( 0 < ( ( F ` C ) ` 1 ) <-> 0 < ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) ) )
39	30, 38	mtbird 680	. . . . 5 |- ( ( C e. O /\ -. 1 e. C ) -> -. 0 < ( ( F ` C ) ` 1 ) )
40		fveq2 5676	. . . . . . . 8 |- ( i = 1 -> ( ( F ` C ) ` i ) = ( ( F ` C ) ` 1 ) )
41	40	breq2d 4127	. . . . . . 7 |- ( i = 1 -> ( 0 < ( ( F ` C ) ` i ) <-> 0 < ( ( F ` C ) ` 1 ) ) )
42	41	notbid 673	. . . . . 6 |- ( i = 1 -> ( -. 0 < ( ( F ` C ) ` i ) <-> -. 0 < ( ( F ` C ) ` 1 ) ) )
43	42	rspcev 2923	. . . . 5 |- ( ( 1 e. ( 1 ... ( M + N ) ) /\ -. 0 < ( ( F ` C ) ` 1 ) ) -> E. i e. ( 1 ... ( M + N ) ) -. 0 < ( ( F ` C ) ` i ) )
44	8, 39, 43	sylancr 414	. . . 4 |- ( ( C e. O /\ -. 1 e. C ) -> E. i e. ( 1 ... ( M + N ) ) -. 0 < ( ( F ` C ) ` i ) )
45		rexnalim 2533	. . . 4 |- ( E. i e. ( 1 ... ( M + N ) ) -. 0 < ( ( F ` C ) ` i ) -> -. A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) )
46	44, 45	syl 14	. . 3 |- ( ( C e. O /\ -. 1 e. C ) -> -. A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) )
47		ballotth.e	. . . . 5 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
48	1, 2, 22, 23, 24, 47	ballotfileme 13185	. . . 4 |- ( C e. E <-> ( C e. O /\ A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) ) )
49	48	simprbi 275	. . 3 |- ( C e. E -> A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) )
50	46, 49	nsyl 633	. 2 |- ( ( C e. O /\ -. 1 e. C ) -> -. C e. E )
51	50	ex 115	1 |- ( C e. O -> ( -. 1 e. C -> -. C e. E ) )

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   ~Pcpw 3675   class class class wbr 4115   |-> cmpt 4177   ` cfv 5358  (class class class)co 6059   Fincfn 6989   RRcr 8143   0cc0 8144   1c1 8145   + caddc 8147   < clt 8325   <_ cle 8326   - cmin 8462   / cdiv 8967   NNcn 9258   ZZcz 9598   ZZ>=cuz 9875   ...cfz 10365  ♯chash 11167

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 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-iinf 4716  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-addcom 8244  ax-addass 8246  ax-distr 8248  ax-i2m1 8249  ax-0lt1 8250  ax-0id 8252  ax-rnegex 8253  ax-cnre 8255  ax-pre-ltirr 8256  ax-pre-ltwlin 8257  ax-pre-lttrn 8258  ax-pre-apti 8259  ax-pre-ltadd 8260

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-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 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4719  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1st 6348  df-2nd 6349  df-recs 6550  df-irdg 6615  df-frec 6636  df-1o 6661  df-oadd 6665  df-er 6781  df-en 6990  df-dom 6991  df-fin 6992  df-pnf 8327  df-mnf 8328  df-xr 8329  df-ltxr 8330  df-le 8331  df-sub 8464  df-neg 8465  df-inn 9259  df-n0 9518  df-z 9599  df-uz 9876  df-fz 10366  df-ihash 11168

This theorem is referenced by:  ballotfilemth  13230