Theorem ballotfilem4 13219

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 9303	. . . . . . . 8 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
4	1, 2, 3	mp2an 430	. . . . . . 7 |- ( M + N ) e. NN
5		elnnuz 9938	. . . . . . 7 |- ( ( M + N ) e. NN <-> ( M + N ) e. ( ZZ>= ` 1 ) )
6	4, 5	mpbi 145	. . . . . 6 |- ( M + N ) e. ( ZZ>= ` 1 )
7		eluzfz1 10414	. . . . . 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 8799	. . . . . . . . . 10 |- 0 <_ 1
10		0re 8316	. . . . . . . . . . 11 |- 0 e. RR
11		1re 8315	. . . . . . . . . . 11 |- 1 e. RR
12	10, 11	lenlti 8416	. . . . . . . . . 10 |- ( 0 <_ 1 <-> -. 1 < 0 )
13	9, 12	mpbi 145	. . . . . . . . 9 |- -. 1 < 0
14		ltsub13 8761	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 0 e. RR /\ 1 e. RR ) -> ( 0 < ( 0 - 1 ) <-> 1 < ( 0 - 0 ) ) )
15	10, 10, 11, 14	mp3an 1378	. . . . . . . . . 10 |- ( 0 < ( 0 - 1 ) <-> 1 < ( 0 - 0 ) )
16		0m0e0 9395	. . . . . . . . . . 11 |- ( 0 - 0 ) = 0
17	16	breq2i 4133	. . . . . . . . . 10 |- ( 1 < ( 0 - 0 ) <-> 1 < 0 )
18	15, 17	bitri 184	. . . . . . . . 9 |- ( 0 < ( 0 - 1 ) <-> 1 < 0 )
19	13, 18	mtbir 682	. . . . . . . 8 |- -. 0 < ( 0 - 1 )
20		1m1e0 9352	. . . . . . . . . . . 12 |- ( 1 - 1 ) = 0
21	20	fveq2i 5693	. . . . . . . . . . 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 13213	. . . . . . . . . . 11 |- ( C e. O -> ( ( F ` C ) ` 0 ) = 0 )
26	21, 25	eqtrid 2283	. . . . . . . . . 10 |- ( C e. O -> ( ( F ` C ) ` ( 1 - 1 ) ) = 0 )
27	26	oveq1d 6090	. . . . . . . . 9 |- ( C e. O -> ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) = ( 0 - 1 ) )
28	27	breq2d 4137	. . . . . . . 8 |- ( C e. O -> ( 0 < ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) <-> 0 < ( 0 - 1 ) ) )
29	19, 28	mtbiri 686	. . . . . . 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 9294	. . . . . . . . . . . 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 13209	. . . . . . . . . 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 429	. . . . . . . 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 4137	. . . . . 6 |- ( ( C e. O /\ -. 1 e. C ) -> ( 0 < ( ( F ` C ) ` 1 ) <-> 0 < ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) ) )
39	30, 38	mtbird 684	. . . . 5 |- ( ( C e. O /\ -. 1 e. C ) -> -. 0 < ( ( F ` C ) ` 1 ) )
40		fveq2 5690	. . . . . . . 8 |- ( i = 1 -> ( ( F ` C ) ` i ) = ( ( F ` C ) ` 1 ) )
41	40	breq2d 4137	. . . . . . 7 |- ( i = 1 -> ( 0 < ( ( F ` C ) ` i ) <-> 0 < ( ( F ` C ) ` 1 ) ) )
42	41	notbid 677	. . . . . 6 |- ( i = 1 -> ( -. 0 < ( ( F ` C ) ` i ) <-> -. 0 < ( ( F ` C ) ` 1 ) ) )
43	42	rspcev 2929	. . . . 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 418	. . . 4 |- ( ( C e. O /\ -. 1 e. C ) -> E. i e. ( 1 ... ( M + N ) ) -. 0 < ( ( F ` C ) ` i ) )
45		rexnalim 2539	. . . 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 13214	. . . 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 637	. 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 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   \ cdif 3217   i^i cin 3219   ~Pcpw 3685   class class class wbr 4125   |-> cmpt 4187   ` cfv 5372  (class class class)co 6075   Fincfn 7012   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   < clt 8350   <_ cle 8351   - cmin 8487   / cdiv 8992   NNcn 9283   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390  ♯chash 11192

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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-ihash 11193

This theorem is referenced by:  ballotfilemth  13259