Theorem ballotlem4 30560

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
ballotth.o	|- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
ballotth.p	|- P = ( x e. ~P O |-> ( ( # ` 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
ballotlem4	|- ( 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 ballotlem4

Step	Hyp	Ref	Expression
1		ballotth.m	. . . . . . . 8 |- M e. NN
2		ballotth.n	. . . . . . . 8 |- N e. NN
3		nnaddcl 11042	. . . . . . . 8 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
4	1, 2, 3	mp2an 708	. . . . . . 7 |- ( M + N ) e. NN
5		elnnuz 11724	. . . . . . 7 |- ( ( M + N ) e. NN <-> ( M + N ) e. ( ZZ>= ` 1 ) )
6	4, 5	mpbi 220	. . . . . 6 |- ( M + N ) e. ( ZZ>= ` 1 )
7		eluzfz1 12348	. . . . . 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 10551	. . . . . . . . . 10 |- 0 <_ 1
10		0re 10040	. . . . . . . . . . 11 |- 0 e. RR
11		1re 10039	. . . . . . . . . . 11 |- 1 e. RR
12	10, 11	lenlti 10157	. . . . . . . . . 10 |- ( 0 <_ 1 <-> -. 1 < 0 )
13	9, 12	mpbi 220	. . . . . . . . 9 |- -. 1 < 0
14		ltsub13 10509	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 0 e. RR /\ 1 e. RR ) -> ( 0 < ( 0 - 1 ) <-> 1 < ( 0 - 0 ) ) )
15	10, 10, 11, 14	mp3an 1424	. . . . . . . . . 10 |- ( 0 < ( 0 - 1 ) <-> 1 < ( 0 - 0 ) )
16		0m0e0 11130	. . . . . . . . . . 11 |- ( 0 - 0 ) = 0
17	16	breq2i 4661	. . . . . . . . . 10 |- ( 1 < ( 0 - 0 ) <-> 1 < 0 )
18	15, 17	bitri 264	. . . . . . . . 9 |- ( 0 < ( 0 - 1 ) <-> 1 < 0 )
19	13, 18	mtbir 313	. . . . . . . 8 |- -. 0 < ( 0 - 1 )
20		1m1e0 11089	. . . . . . . . . . . 12 |- ( 1 - 1 ) = 0
21	20	fveq2i 6194	. . . . . . . . . . 11 |- ( ( F ` C ) ` ( 1 - 1 ) ) = ( ( F ` C ) ` 0 )
22		ballotth.o	. . . . . . . . . . . 12 |- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
23		ballotth.p	. . . . . . . . . . . 12 |- P = ( x e. ~P O |-> ( ( # ` 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	ballotlemfval0 30557	. . . . . . . . . . 11 |- ( C e. O -> ( ( F ` C ) ` 0 ) = 0 )
26	21, 25	syl5eq 2668	. . . . . . . . . 10 |- ( C e. O -> ( ( F ` C ) ` ( 1 - 1 ) ) = 0 )
27	26	oveq1d 6665	. . . . . . . . 9 |- ( C e. O -> ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) = ( 0 - 1 ) )
28	27	breq2d 4665	. . . . . . . 8 |- ( C e. O -> ( 0 < ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) <-> 0 < ( 0 - 1 ) ) )
29	19, 28	mtbiri 317	. . . . . . 7 |- ( C e. O -> -. 0 < ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) )
30	29	adantr 481	. . . . . 6 |- ( ( C e. O /\ -. 1 e. C ) -> -. 0 < ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) )
31		simpl 473	. . . . . . . . . . 11 |- ( ( C e. O /\ 1 e. ( 1 ... ( M + N ) ) ) -> C e. O )
32		1nn 11031	. . . . . . . . . . . 12 |- 1 e. NN
33	32	a1i 11	. . . . . . . . . . 11 |- ( ( C e. O /\ 1 e. ( 1 ... ( M + N ) ) ) -> 1 e. NN )
34	1, 2, 22, 23, 24, 31, 33	ballotlemfp1 30553	. . . . . . . . . 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 475	. . . . . . . . 9 |- ( ( C e. O /\ 1 e. ( 1 ... ( M + N ) ) ) -> ( -. 1 e. C -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) ) )
36	8, 35	mpan2 707	. . . . . . . 8 |- ( C e. O -> ( -. 1 e. C -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) ) )
37	36	imp 445	. . . . . . 7 |- ( ( C e. O /\ -. 1 e. C ) -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) )
38	37	breq2d 4665	. . . . . 6 |- ( ( C e. O /\ -. 1 e. C ) -> ( 0 < ( ( F ` C ) ` 1 ) <-> 0 < ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) ) )
39	30, 38	mtbird 315	. . . . 5 |- ( ( C e. O /\ -. 1 e. C ) -> -. 0 < ( ( F ` C ) ` 1 ) )
40		fveq2 6191	. . . . . . . 8 |- ( i = 1 -> ( ( F ` C ) ` i ) = ( ( F ` C ) ` 1 ) )
41	40	breq2d 4665	. . . . . . 7 |- ( i = 1 -> ( 0 < ( ( F ` C ) ` i ) <-> 0 < ( ( F ` C ) ` 1 ) ) )
42	41	notbid 308	. . . . . 6 |- ( i = 1 -> ( -. 0 < ( ( F ` C ) ` i ) <-> -. 0 < ( ( F ` C ) ` 1 ) ) )
43	42	rspcev 3309	. . . . 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 695	. . . 4 |- ( ( C e. O /\ -. 1 e. C ) -> E. i e. ( 1 ... ( M + N ) ) -. 0 < ( ( F ` C ) ` i ) )
45		rexnal 2995	. . . 4 |- ( E. i e. ( 1 ... ( M + N ) ) -. 0 < ( ( F ` C ) ` i ) <-> -. A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) )
46	44, 45	sylib 208	. . 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	ballotleme 30558	. . . 4 |- ( C e. E <-> ( C e. O /\ A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) ) )
49	48	simprbi 480	. . 3 |- ( C e. E -> A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) )
50	46, 49	nsyl 135	. 2 |- ( ( C e. O /\ -. 1 e. C ) -> -. C e. E )
51	50	ex 450	1 |- ( C e. O -> ( -. 1 e. C -> -. C e. E ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   \ cdif 3571   i^i cin 3573   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by:  ballotth  30599