Theorem ballotfilemi1 13247

Description: The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 12-Mar-2017.)

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
ballotfilemi1	|- ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( I ` C ) =/= 1 )

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 ballotfilemi1

Step	Hyp	Ref	Expression
1		0re 8326	. . . . . . 7 |- 0 e. RR
2		1re 8325	. . . . . . 7 |- 1 e. RR
3	1, 2	resubcli 8589	. . . . . 6 |- ( 0 - 1 ) e. RR
4		0lt1 8453	. . . . . . 7 |- 0 < 1
5		ltsub23 8770	. . . . . . . . 9 |- ( ( 0 e. RR /\ 1 e. RR /\ 0 e. RR ) -> ( ( 0 - 1 ) < 0 <-> ( 0 - 0 ) < 1 ) )
6	1, 2, 1, 5	mp3an 1378	. . . . . . . 8 |- ( ( 0 - 1 ) < 0 <-> ( 0 - 0 ) < 1 )
7		0m0e0 9417	. . . . . . . . 9 |- ( 0 - 0 ) = 0
8	7	breq1i 4137	. . . . . . . 8 |- ( ( 0 - 0 ) < 1 <-> 0 < 1 )
9	6, 8	bitr2i 185	. . . . . . 7 |- ( 0 < 1 <-> ( 0 - 1 ) < 0 )
10	4, 9	mpbi 145	. . . . . 6 |- ( 0 - 1 ) < 0
11	3, 10	gtneii 8421	. . . . 5 |- 0 =/= ( 0 - 1 )
12	11	nesymi 2466	. . . 4 |- -. ( 0 - 1 ) = 0
13		ballotth.m	. . . . . . . . 9 |- M e. NN
14		ballotth.n	. . . . . . . . 9 |- N e. NN
15		ballotfilem.o	. . . . . . . . 9 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
16		ballotfilem.p	. . . . . . . . 9 |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
17		ballotth.f	. . . . . . . . 9 |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
18		eldifi 3351	. . . . . . . . 9 |- ( C e. ( O \ E ) -> C e. O )
19		1nn 9316	. . . . . . . . . 10 |- 1 e. NN
20	19	a1i 9	. . . . . . . . 9 |- ( C e. ( O \ E ) -> 1 e. NN )
21	13, 14, 15, 16, 17, 18, 20	ballotfilemfp1 13233	. . . . . . . 8 |- ( C e. ( O \ E ) -> ( ( -. 1 e. C -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) ) /\ ( 1 e. C -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) + 1 ) ) ) )
22	21	simpld 112	. . . . . . 7 |- ( C e. ( O \ E ) -> ( -. 1 e. C -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) ) )
23	22	imp 124	. . . . . 6 |- ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) )
24		1m1e0 9374	. . . . . . . . 9 |- ( 1 - 1 ) = 0
25	24	fveq2i 5698	. . . . . . . 8 |- ( ( F ` C ) ` ( 1 - 1 ) ) = ( ( F ` C ) ` 0 )
26	25	oveq1i 6095	. . . . . . 7 |- ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) = ( ( ( F ` C ) ` 0 ) - 1 )
27	26	a1i 9	. . . . . 6 |- ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) = ( ( ( F ` C ) ` 0 ) - 1 ) )
28	13, 14, 15, 16, 17	ballotfilemfval0 13237	. . . . . . . . 9 |- ( C e. O -> ( ( F ` C ) ` 0 ) = 0 )
29	18, 28	syl 14	. . . . . . . 8 |- ( C e. ( O \ E ) -> ( ( F ` C ) ` 0 ) = 0 )
30	29	adantr 276	. . . . . . 7 |- ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( ( F ` C ) ` 0 ) = 0 )
31	30	oveq1d 6100	. . . . . 6 |- ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( ( ( F ` C ) ` 0 ) - 1 ) = ( 0 - 1 ) )
32	23, 27, 31	3eqtrrd 2276	. . . . 5 |- ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( 0 - 1 ) = ( ( F ` C ) ` 1 ) )
33	32	eqeq1d 2247	. . . 4 |- ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( ( 0 - 1 ) = 0 <-> ( ( F ` C ) ` 1 ) = 0 ) )
34	12, 33	mtbii 685	. . 3 |- ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> -. ( ( F ` C ) ` 1 ) = 0 )
35		ballotth.e	. . . . . . 7 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
36		ballotth.mgtn	. . . . . . 7 |- N < M
37		ballotth.i	. . . . . . 7 |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
38	13, 14, 15, 16, 17, 35, 36, 37	ballotfilemiex 13246	. . . . . 6 |- ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
39	38	simprd 114	. . . . 5 |- ( C e. ( O \ E ) -> ( ( F ` C ) ` ( I ` C ) ) = 0 )
40	39	ad2antrr 492	. . . 4 |- ( ( ( C e. ( O \ E ) /\ -. 1 e. C ) /\ ( I ` C ) = 1 ) -> ( ( F ` C ) ` ( I ` C ) ) = 0 )
41		fveqeq2 5704	. . . . 5 |- ( ( I ` C ) = 1 -> ( ( ( F ` C ) ` ( I ` C ) ) = 0 <-> ( ( F ` C ) ` 1 ) = 0 ) )
42	41	adantl 277	. . . 4 |- ( ( ( C e. ( O \ E ) /\ -. 1 e. C ) /\ ( I ` C ) = 1 ) -> ( ( ( F ` C ) ` ( I ` C ) ) = 0 <-> ( ( F ` C ) ` 1 ) = 0 ) )
43	40, 42	mpbid 147	. . 3 |- ( ( ( C e. ( O \ E ) /\ -. 1 e. C ) /\ ( I ` C ) = 1 ) -> ( ( F ` C ) ` 1 ) = 0 )
44	34, 43	mtand 675	. 2 |- ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> -. ( I ` C ) = 1 )
45	44	neqned 2427	1 |- ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( I ` C ) =/= 1 )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   {crab 2532   \ cdif 3217   i^i cin 3219   ~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   - cmin 8497   / cdiv 9003   NNcn 9305   ZZcz 9646   ...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