Theorem ballotfilemi1 13304

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 8333	. . . . . . 7 |- 0 e. RR
2		1re 8332	. . . . . . 7 |- 1 e. RR
3	1, 2	resubcli 8597	. . . . . 6 |- ( 0 - 1 ) e. RR
4		0lt1 8461	. . . . . . 7 |- 0 < 1
5		ltsub23 8778	. . . . . . . . 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 9426	. . . . . . . . 9 |- ( 0 - 0 ) = 0
8	7	breq1i 4138	. . . . . . . 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 8429	. . . . 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 9325	. . . . . . . . . 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 13290	. . . . . . . 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 9383	. . . . . . . . 9 |- ( 1 - 1 ) = 0
25	24	fveq2i 5701	. . . . . . . 8 |- ( ( F ` C ) ` ( 1 - 1 ) ) = ( ( F ` C ) ` 0 )
26	25	oveq1i 6098	. . . . . . 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 13294	. . . . . . . . 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 6103	. . . . . 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 13303	. . . . . 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 5707	. . . . 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 4131   |-> cmpt 4193   ` cfv 5380  (class class class)co 6088   Fincfn 7027  infcinf 7330   RRcr 8185   0cc0 8186   1c1 8187   + caddc 8189   < clt 8367   - cmin 8505   / cdiv 9011   NNcn 9314   ZZcz 9656   ...cfz 10429  ♯chash 11237

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7028  df-dom 7029  df-fin 7030  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-fz 10430  df-fzo 10568  df-ihash 11238

This theorem is used by:  ballotfilemic  13309