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Theorem signstfveq0a 28817
Description: Lemma for signstfveq0 28818 (Contributed by Thierry Arnoux, 11-Oct-2018.)
Hypotheses
Ref Expression
signsv.p  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
signsv.w  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
signsv.t  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
signsv.v  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
signstfveq0.1  |-  N  =  ( # `  F
)
Assertion
Ref Expression
signstfveq0a  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  ( ZZ>= `  2 )
)
Distinct variable groups:    a, b,  .+^    f, i, n, F    f, W, i, n    F, a, b, f, i, n    N, a    f, b, i, n, N    T, a,
b
Allowed substitution hints:    .+^ ( f, i,
j, n)    T( f,
i, j, n)    F( j)    N( j)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signstfveq0a
StepHypRef Expression
1 simpll 753 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  F  e.  (Word  RR  \  { (/) } ) )
21eldifad 3483 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  F  e. Word  RR )
3 signstfveq0.1 . . . . 5  |-  N  =  ( # `  F
)
4 lencl 12569 . . . . 5  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  NN0 )
53, 4syl5eqel 2549 . . . 4  |-  ( F  e. Word  RR  ->  N  e. 
NN0 )
62, 5syl 16 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  NN0 )
7 eldifsn 4157 . . . . 5  |-  ( F  e.  (Word  RR  \  { (/) } )  <->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
81, 7sylib 196 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
9 hasheq0 12436 . . . . . . 7  |-  ( F  e. Word  RR  ->  ( (
# `  F )  =  0  <->  F  =  (/) ) )
109necon3bid 2715 . . . . . 6  |-  ( F  e. Word  RR  ->  ( (
# `  F )  =/=  0  <->  F  =/=  (/) ) )
1110biimpar 485 . . . . 5  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  ( # `
 F )  =/=  0 )
123neeq1i 2742 . . . . 5  |-  ( N  =/=  0  <->  ( # `  F
)  =/=  0 )
1311, 12sylibr 212 . . . 4  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  N  =/=  0 )
148, 13syl 16 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  =/=  0 )
15 elnnne0 10830 . . 3  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
166, 14, 15sylanbrc 664 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  NN )
17 simplr 755 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F `  0 )  =/=  0 )
18 simpr 461 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F `  ( N  -  1 ) )  =  0 )
1917, 18neeqtrrd 2757 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F `  0 )  =/=  ( F `  ( N  -  1 ) ) )
2019necomd 2728 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F `  ( N  -  1 ) )  =/=  ( F `  0 )
)
21 oveq1 6303 . . . . . 6  |-  ( N  =  1  ->  ( N  -  1 )  =  ( 1  -  1 ) )
22 1m1e0 10625 . . . . . 6  |-  ( 1  -  1 )  =  0
2321, 22syl6eq 2514 . . . . 5  |-  ( N  =  1  ->  ( N  -  1 )  =  0 )
2423fveq2d 5876 . . . 4  |-  ( N  =  1  ->  ( F `  ( N  -  1 ) )  =  ( F ` 
0 ) )
2524necon3i 2697 . . 3  |-  ( ( F `  ( N  -  1 ) )  =/=  ( F ` 
0 )  ->  N  =/=  1 )
2620, 25syl 16 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  =/=  1 )
27 eluz2b3 11180 . 2  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  N  =/=  1 ) )
2816, 26, 27sylanbrc 664 1  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  ( ZZ>= `  2 )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652    \ cdif 3468   (/)c0 3793   ifcif 3944   {csn 4032   {cpr 4034   {ctp 4036   <.cop 4038    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   RRcr 9508   0cc0 9509   1c1 9510    - cmin 9824   -ucneg 9825   NNcn 10556   2c2 10606   NN0cn0 10816   ZZ>=cuz 11106   ...cfz 11697  ..^cfzo 11821   #chash 12408  Word cword 12538  sgncsgn 13022   sum_csu 13611   ndxcnx 14732   Basecbs 14735   +g cplusg 14803    gsumg cgsu 14949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546
This theorem is referenced by:  signstfveq0  28818
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