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Theorem signstfveq0a 29471
Description: Lemma for signstfveq0 29472. (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 765 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  F  e.  (Word  RR  \  { (/) } ) )
21eldifad 3384 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  F  e. Word  RR )
3 signstfveq0.1 . . . . 5  |-  N  =  ( # `  F
)
4 lencl 12684 . . . . 5  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  NN0 )
53, 4syl5eqel 2534 . . . 4  |-  ( F  e. Word  RR  ->  N  e. 
NN0 )
62, 5syl 17 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  NN0 )
7 eldifsn 4066 . . . . 5  |-  ( F  e.  (Word  RR  \  { (/) } )  <->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
81, 7sylib 201 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
9 hasheq0 12538 . . . . . . 7  |-  ( F  e. Word  RR  ->  ( (
# `  F )  =  0  <->  F  =  (/) ) )
109necon3bid 2668 . . . . . 6  |-  ( F  e. Word  RR  ->  ( (
# `  F )  =/=  0  <->  F  =/=  (/) ) )
1110biimpar 492 . . . . 5  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  ( # `
 F )  =/=  0 )
123neeq1i 2688 . . . . 5  |-  ( N  =/=  0  <->  ( # `  F
)  =/=  0 )
1311, 12sylibr 217 . . . 4  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  N  =/=  0 )
148, 13syl 17 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  =/=  0 )
15 elnnne0 10873 . . 3  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
166, 14, 15sylanbrc 675 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  NN )
17 simplr 767 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F `  0 )  =/=  0 )
18 simpr 467 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F `  ( N  -  1 ) )  =  0 )
1917, 18neeqtrrd 2698 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F `  0 )  =/=  ( F `  ( N  -  1 ) ) )
2019necomd 2679 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F `  ( N  -  1 ) )  =/=  ( F `  0 )
)
21 oveq1 6283 . . . . . 6  |-  ( N  =  1  ->  ( N  -  1 )  =  ( 1  -  1 ) )
22 1m1e0 10667 . . . . . 6  |-  ( 1  -  1 )  =  0
2321, 22syl6eq 2502 . . . . 5  |-  ( N  =  1  ->  ( N  -  1 )  =  0 )
2423fveq2d 5852 . . . 4  |-  ( N  =  1  ->  ( F `  ( N  -  1 ) )  =  ( F ` 
0 ) )
2524necon3i 2656 . . 3  |-  ( ( F `  ( N  -  1 ) )  =/=  ( F ` 
0 )  ->  N  =/=  1 )
2620, 25syl 17 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  =/=  1 )
27 eluz2b3 11222 . 2  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  N  =/=  1 ) )
2816, 26, 27sylanbrc 675 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 375    = wceq 1448    e. wcel 1891    =/= wne 2622    \ cdif 3369   (/)c0 3699   ifcif 3849   {csn 3936   {cpr 3938   {ctp 3940   <.cop 3942    |-> cmpt 4433   ` cfv 5561  (class class class)co 6276    |-> cmpt2 6278   RRcr 9525   0cc0 9526   1c1 9527    - cmin 9847   -ucneg 9848   NNcn 10598   2c2 10648   NN0cn0 10859   ZZ>=cuz 11149   ...cfz 11775  ..^cfzo 11908   #chash 12509  Word cword 12651  sgncsgn 13160   sum_csu 13763   ndxcnx 15129   Basecbs 15132   +g cplusg 15201    gsumg cgsu 15350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571  ax-cnex 9582  ax-resscn 9583  ax-1cn 9584  ax-icn 9585  ax-addcl 9586  ax-addrcl 9587  ax-mulcl 9588  ax-mulrcl 9589  ax-mulcom 9590  ax-addass 9591  ax-mulass 9592  ax-distr 9593  ax-i2m1 9594  ax-1ne0 9595  ax-1rid 9596  ax-rnegex 9597  ax-rrecex 9598  ax-cnre 9599  ax-pre-lttri 9600  ax-pre-lttrn 9601  ax-pre-ltadd 9602  ax-pre-mulgt0 9603
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-int 4205  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-tr 4470  df-eprel 4723  df-id 4727  df-po 4733  df-so 4734  df-fr 4771  df-we 4773  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-pred 5359  df-ord 5405  df-on 5406  df-lim 5407  df-suc 5408  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-riota 6238  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-om 6681  df-1st 6781  df-2nd 6782  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-card 8360  df-cda 8585  df-pnf 9664  df-mnf 9665  df-xr 9666  df-ltxr 9667  df-le 9668  df-sub 9849  df-neg 9850  df-nn 10599  df-2 10657  df-n0 10860  df-z 10928  df-uz 11150  df-fz 11776  df-fzo 11909  df-hash 12510  df-word 12659
This theorem is referenced by:  signstfveq0  29472
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