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Theorem signstres 28796
Description: Restriction of a zero skipping sign to a subword. (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 ) )
Assertion
Ref Expression
signstres  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( ( T `  F )  |`  (
0..^ N ) )  =  ( T `  ( F  |`  ( 0..^ N ) ) ) )
Distinct variable groups:    a, b,  .+^    f, i, n, F    f, W, i, n    f, N, i, n
Allowed substitution hints:    .+^ ( f, i,
j, n)    T( f,
i, j, n, a, b)    F( j, a, b)    N( j, a, b)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signstres
Dummy variables  g  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 signsv.p . . . . . . . 8  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
2 signsv.w . . . . . . . 8  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
3 signsv.t . . . . . . . 8  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
4 signsv.v . . . . . . . 8  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
51, 2, 3, 4signstf 28787 . . . . . . 7  |-  ( F  e. Word  RR  ->  ( T `
 F )  e. Word  RR )
6 wrdf 12538 . . . . . . 7  |-  ( ( T `  F )  e. Word  RR  ->  ( T `
 F ) : ( 0..^ ( # `  ( T `  F
) ) ) --> RR )
7 ffn 5713 . . . . . . 7  |-  ( ( T `  F ) : ( 0..^ (
# `  ( T `  F ) ) ) --> RR  ->  ( T `  F )  Fn  (
0..^ ( # `  ( T `  F )
) ) )
85, 6, 73syl 20 . . . . . 6  |-  ( F  e. Word  RR  ->  ( T `
 F )  Fn  ( 0..^ ( # `  ( T `  F
) ) ) )
91, 2, 3, 4signstlen 28788 . . . . . . . 8  |-  ( F  e. Word  RR  ->  ( # `  ( T `  F
) )  =  (
# `  F )
)
109oveq2d 6286 . . . . . . 7  |-  ( F  e. Word  RR  ->  ( 0..^ ( # `  ( T `  F )
) )  =  ( 0..^ ( # `  F
) ) )
1110fneq2d 5654 . . . . . 6  |-  ( F  e. Word  RR  ->  ( ( T `  F )  Fn  ( 0..^ (
# `  ( T `  F ) ) )  <-> 
( T `  F
)  Fn  ( 0..^ ( # `  F
) ) ) )
128, 11mpbid 210 . . . . 5  |-  ( F  e. Word  RR  ->  ( T `
 F )  Fn  ( 0..^ ( # `  F ) ) )
13 fnresin 27689 . . . . 5  |-  ( ( T `  F )  Fn  ( 0..^ (
# `  F )
)  ->  ( ( T `  F )  |`  ( 0..^ N ) )  Fn  ( ( 0..^ ( # `  F
) )  i^i  (
0..^ N ) ) )
1412, 13syl 16 . . . 4  |-  ( F  e. Word  RR  ->  ( ( T `  F )  |`  ( 0..^ N ) )  Fn  ( ( 0..^ ( # `  F
) )  i^i  (
0..^ N ) ) )
1514adantr 463 . . 3  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( ( T `  F )  |`  (
0..^ N ) )  Fn  ( ( 0..^ ( # `  F
) )  i^i  (
0..^ N ) ) )
16 elfzuz3 11688 . . . . . 6  |-  ( N  e.  ( 0 ... ( # `  F
) )  ->  ( # `
 F )  e.  ( ZZ>= `  N )
)
17 fzoss2 11830 . . . . . 6  |-  ( (
# `  F )  e.  ( ZZ>= `  N )  ->  ( 0..^ N ) 
C_  ( 0..^ (
# `  F )
) )
1816, 17syl 16 . . . . 5  |-  ( N  e.  ( 0 ... ( # `  F
) )  ->  (
0..^ N )  C_  ( 0..^ ( # `  F
) ) )
1918adantl 464 . . . 4  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( 0..^ N ) 
C_  ( 0..^ (
# `  F )
) )
20 incom 3677 . . . . . 6  |-  ( ( 0..^ N )  i^i  ( 0..^ ( # `  F ) ) )  =  ( ( 0..^ ( # `  F
) )  i^i  (
0..^ N ) )
21 df-ss 3475 . . . . . . 7  |-  ( ( 0..^ N )  C_  ( 0..^ ( # `  F
) )  <->  ( (
0..^ N )  i^i  ( 0..^ ( # `  F ) ) )  =  ( 0..^ N ) )
2221biimpi 194 . . . . . 6  |-  ( ( 0..^ N )  C_  ( 0..^ ( # `  F
) )  ->  (
( 0..^ N )  i^i  ( 0..^ (
# `  F )
) )  =  ( 0..^ N ) )
2320, 22syl5eqr 2509 . . . . 5  |-  ( ( 0..^ N )  C_  ( 0..^ ( # `  F
) )  ->  (
( 0..^ ( # `  F ) )  i^i  ( 0..^ N ) )  =  ( 0..^ N ) )
2423fneq2d 5654 . . . 4  |-  ( ( 0..^ N )  C_  ( 0..^ ( # `  F
) )  ->  (
( ( T `  F )  |`  (
0..^ N ) )  Fn  ( ( 0..^ ( # `  F
) )  i^i  (
0..^ N ) )  <-> 
( ( T `  F )  |`  (
0..^ N ) )  Fn  ( 0..^ N ) ) )
2519, 24syl 16 . . 3  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( ( ( T `
 F )  |`  ( 0..^ N ) )  Fn  ( ( 0..^ ( # `  F
) )  i^i  (
0..^ N ) )  <-> 
( ( T `  F )  |`  (
0..^ N ) )  Fn  ( 0..^ N ) ) )
2615, 25mpbid 210 . 2  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( ( T `  F )  |`  (
0..^ N ) )  Fn  ( 0..^ N ) )
27 wrdres 28758 . . . 4  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( F  |`  (
0..^ N ) )  e. Word  RR )
281, 2, 3, 4signstf 28787 . . . 4  |-  ( ( F  |`  ( 0..^ N ) )  e. Word  RR  ->  ( T `  ( F  |`  ( 0..^ N ) ) )  e. Word  RR )
29 wrdf 12538 . . . 4  |-  ( ( T `  ( F  |`  ( 0..^ N ) ) )  e. Word  RR  ->  ( T `  ( F  |`  ( 0..^ N ) ) ) : ( 0..^ ( # `  ( T `  ( F  |`  ( 0..^ N ) ) ) ) ) --> RR )
30 ffn 5713 . . . 4  |-  ( ( T `  ( F  |`  ( 0..^ N ) ) ) : ( 0..^ ( # `  ( T `  ( F  |`  ( 0..^ N ) ) ) ) ) --> RR  ->  ( T `  ( F  |`  (
0..^ N ) ) )  Fn  ( 0..^ ( # `  ( T `  ( F  |`  ( 0..^ N ) ) ) ) ) )
3127, 28, 29, 304syl 21 . . 3  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( T `  ( F  |`  ( 0..^ N ) ) )  Fn  ( 0..^ ( # `  ( T `  ( F  |`  ( 0..^ N ) ) ) ) ) )
321, 2, 3, 4signstlen 28788 . . . . . . 7  |-  ( ( F  |`  ( 0..^ N ) )  e. Word  RR  ->  ( # `  ( T `  ( F  |`  ( 0..^ N ) ) ) )  =  ( # `  ( F  |`  ( 0..^ N ) ) ) )
3327, 32syl 16 . . . . . 6  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( # `  ( T `
 ( F  |`  ( 0..^ N ) ) ) )  =  (
# `  ( F  |`  ( 0..^ N ) ) ) )
34 wrdfn 12547 . . . . . . . 8  |-  ( F  e. Word  RR  ->  F  Fn  ( 0..^ ( # `  F
) ) )
35 fnssres 5676 . . . . . . . 8  |-  ( ( F  Fn  ( 0..^ ( # `  F
) )  /\  (
0..^ N )  C_  ( 0..^ ( # `  F
) ) )  -> 
( F  |`  (
0..^ N ) )  Fn  ( 0..^ N ) )
3634, 18, 35syl2an 475 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( F  |`  (
0..^ N ) )  Fn  ( 0..^ N ) )
37 hashfn 12426 . . . . . . 7  |-  ( ( F  |`  ( 0..^ N ) )  Fn  ( 0..^ N )  ->  ( # `  ( F  |`  ( 0..^ N ) ) )  =  ( # `  (
0..^ N ) ) )
3836, 37syl 16 . . . . . 6  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( # `  ( F  |`  ( 0..^ N ) ) )  =  (
# `  ( 0..^ N ) ) )
39 elfznn0 11775 . . . . . . . 8  |-  ( N  e.  ( 0 ... ( # `  F
) )  ->  N  e.  NN0 )
40 hashfzo0 12472 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( # `  ( 0..^ N ) )  =  N )
4139, 40syl 16 . . . . . . 7  |-  ( N  e.  ( 0 ... ( # `  F
) )  ->  ( # `
 ( 0..^ N ) )  =  N )
4241adantl 464 . . . . . 6  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( # `  ( 0..^ N ) )  =  N )
4333, 38, 423eqtrd 2499 . . . . 5  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( # `  ( T `
 ( F  |`  ( 0..^ N ) ) ) )  =  N )
4443oveq2d 6286 . . . 4  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( 0..^ ( # `  ( T `  ( F  |`  ( 0..^ N ) ) ) ) )  =  ( 0..^ N ) )
4544fneq2d 5654 . . 3  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( ( T `  ( F  |`  ( 0..^ N ) ) )  Fn  ( 0..^ (
# `  ( T `  ( F  |`  (
0..^ N ) ) ) ) )  <->  ( T `  ( F  |`  (
0..^ N ) ) )  Fn  ( 0..^ N ) ) )
4631, 45mpbid 210 . 2  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( T `  ( F  |`  ( 0..^ N ) ) )  Fn  ( 0..^ N ) )
47 fvres 5862 . . . . 5  |-  ( m  e.  ( 0..^ N )  ->  ( (
( T `  F
)  |`  ( 0..^ N ) ) `  m
)  =  ( ( T `  F ) `
 m ) )
4847ad3antlr 728 . . . 4  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) ++  g ) )  -> 
( ( ( T `
 F )  |`  ( 0..^ N ) ) `
 m )  =  ( ( T `  F ) `  m
) )
49 simpr 459 . . . . . 6  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) ++  g ) )  ->  F  =  ( ( F  |`  ( 0..^ N ) ) ++  g ) )
5049fveq2d 5852 . . . . 5  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) ++  g ) )  -> 
( T `  F
)  =  ( T `
 ( ( F  |`  ( 0..^ N ) ) ++  g ) ) )
5150fveq1d 5850 . . . 4  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) ++  g ) )  -> 
( ( T `  F ) `  m
)  =  ( ( T `  ( ( F  |`  ( 0..^ N ) ) ++  g ) ) `  m
) )
5227ad3antrrr 727 . . . . 5  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) ++  g ) )  -> 
( F  |`  (
0..^ N ) )  e. Word  RR )
53 simplr 753 . . . . 5  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) ++  g ) )  -> 
g  e. Word  RR )
5438, 42eqtrd 2495 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( # `  ( F  |`  ( 0..^ N ) ) )  =  N )
5554oveq2d 6286 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( 0..^ ( # `  ( F  |`  (
0..^ N ) ) ) )  =  ( 0..^ N ) )
5655eleq2d 2524 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( m  e.  ( 0..^ ( # `  ( F  |`  ( 0..^ N ) ) ) )  <-> 
m  e.  ( 0..^ N ) ) )
5756biimpar 483 . . . . . 6  |-  ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  /\  m  e.  ( 0..^ N ) )  ->  m  e.  ( 0..^ ( # `  ( F  |`  ( 0..^ N ) ) ) ) )
5857ad2antrr 723 . . . . 5  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) ++  g ) )  ->  m  e.  ( 0..^ ( # `  ( F  |`  ( 0..^ N ) ) ) ) )
591, 2, 3, 4signstfvc 28795 . . . . 5  |-  ( ( ( F  |`  (
0..^ N ) )  e. Word  RR  /\  g  e. Word  RR  /\  m  e.  ( 0..^ ( # `  ( F  |`  (
0..^ N ) ) ) ) )  -> 
( ( T `  ( ( F  |`  ( 0..^ N ) ) ++  g ) ) `  m )  =  ( ( T `  ( F  |`  ( 0..^ N ) ) ) `  m ) )
6052, 53, 58, 59syl3anc 1226 . . . 4  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) ++  g ) )  -> 
( ( T `  ( ( F  |`  ( 0..^ N ) ) ++  g ) ) `  m )  =  ( ( T `  ( F  |`  ( 0..^ N ) ) ) `  m ) )
6148, 51, 603eqtrd 2499 . . 3  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) ++  g ) )  -> 
( ( ( T `
 F )  |`  ( 0..^ N ) ) `
 m )  =  ( ( T `  ( F  |`  ( 0..^ N ) ) ) `
 m ) )
62 wrdsplex 28759 . . . 4  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  ->  E. g  e. Word  RR F  =  ( ( F  |`  ( 0..^ N ) ) ++  g ) )
6362adantr 463 . . 3  |-  ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  /\  m  e.  ( 0..^ N ) )  ->  E. g  e. Word  RR F  =  ( ( F  |`  ( 0..^ N ) ) ++  g ) )
6461, 63r19.29a 2996 . 2  |-  ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  /\  m  e.  ( 0..^ N ) )  -> 
( ( ( T `
 F )  |`  ( 0..^ N ) ) `
 m )  =  ( ( T `  ( F  |`  ( 0..^ N ) ) ) `
 m ) )
6526, 46, 64eqfnfvd 5960 1  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( ( T `  F )  |`  (
0..^ N ) )  =  ( T `  ( F  |`  ( 0..^ N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805    i^i cin 3460    C_ wss 3461   ifcif 3929   {cpr 4018   {ctp 4020   <.cop 4022    |-> cmpt 4497    |` cres 4990    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   RRcr 9480   0cc0 9481   1c1 9482    - cmin 9796   -ucneg 9797   NN0cn0 10791   ZZ>=cuz 11082   ...cfz 11675  ..^cfzo 11799   #chash 12387  Word cword 12518   ++ cconcat 12520  sgncsgn 13001   sum_csu 13590   ndxcnx 14713   Basecbs 14716   +g cplusg 14784    gsumg cgsu 14930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-word 12526  df-lsw 12527  df-concat 12528  df-s1 12529  df-substr 12530  df-sgn 13002  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-plusg 14797  df-0g 14931  df-gsum 14932  df-mgm 16071  df-sgrp 16110  df-mnd 16120
This theorem is referenced by: (None)
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