Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ccats1rev Structured version   Unicode version

Theorem ccats1rev 30409
Description: The first symbol of the concatenation of a word with two single symbols. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
Assertion
Ref Expression
ccats1rev  |-  ( ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  /\  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) ) )  ->  ( W  =  ( U substr  <.
0 ,  ( # `  W ) >. )  ->  U  =  ( W concat  <" S "> ) ) )
Distinct variable groups:    S, s    x, U    V, s, x    W, s, x    X, s, x
Allowed substitution hints:    S( x)    U( s)

Proof of Theorem ccats1rev
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 eleq1 2526 . . . . . . . . 9  |-  ( x  =  U  ->  (
x  e. Word  V  <->  U  e. Word  V ) )
2 fveq2 5800 . . . . . . . . . 10  |-  ( x  =  U  ->  ( # `
 x )  =  ( # `  U
) )
32eqeq1d 2456 . . . . . . . . 9  |-  ( x  =  U  ->  (
( # `  x )  =  ( ( # `  W )  +  1 )  <->  ( # `  U
)  =  ( (
# `  W )  +  1 ) ) )
41, 3anbi12d 710 . . . . . . . 8  |-  ( x  =  U  ->  (
( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) )  <-> 
( U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) ) ) )
54rspcva 3177 . . . . . . 7  |-  ( ( U  e.  X  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x )  =  ( ( # `  W )  +  1 ) ) )  -> 
( U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) ) )
65expcom 435 . . . . . 6  |-  ( A. x  e.  X  (
x  e. Word  V  /\  ( # `  x )  =  ( ( # `  W )  +  1 ) )  ->  ( U  e.  X  ->  ( U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) ) ) )
76adantl 466 . . . . 5  |-  ( ( A. s  e.  V  ( ( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) )  ->  ( U  e.  X  ->  ( U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) ) ) )
87com12 31 . . . 4  |-  ( U  e.  X  ->  (
( A. s  e.  V  ( ( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x )  =  ( ( # `  W )  +  1 ) ) )  -> 
( U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) ) ) )
983ad2ant2 1010 . . 3  |-  ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  ->  ( ( A. s  e.  V  ( ( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) )  ->  ( U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W
)  +  1 ) ) ) )
109imp 429 . 2  |-  ( ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  /\  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) ) )  ->  ( U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) ) )
11 3anass 969 . . . . . . . . . 10  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  <->  ( W  e. Word  V  /\  ( U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) ) ) )
1211simplbi2 625 . . . . . . . . 9  |-  ( W  e. Word  V  ->  (
( U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) )  ->  ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) ) ) )
13123ad2ant1 1009 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  ->  ( ( U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) ) ) )
1413adantr 465 . . . . . . 7  |-  ( ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  /\  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) ) )  ->  (
( U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) )  ->  ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) ) ) )
1514impcom 430 . . . . . 6  |-  ( ( ( U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) )  /\  ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  /\  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) ) ) )  -> 
( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) ) )
16 ccats1swrdeqrex 30408 . . . . . 6  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( W  =  ( U substr  <. 0 ,  ( # `  W
) >. )  ->  E. u  e.  V  U  =  ( W concat  <" u "> ) ) )
1715, 16syl 16 . . . . 5  |-  ( ( ( U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) )  /\  ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  /\  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) ) ) )  -> 
( W  =  ( U substr  <. 0 ,  (
# `  W ) >. )  ->  E. u  e.  V  U  =  ( W concat  <" u "> ) ) )
1817imp 429 . . . 4  |-  ( ( ( ( U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) )  /\  ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  /\  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) ) ) )  /\  W  =  ( U substr  <.
0 ,  ( # `  W ) >. )
)  ->  E. u  e.  V  U  =  ( W concat  <" u "> ) )
19 s1eq 12410 . . . . . . . . . . . . . . . . . . 19  |-  ( s  =  u  ->  <" s ">  =  <" u "> )
2019oveq2d 6217 . . . . . . . . . . . . . . . . . 18  |-  ( s  =  u  ->  ( W concat  <" s "> )  =  ( W concat  <" u "> ) )
2120eleq1d 2523 . . . . . . . . . . . . . . . . 17  |-  ( s  =  u  ->  (
( W concat  <" s "> )  e.  X  <->  ( W concat  <" u "> )  e.  X
) )
22 eqeq2 2469 . . . . . . . . . . . . . . . . 17  |-  ( s  =  u  ->  ( S  =  s  <->  S  =  u ) )
2321, 22imbi12d 320 . . . . . . . . . . . . . . . 16  |-  ( s  =  u  ->  (
( ( W concat  <" s "> )  e.  X  ->  S  =  s )  <-> 
( ( W concat  <" u "> )  e.  X  ->  S  =  u ) ) )
2423rspcva 3177 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  V  /\  A. s  e.  V  ( ( W concat  <" s "> )  e.  X  ->  S  =  s ) )  ->  ( ( W concat  <" u "> )  e.  X  ->  S  =  u ) )
25 eleq1 2526 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( W concat  <" u "> )  =  U  ->  ( ( W concat  <" u "> )  e.  X  <->  U  e.  X ) )
2625eqcoms 2466 . . . . . . . . . . . . . . . . . . . 20  |-  ( U  =  ( W concat  <" u "> )  ->  (
( W concat  <" u "> )  e.  X  <->  U  e.  X ) )
2726adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  X  /\  U  =  ( W concat  <" u "> ) )  ->  (
( W concat  <" u "> )  e.  X  <->  U  e.  X ) )
2827imbi1d 317 . . . . . . . . . . . . . . . . . 18  |-  ( ( U  e.  X  /\  U  =  ( W concat  <" u "> ) )  ->  (
( ( W concat  <" u "> )  e.  X  ->  S  =  u )  <-> 
( U  e.  X  ->  S  =  u ) ) )
29 pm3.35 587 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( U  e.  X  /\  ( U  e.  X  ->  S  =  u ) )  ->  S  =  u )
30 s1eq 12410 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( u  =  S  ->  <" u ">  =  <" S "> )
3130eqcoms 2466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( S  =  u  ->  <" u ">  =  <" S "> )
3231oveq2d 6217 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( S  =  u  ->  ( W concat  <" u "> )  =  ( W concat  <" S "> ) )
3332eqeq2d 2468 . . . . . . . . . . . . . . . . . . . . 21  |-  ( S  =  u  ->  ( U  =  ( W concat  <" u "> ) 
<->  U  =  ( W concat  <" S "> ) ) )
3433biimpd 207 . . . . . . . . . . . . . . . . . . . 20  |-  ( S  =  u  ->  ( U  =  ( W concat  <" u "> )  ->  U  =  ( W concat  <" S "> ) ) )
3529, 34syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  X  /\  ( U  e.  X  ->  S  =  u ) )  ->  ( U  =  ( W concat  <" u "> )  ->  U  =  ( W concat  <" S "> ) ) )
3635impancom 440 . . . . . . . . . . . . . . . . . 18  |-  ( ( U  e.  X  /\  U  =  ( W concat  <" u "> ) )  ->  (
( U  e.  X  ->  S  =  u )  ->  U  =  ( W concat  <" S "> ) ) )
3728, 36sylbid 215 . . . . . . . . . . . . . . . . 17  |-  ( ( U  e.  X  /\  U  =  ( W concat  <" u "> ) )  ->  (
( ( W concat  <" u "> )  e.  X  ->  S  =  u )  ->  U  =  ( W concat  <" S "> ) ) )
3837com12 31 . . . . . . . . . . . . . . . 16  |-  ( ( ( W concat  <" u "> )  e.  X  ->  S  =  u )  ->  ( ( U  e.  X  /\  U  =  ( W concat  <" u "> ) )  ->  U  =  ( W concat  <" S "> ) ) )
3938expd 436 . . . . . . . . . . . . . . 15  |-  ( ( ( W concat  <" u "> )  e.  X  ->  S  =  u )  ->  ( U  e.  X  ->  ( U  =  ( W concat  <" u "> )  ->  U  =  ( W concat  <" S "> ) ) ) )
4024, 39syl 16 . . . . . . . . . . . . . 14  |-  ( ( u  e.  V  /\  A. s  e.  V  ( ( W concat  <" s "> )  e.  X  ->  S  =  s ) )  ->  ( U  e.  X  ->  ( U  =  ( W concat  <" u "> )  ->  U  =  ( W concat  <" S "> ) ) ) )
4140ex 434 . . . . . . . . . . . . 13  |-  ( u  e.  V  ->  ( A. s  e.  V  ( ( W concat  <" s "> )  e.  X  ->  S  =  s )  ->  ( U  e.  X  ->  ( U  =  ( W concat  <" u "> )  ->  U  =  ( W concat  <" S "> ) ) ) ) )
4241com3l 81 . . . . . . . . . . . 12  |-  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  ->  ( U  e.  X  ->  ( u  e.  V  ->  ( U  =  ( W concat  <" u "> )  ->  U  =  ( W concat  <" S "> ) ) ) ) )
4342adantr 465 . . . . . . . . . . 11  |-  ( ( A. s  e.  V  ( ( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) )  ->  ( U  e.  X  ->  ( u  e.  V  ->  ( U  =  ( W concat  <" u "> )  ->  U  =  ( W concat  <" S "> ) ) ) ) )
4443com12 31 . . . . . . . . . 10  |-  ( U  e.  X  ->  (
( A. s  e.  V  ( ( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x )  =  ( ( # `  W )  +  1 ) ) )  -> 
( u  e.  V  ->  ( U  =  ( W concat  <" u "> )  ->  U  =  ( W concat  <" S "> ) ) ) ) )
45443ad2ant2 1010 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  ->  ( ( A. s  e.  V  ( ( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) )  ->  ( u  e.  V  ->  ( U  =  ( W concat  <" u "> )  ->  U  =  ( W concat  <" S "> ) ) ) ) )
4645imp 429 . . . . . . . 8  |-  ( ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  /\  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) ) )  ->  (
u  e.  V  -> 
( U  =  ( W concat  <" u "> )  ->  U  =  ( W concat  <" S "> ) ) ) )
4746adantl 466 . . . . . . 7  |-  ( ( ( U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) )  /\  ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  /\  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) ) ) )  -> 
( u  e.  V  ->  ( U  =  ( W concat  <" u "> )  ->  U  =  ( W concat  <" S "> ) ) ) )
4847adantr 465 . . . . . 6  |-  ( ( ( ( U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) )  /\  ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  /\  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) ) ) )  /\  W  =  ( U substr  <.
0 ,  ( # `  W ) >. )
)  ->  ( u  e.  V  ->  ( U  =  ( W concat  <" u "> )  ->  U  =  ( W concat  <" S "> ) ) ) )
4948imp 429 . . . . 5  |-  ( ( ( ( ( U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  /\  ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  /\  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) ) ) )  /\  W  =  ( U substr  <.
0 ,  ( # `  W ) >. )
)  /\  u  e.  V )  ->  ( U  =  ( W concat  <" u "> )  ->  U  =  ( W concat  <" S "> ) ) )
5049rexlimdva 2947 . . . 4  |-  ( ( ( ( U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) )  /\  ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  /\  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) ) ) )  /\  W  =  ( U substr  <.
0 ,  ( # `  W ) >. )
)  ->  ( E. u  e.  V  U  =  ( W concat  <" u "> )  ->  U  =  ( W concat  <" S "> ) ) )
5118, 50mpd 15 . . 3  |-  ( ( ( ( U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) )  /\  ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  /\  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) ) ) )  /\  W  =  ( U substr  <.
0 ,  ( # `  W ) >. )
)  ->  U  =  ( W concat  <" S "> ) )
5251ex 434 . 2  |-  ( ( ( U  e. Word  V  /\  ( # `  U
)  =  ( (
# `  W )  +  1 ) )  /\  ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  /\  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) ) ) )  -> 
( W  =  ( U substr  <. 0 ,  (
# `  W ) >. )  ->  U  =  ( W concat  <" S "> ) ) )
5310, 52mpancom 669 1  |-  ( ( ( W  e. Word  V  /\  U  e.  X  /\  ( W concat  <" S "> )  e.  X
)  /\  ( A. s  e.  V  (
( W concat  <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x
)  =  ( (
# `  W )  +  1 ) ) ) )  ->  ( W  =  ( U substr  <.
0 ,  ( # `  W ) >. )  ->  U  =  ( W concat  <" S "> ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800   <.cop 3992   ` cfv 5527  (class class class)co 6201   0cc0 9394   1c1 9395    + caddc 9397   #chash 12221  Word cword 12340   concat cconcat 12342   <"cs1 12343   substr csubstr 12344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-hash 12222  df-word 12348  df-lsw 12349  df-concat 12350  df-s1 12351  df-substr 12352
This theorem is referenced by:  reuccats1  30410
  Copyright terms: Public domain W3C validator