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Theorem ccats1rev 30213
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 2498 . . . . . . . . 9  |-  ( x  =  U  ->  (
x  e. Word  V  <->  U  e. Word  V ) )
2 fveq2 5686 . . . . . . . . . 10  |-  ( x  =  U  ->  ( # `
 x )  =  ( # `  U
) )
32eqeq1d 2446 . . . . . . . . 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 3066 . . . . . . 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 30212 . . . . . 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 12283 . . . . . . . . . . . . . . . . . . 19  |-  ( s  =  u  ->  <" s ">  =  <" u "> )
2019oveq2d 6102 . . . . . . . . . . . . . . . . . 18  |-  ( s  =  u  ->  ( W concat  <" s "> )  =  ( W concat  <" u "> ) )
2120eleq1d 2504 . . . . . . . . . . . . . . . . 17  |-  ( s  =  u  ->  (
( W concat  <" s "> )  e.  X  <->  ( W concat  <" u "> )  e.  X
) )
22 eqeq2 2447 . . . . . . . . . . . . . . . . 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 3066 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  V  /\  A. s  e.  V  ( ( W concat  <" s "> )  e.  X  ->  S  =  s ) )  ->  ( ( W concat  <" u "> )  e.  X  ->  S  =  u ) )
25 eleq1 2498 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( W concat  <" u "> )  =  U  ->  ( ( W concat  <" u "> )  e.  X  <->  U  e.  X ) )
2625eqcoms 2441 . . . . . . . . . . . . . . . . . . . 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 12283 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( u  =  S  ->  <" u ">  =  <" S "> )
3130eqcoms 2441 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( S  =  u  ->  <" u ">  =  <" S "> )
3231oveq2d 6102 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( S  =  u  ->  ( W concat  <" u "> )  =  ( W concat  <" S "> ) )
3332eqeq2d 2449 . . . . . . . . . . . . . . . . . . . . 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 2836 . . . 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 1369    e. wcel 1756   A.wral 2710   E.wrex 2711   <.cop 3878   ` cfv 5413  (class class class)co 6086   0cc0 9274   1c1 9275    + caddc 9277   #chash 12095  Word cword 12213   concat cconcat 12215   <"cs1 12216   substr csubstr 12217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-hash 12096  df-word 12221  df-lsw 12222  df-concat 12223  df-s1 12224  df-substr 12225
This theorem is referenced by:  reuccats1  30214
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