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Theorem repswccat 12423
Description: The concatenation of two "repeated symbol words" with the same symbol is again a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
Assertion
Ref Expression
repswccat  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
( S repeatS  N ) concat  ( S repeatS  M ) )  =  ( S repeatS  ( N  +  M ) ) )

Proof of Theorem repswccat
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 repswlen 12414 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( # `  ( S repeatS  N ) )  =  N )
213adant3 1008 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  ( # `
 ( S repeatS  N
) )  =  N )
3 repswlen 12414 . . . . . 6  |-  ( ( S  e.  V  /\  M  e.  NN0 )  -> 
( # `  ( S repeatS  M ) )  =  M )
433adant2 1007 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  ( # `
 ( S repeatS  M
) )  =  M )
52, 4oveq12d 6109 . . . 4  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
( # `  ( S repeatS  N ) )  +  ( # `  ( S repeatS  M ) ) )  =  ( N  +  M ) )
65oveq2d 6107 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
0..^ ( ( # `  ( S repeatS  N )
)  +  ( # `  ( S repeatS  M )
) ) )  =  ( 0..^ ( N  +  M ) ) )
7 simp1 988 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  S  e.  V )
87adantr 465 . . . . . . 7  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  /\  x  e.  (
0..^ ( # `  ( S repeatS  N ) ) ) )  ->  S  e.  V )
9 simpl2 992 . . . . . . 7  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  /\  x  e.  (
0..^ ( # `  ( S repeatS  N ) ) ) )  ->  N  e.  NN0 )
102oveq2d 6107 . . . . . . . . 9  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
0..^ ( # `  ( S repeatS  N ) ) )  =  ( 0..^ N ) )
1110eleq2d 2510 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
x  e.  ( 0..^ ( # `  ( S repeatS  N ) ) )  <-> 
x  e.  ( 0..^ N ) ) )
1211biimpa 484 . . . . . . 7  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  /\  x  e.  (
0..^ ( # `  ( S repeatS  N ) ) ) )  ->  x  e.  ( 0..^ N ) )
138, 9, 123jca 1168 . . . . . 6  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  /\  x  e.  (
0..^ ( # `  ( S repeatS  N ) ) ) )  ->  ( S  e.  V  /\  N  e. 
NN0  /\  x  e.  ( 0..^ N ) ) )
1413adantlr 714 . . . . 5  |-  ( ( ( ( S  e.  V  /\  N  e. 
NN0  /\  M  e.  NN0 )  /\  x  e.  ( 0..^ ( (
# `  ( S repeatS  N ) )  +  (
# `  ( S repeatS  M ) ) ) ) )  /\  x  e.  ( 0..^ ( # `  ( S repeatS  N )
) ) )  -> 
( S  e.  V  /\  N  e.  NN0  /\  x  e.  ( 0..^ N ) ) )
15 repswsymb 12412 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  x  e.  ( 0..^ N ) )  ->  ( ( S repeatS  N ) `  x
)  =  S )
1614, 15syl 16 . . . 4  |-  ( ( ( ( S  e.  V  /\  N  e. 
NN0  /\  M  e.  NN0 )  /\  x  e.  ( 0..^ ( (
# `  ( S repeatS  N ) )  +  (
# `  ( S repeatS  M ) ) ) ) )  /\  x  e.  ( 0..^ ( # `  ( S repeatS  N )
) ) )  -> 
( ( S repeatS  N
) `  x )  =  S )
177ad2antrr 725 . . . . 5  |-  ( ( ( ( S  e.  V  /\  N  e. 
NN0  /\  M  e.  NN0 )  /\  x  e.  ( 0..^ ( (
# `  ( S repeatS  N ) )  +  (
# `  ( S repeatS  M ) ) ) ) )  /\  -.  x  e.  ( 0..^ ( # `  ( S repeatS  N )
) ) )  ->  S  e.  V )
18 simpll3 1029 . . . . 5  |-  ( ( ( ( S  e.  V  /\  N  e. 
NN0  /\  M  e.  NN0 )  /\  x  e.  ( 0..^ ( (
# `  ( S repeatS  N ) )  +  (
# `  ( S repeatS  M ) ) ) ) )  /\  -.  x  e.  ( 0..^ ( # `  ( S repeatS  N )
) ) )  ->  M  e.  NN0 )
192, 4jca 532 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M ) )
20 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN0  /\  M  e.  NN0 )  /\  x  e.  (
0..^ ( N  +  M ) ) )  ->  x  e.  ( 0..^ ( N  +  M ) ) )
2120anim1i 568 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e. 
NN0  /\  M  e.  NN0 )  /\  x  e.  ( 0..^ ( N  +  M ) ) )  /\  -.  x  e.  ( 0..^ N ) )  ->  ( x  e.  ( 0..^ ( N  +  M ) )  /\  -.  x  e.  ( 0..^ N ) ) )
22 nn0z 10669 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  N  e.  ZZ )
23 nn0z 10669 . . . . . . . . . . . . . . 15  |-  ( M  e.  NN0  ->  M  e.  ZZ )
2422, 23anim12i 566 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  -> 
( N  e.  ZZ  /\  M  e.  ZZ ) )
2524ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e. 
NN0  /\  M  e.  NN0 )  /\  x  e.  ( 0..^ ( N  +  M ) ) )  /\  -.  x  e.  ( 0..^ N ) )  ->  ( N  e.  ZZ  /\  M  e.  ZZ ) )
26 fzocatel 11602 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ( 0..^ ( N  +  M ) )  /\  -.  x  e.  (
0..^ N ) )  /\  ( N  e.  ZZ  /\  M  e.  ZZ ) )  -> 
( x  -  N
)  e.  ( 0..^ M ) )
2721, 25, 26syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
NN0  /\  M  e.  NN0 )  /\  x  e.  ( 0..^ ( N  +  M ) ) )  /\  -.  x  e.  ( 0..^ N ) )  ->  ( x  -  N )  e.  ( 0..^ M ) )
2827ex 434 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  M  e.  NN0 )  /\  x  e.  (
0..^ ( N  +  M ) ) )  ->  ( -.  x  e.  ( 0..^ N )  ->  ( x  -  N )  e.  ( 0..^ M ) ) )
2928ex 434 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  -> 
( x  e.  ( 0..^ ( N  +  M ) )  -> 
( -.  x  e.  ( 0..^ N )  ->  ( x  -  N )  e.  ( 0..^ M ) ) ) )
30293adant1 1006 . . . . . . . . 9  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
x  e.  ( 0..^ ( N  +  M
) )  ->  ( -.  x  e.  (
0..^ N )  -> 
( x  -  N
)  e.  ( 0..^ M ) ) ) )
31 oveq12 6100 . . . . . . . . . . . 12  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( ( # `  ( S repeatS  N )
)  +  ( # `  ( S repeatS  M )
) )  =  ( N  +  M ) )
3231oveq2d 6107 . . . . . . . . . . 11  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( 0..^ ( ( # `  ( S repeatS  N ) )  +  ( # `  ( S repeatS  M ) ) ) )  =  ( 0..^ ( N  +  M
) ) )
3332eleq2d 2510 . . . . . . . . . 10  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( x  e.  ( 0..^ ( (
# `  ( S repeatS  N ) )  +  (
# `  ( S repeatS  M ) ) ) )  <-> 
x  e.  ( 0..^ ( N  +  M
) ) ) )
34 oveq2 6099 . . . . . . . . . . . . . 14  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( 0..^ (
# `  ( S repeatS  N ) ) )  =  ( 0..^ N ) )
3534eleq2d 2510 . . . . . . . . . . . . 13  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( x  e.  ( 0..^ ( # `  ( S repeatS  N )
) )  <->  x  e.  ( 0..^ N ) ) )
3635notbid 294 . . . . . . . . . . . 12  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( -.  x  e.  ( 0..^ ( # `  ( S repeatS  N )
) )  <->  -.  x  e.  ( 0..^ N ) ) )
3736adantr 465 . . . . . . . . . . 11  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( -.  x  e.  ( 0..^ ( # `  ( S repeatS  N )
) )  <->  -.  x  e.  ( 0..^ N ) ) )
38 oveq2 6099 . . . . . . . . . . . . 13  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( x  -  ( # `  ( S repeatS  N ) ) )  =  ( x  -  N ) )
3938eleq1d 2509 . . . . . . . . . . . 12  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( ( x  -  ( # `  ( S repeatS  N ) ) )  e.  ( 0..^ M )  <->  ( x  -  N )  e.  ( 0..^ M ) ) )
4039adantr 465 . . . . . . . . . . 11  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( ( x  -  ( # `  ( S repeatS  N ) ) )  e.  ( 0..^ M )  <->  ( x  -  N )  e.  ( 0..^ M ) ) )
4137, 40imbi12d 320 . . . . . . . . . 10  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( ( -.  x  e.  ( 0..^ ( # `  ( S repeatS  N ) ) )  ->  ( x  -  ( # `  ( S repeatS  N ) ) )  e.  ( 0..^ M ) )  <->  ( -.  x  e.  ( 0..^ N )  ->  (
x  -  N )  e.  ( 0..^ M ) ) ) )
4233, 41imbi12d 320 . . . . . . . . 9  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( ( x  e.  ( 0..^ ( ( # `  ( S repeatS  N ) )  +  ( # `  ( S repeatS  M ) ) ) )  ->  ( -.  x  e.  ( 0..^ ( # `  ( S repeatS  N ) ) )  ->  ( x  -  ( # `  ( S repeatS  N ) ) )  e.  ( 0..^ M ) ) )  <->  ( x  e.  ( 0..^ ( N  +  M ) )  ->  ( -.  x  e.  ( 0..^ N )  ->  ( x  -  N )  e.  ( 0..^ M ) ) ) ) )
4330, 42syl5ibr 221 . . . . . . . 8  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e. 
NN0 )  ->  (
x  e.  ( 0..^ ( ( # `  ( S repeatS  N ) )  +  ( # `  ( S repeatS  M ) ) ) )  ->  ( -.  x  e.  ( 0..^ ( # `  ( S repeatS  N ) ) )  ->  ( x  -  ( # `  ( S repeatS  N ) ) )  e.  ( 0..^ M ) ) ) ) )
4419, 43mpcom 36 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
x  e.  ( 0..^ ( ( # `  ( S repeatS  N ) )  +  ( # `  ( S repeatS  M ) ) ) )  ->  ( -.  x  e.  ( 0..^ ( # `  ( S repeatS  N ) ) )  ->  ( x  -  ( # `  ( S repeatS  N ) ) )  e.  ( 0..^ M ) ) ) )
4544imp 429 . . . . . 6  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  /\  x  e.  (
0..^ ( ( # `  ( S repeatS  N )
)  +  ( # `  ( S repeatS  M )
) ) ) )  ->  ( -.  x  e.  ( 0..^ ( # `  ( S repeatS  N )
) )  ->  (
x  -  ( # `  ( S repeatS  N )
) )  e.  ( 0..^ M ) ) )
4645imp 429 . . . . 5  |-  ( ( ( ( S  e.  V  /\  N  e. 
NN0  /\  M  e.  NN0 )  /\  x  e.  ( 0..^ ( (
# `  ( S repeatS  N ) )  +  (
# `  ( S repeatS  M ) ) ) ) )  /\  -.  x  e.  ( 0..^ ( # `  ( S repeatS  N )
) ) )  -> 
( x  -  ( # `
 ( S repeatS  N
) ) )  e.  ( 0..^ M ) )
47 repswsymb 12412 . . . . 5  |-  ( ( S  e.  V  /\  M  e.  NN0  /\  (
x  -  ( # `  ( S repeatS  N )
) )  e.  ( 0..^ M ) )  ->  ( ( S repeatS  M ) `  (
x  -  ( # `  ( S repeatS  N )
) ) )  =  S )
4817, 18, 46, 47syl3anc 1218 . . . 4  |-  ( ( ( ( S  e.  V  /\  N  e. 
NN0  /\  M  e.  NN0 )  /\  x  e.  ( 0..^ ( (
# `  ( S repeatS  N ) )  +  (
# `  ( S repeatS  M ) ) ) ) )  /\  -.  x  e.  ( 0..^ ( # `  ( S repeatS  N )
) ) )  -> 
( ( S repeatS  M
) `  ( x  -  ( # `  ( S repeatS  N ) ) ) )  =  S )
4916, 48ifeqda 3822 . . 3  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  /\  x  e.  (
0..^ ( ( # `  ( S repeatS  N )
)  +  ( # `  ( S repeatS  M )
) ) ) )  ->  if ( x  e.  ( 0..^ (
# `  ( S repeatS  N ) ) ) ,  ( ( S repeatS  N
) `  x ) ,  ( ( S repeatS  M ) `  (
x  -  ( # `  ( S repeatS  N )
) ) ) )  =  S )
506, 49mpteq12dva 4369 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
x  e.  ( 0..^ ( ( # `  ( S repeatS  N ) )  +  ( # `  ( S repeatS  M ) ) ) )  |->  if ( x  e.  ( 0..^ (
# `  ( S repeatS  N ) ) ) ,  ( ( S repeatS  N
) `  x ) ,  ( ( S repeatS  M ) `  (
x  -  ( # `  ( S repeatS  N )
) ) ) ) )  =  ( x  e.  ( 0..^ ( N  +  M ) )  |->  S ) )
51 ovex 6116 . . . 4  |-  ( S repeatS  N )  e.  _V
52 ovex 6116 . . . 4  |-  ( S repeatS  M )  e.  _V
5351, 52pm3.2i 455 . . 3  |-  ( ( S repeatS  N )  e.  _V  /\  ( S repeatS  M )  e.  _V )
54 ccatfval 12273 . . 3  |-  ( ( ( S repeatS  N )  e.  _V  /\  ( S repeatS  M )  e.  _V )  ->  ( ( S repeatS  N ) concat  ( S repeatS  M
) )  =  ( x  e.  ( 0..^ ( ( # `  ( S repeatS  N ) )  +  ( # `  ( S repeatS  M ) ) ) )  |->  if ( x  e.  ( 0..^ (
# `  ( S repeatS  N ) ) ) ,  ( ( S repeatS  N
) `  x ) ,  ( ( S repeatS  M ) `  (
x  -  ( # `  ( S repeatS  N )
) ) ) ) ) )
5553, 54mp1i 12 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
( S repeatS  N ) concat  ( S repeatS  M ) )  =  ( x  e.  ( 0..^ ( ( # `  ( S repeatS  N )
)  +  ( # `  ( S repeatS  M )
) ) )  |->  if ( x  e.  ( 0..^ ( # `  ( S repeatS  N ) ) ) ,  ( ( S repeatS  N ) `  x
) ,  ( ( S repeatS  M ) `  (
x  -  ( # `  ( S repeatS  N )
) ) ) ) ) )
56 nn0addcl 10615 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  -> 
( N  +  M
)  e.  NN0 )
57563adant1 1006 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  ( N  +  M )  e.  NN0 )
58 reps 12408 . . 3  |-  ( ( S  e.  V  /\  ( N  +  M
)  e.  NN0 )  ->  ( S repeatS  ( N  +  M ) )  =  ( x  e.  ( 0..^ ( N  +  M ) )  |->  S ) )
597, 57, 58syl2anc 661 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  ( S repeatS  ( N  +  M
) )  =  ( x  e.  ( 0..^ ( N  +  M
) )  |->  S ) )
6050, 55, 593eqtr4d 2485 1  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
( S repeatS  N ) concat  ( S repeatS  M ) )  =  ( S repeatS  ( N  +  M ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2972   ifcif 3791    e. cmpt 4350   ` cfv 5418  (class class class)co 6091   0cc0 9282    + caddc 9285    - cmin 9595   NN0cn0 10579   ZZcz 10646  ..^cfzo 11548   #chash 12103   concat cconcat 12223   repeatS creps 12228
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-concat 12231  df-reps 12236
This theorem is referenced by:  repswcshw  12446  repsw2  12550  repsw3  12551
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