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Theorem repswccat 12769
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 ) ++  ( S repeatS  M ) )  =  ( S repeatS  ( N  +  M )
) )

Proof of Theorem repswccat
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 repswlen 12760 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( # `  ( S repeatS  N ) )  =  N )
213adant3 1016 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  ( # `
 ( S repeatS  N
) )  =  N )
3 repswlen 12760 . . . . . 6  |-  ( ( S  e.  V  /\  M  e.  NN0 )  -> 
( # `  ( S repeatS  M ) )  =  M )
433adant2 1015 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  ( # `
 ( S repeatS  M
) )  =  M )
52, 4oveq12d 6314 . . . 4  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
( # `  ( S repeatS  N ) )  +  ( # `  ( S repeatS  M ) ) )  =  ( N  +  M ) )
65oveq2d 6312 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
0..^ ( ( # `  ( S repeatS  N )
)  +  ( # `  ( S repeatS  M )
) ) )  =  ( 0..^ ( N  +  M ) ) )
7 simp1 996 . . . . . . . 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 1000 . . . . . . 7  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  /\  x  e.  (
0..^ ( # `  ( S repeatS  N ) ) ) )  ->  N  e.  NN0 )
102oveq2d 6312 . . . . . . . . 9  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
0..^ ( # `  ( S repeatS  N ) ) )  =  ( 0..^ N ) )
1110eleq2d 2527 . . . . . . . 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 1176 . . . . . 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 12758 . . . . 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 1037 . . . . 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 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M ) )
20 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN0  /\  M  e.  NN0 )  /\  x  e.  (
0..^ ( N  +  M ) ) )  ->  x  e.  ( 0..^ ( N  +  M ) ) )
2120anim1i 568 . . . . . . . . . . 11  |-  ( ( ( ( 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 10908 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  ZZ )
23 nn0z 10908 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  M  e.  ZZ )
2422, 23anim12i 566 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  -> 
( N  e.  ZZ  /\  M  e.  ZZ ) )
2524ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( N  e. 
NN0  /\  M  e.  NN0 )  /\  x  e.  ( 0..^ ( N  +  M ) ) )  /\  -.  x  e.  ( 0..^ N ) )  ->  ( N  e.  ZZ  /\  M  e.  ZZ ) )
26 fzocatel 11883 . . . . . . . . . . 11  |-  ( ( ( x  e.  ( 0..^ ( N  +  M ) )  /\  -.  x  e.  (
0..^ N ) )  /\  ( N  e.  ZZ  /\  M  e.  ZZ ) )  -> 
( x  -  N
)  e.  ( 0..^ M ) )
2721, 25, 26syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
NN0  /\  M  e.  NN0 )  /\  x  e.  ( 0..^ ( N  +  M ) ) )  /\  -.  x  e.  ( 0..^ N ) )  ->  ( x  -  N )  e.  ( 0..^ M ) )
2827exp31 604 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  -> 
( x  e.  ( 0..^ ( N  +  M ) )  -> 
( -.  x  e.  ( 0..^ N )  ->  ( x  -  N )  e.  ( 0..^ M ) ) ) )
29283adant1 1014 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
x  e.  ( 0..^ ( N  +  M
) )  ->  ( -.  x  e.  (
0..^ N )  -> 
( x  -  N
)  e.  ( 0..^ M ) ) ) )
30 oveq12 6305 . . . . . . . . . . 11  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( ( # `  ( S repeatS  N )
)  +  ( # `  ( S repeatS  M )
) )  =  ( N  +  M ) )
3130oveq2d 6312 . . . . . . . . . 10  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( 0..^ ( ( # `  ( S repeatS  N ) )  +  ( # `  ( S repeatS  M ) ) ) )  =  ( 0..^ ( N  +  M
) ) )
3231eleq2d 2527 . . . . . . . . 9  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( x  e.  ( 0..^ ( (
# `  ( S repeatS  N ) )  +  (
# `  ( S repeatS  M ) ) ) )  <-> 
x  e.  ( 0..^ ( N  +  M
) ) ) )
33 oveq2 6304 . . . . . . . . . . . . 13  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( 0..^ (
# `  ( S repeatS  N ) ) )  =  ( 0..^ N ) )
3433eleq2d 2527 . . . . . . . . . . . 12  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( x  e.  ( 0..^ ( # `  ( S repeatS  N )
) )  <->  x  e.  ( 0..^ N ) ) )
3534notbid 294 . . . . . . . . . . 11  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( -.  x  e.  ( 0..^ ( # `  ( S repeatS  N )
) )  <->  -.  x  e.  ( 0..^ N ) ) )
3635adantr 465 . . . . . . . . . 10  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( -.  x  e.  ( 0..^ ( # `  ( S repeatS  N )
) )  <->  -.  x  e.  ( 0..^ N ) ) )
37 oveq2 6304 . . . . . . . . . . . 12  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( x  -  ( # `  ( S repeatS  N ) ) )  =  ( x  -  N ) )
3837eleq1d 2526 . . . . . . . . . . 11  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( ( x  -  ( # `  ( S repeatS  N ) ) )  e.  ( 0..^ M )  <->  ( x  -  N )  e.  ( 0..^ M ) ) )
3938adantr 465 . . . . . . . . . 10  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( ( x  -  ( # `  ( S repeatS  N ) ) )  e.  ( 0..^ M )  <->  ( x  -  N )  e.  ( 0..^ M ) ) )
4036, 39imbi12d 320 . . . . . . . . 9  |-  ( ( ( # `  ( 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 ) ) ) )
4132, 40imbi12d 320 . . . . . . . 8  |-  ( ( ( # `  ( 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 ) ) ) ) )
4229, 41syl5ibr 221 . . . . . . 7  |-  ( ( ( # `  ( 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 ) ) ) ) )
4319, 42mpcom 36 . . . . . 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 ) ) ) )
4443imp31 432 . . . . 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 ) )
45 repswsymb 12758 . . . . 5  |-  ( ( S  e.  V  /\  M  e.  NN0  /\  (
x  -  ( # `  ( S repeatS  N )
) )  e.  ( 0..^ M ) )  ->  ( ( S repeatS  M ) `  (
x  -  ( # `  ( S repeatS  N )
) ) )  =  S )
4617, 18, 44, 45syl3anc 1228 . . . 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 )
4716, 46ifeqda 3977 . . 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 )
486, 47mpteq12dva 4534 . 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 ) )
49 ovex 6324 . . . 4  |-  ( S repeatS  N )  e.  _V
50 ovex 6324 . . . 4  |-  ( S repeatS  M )  e.  _V
5149, 50pm3.2i 455 . . 3  |-  ( ( S repeatS  N )  e.  _V  /\  ( S repeatS  M )  e.  _V )
52 ccatfval 12601 . . 3  |-  ( ( ( S repeatS  N )  e.  _V  /\  ( S repeatS  M )  e.  _V )  ->  ( ( S repeatS  N ) ++  ( 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 )
) ) ) ) ) )
5351, 52mp1i 12 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
( S repeatS  N ) ++  ( 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
) ) ) ) ) ) )
54 nn0addcl 10852 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  -> 
( N  +  M
)  e.  NN0 )
55543adant1 1014 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  ( N  +  M )  e.  NN0 )
56 reps 12754 . . 3  |-  ( ( S  e.  V  /\  ( N  +  M
)  e.  NN0 )  ->  ( S repeatS  ( N  +  M ) )  =  ( x  e.  ( 0..^ ( N  +  M ) )  |->  S ) )
577, 55, 56syl2anc 661 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  ( S repeatS  ( N  +  M
) )  =  ( x  e.  ( 0..^ ( N  +  M
) )  |->  S ) )
5848, 53, 573eqtr4d 2508 1  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
( S repeatS  N ) ++  ( S repeatS  M ) )  =  ( S repeatS  ( N  +  M )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   _Vcvv 3109   ifcif 3944    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   0cc0 9509    + caddc 9512    - cmin 9824   NN0cn0 10816   ZZcz 10885  ..^cfzo 11821   #chash 12408   ++ cconcat 12540   repeatS creps 12545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-concat 12548  df-reps 12553
This theorem is referenced by:  repswcshw  12792  repsw2  12900  repsw3  12901
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