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Theorem repswccat 12883
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 12874 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( # `  ( S repeatS  N ) )  =  N )
213adant3 1027 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  ( # `
 ( S repeatS  N
) )  =  N )
3 repswlen 12874 . . . . . 6  |-  ( ( S  e.  V  /\  M  e.  NN0 )  -> 
( # `  ( S repeatS  M ) )  =  M )
433adant2 1026 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  ( # `
 ( S repeatS  M
) )  =  M )
52, 4oveq12d 6306 . . . 4  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
( # `  ( S repeatS  N ) )  +  ( # `  ( S repeatS  M ) ) )  =  ( N  +  M ) )
65oveq2d 6304 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
0..^ ( ( # `  ( S repeatS  N )
)  +  ( # `  ( S repeatS  M )
) ) )  =  ( 0..^ ( N  +  M ) ) )
7 simp1 1007 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  S  e.  V )
87adantr 467 . . . . . . 7  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  /\  x  e.  (
0..^ ( # `  ( S repeatS  N ) ) ) )  ->  S  e.  V )
9 simpl2 1011 . . . . . . 7  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  /\  x  e.  (
0..^ ( # `  ( S repeatS  N ) ) ) )  ->  N  e.  NN0 )
102oveq2d 6304 . . . . . . . . 9  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
0..^ ( # `  ( S repeatS  N ) ) )  =  ( 0..^ N ) )
1110eleq2d 2513 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
x  e.  ( 0..^ ( # `  ( S repeatS  N ) ) )  <-> 
x  e.  ( 0..^ N ) ) )
1211biimpa 487 . . . . . . 7  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  /\  x  e.  (
0..^ ( # `  ( S repeatS  N ) ) ) )  ->  x  e.  ( 0..^ N ) )
138, 9, 123jca 1187 . . . . . 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 720 . . . . 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 12872 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  x  e.  ( 0..^ N ) )  ->  ( ( S repeatS  N ) `  x
)  =  S )
1614, 15syl 17 . . . 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 731 . . . . 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 1048 . . . . 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 535 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M ) )
20 simpr 463 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN0  /\  M  e.  NN0 )  /\  x  e.  (
0..^ ( N  +  M ) ) )  ->  x  e.  ( 0..^ ( N  +  M ) ) )
2120anim1i 571 . . . . . . . . . . 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 10957 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  ZZ )
23 nn0z 10957 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  M  e.  ZZ )
2422, 23anim12i 569 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  -> 
( N  e.  ZZ  /\  M  e.  ZZ ) )
2524ad2antrr 731 . . . . . . . . . . 11  |-  ( ( ( ( N  e. 
NN0  /\  M  e.  NN0 )  /\  x  e.  ( 0..^ ( N  +  M ) ) )  /\  -.  x  e.  ( 0..^ N ) )  ->  ( N  e.  ZZ  /\  M  e.  ZZ ) )
26 fzocatel 11975 . . . . . . . . . . 11  |-  ( ( ( x  e.  ( 0..^ ( N  +  M ) )  /\  -.  x  e.  (
0..^ N ) )  /\  ( N  e.  ZZ  /\  M  e.  ZZ ) )  -> 
( x  -  N
)  e.  ( 0..^ M ) )
2721, 25, 26syl2anc 666 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
NN0  /\  M  e.  NN0 )  /\  x  e.  ( 0..^ ( N  +  M ) ) )  /\  -.  x  e.  ( 0..^ N ) )  ->  ( x  -  N )  e.  ( 0..^ M ) )
2827exp31 608 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  -> 
( x  e.  ( 0..^ ( N  +  M ) )  -> 
( -.  x  e.  ( 0..^ N )  ->  ( x  -  N )  e.  ( 0..^ M ) ) ) )
29283adant1 1025 . . . . . . . 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 6297 . . . . . . . . . . 11  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( ( # `  ( S repeatS  N )
)  +  ( # `  ( S repeatS  M )
) )  =  ( N  +  M ) )
3130oveq2d 6304 . . . . . . . . . 10  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( 0..^ ( ( # `  ( S repeatS  N ) )  +  ( # `  ( S repeatS  M ) ) ) )  =  ( 0..^ ( N  +  M
) ) )
3231eleq2d 2513 . . . . . . . . 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 6296 . . . . . . . . . . . . 13  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( 0..^ (
# `  ( S repeatS  N ) ) )  =  ( 0..^ N ) )
3433eleq2d 2513 . . . . . . . . . . . 12  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( x  e.  ( 0..^ ( # `  ( S repeatS  N )
) )  <->  x  e.  ( 0..^ N ) ) )
3534notbid 296 . . . . . . . . . . 11  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( -.  x  e.  ( 0..^ ( # `  ( S repeatS  N )
) )  <->  -.  x  e.  ( 0..^ N ) ) )
3635adantr 467 . . . . . . . . . 10  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( -.  x  e.  ( 0..^ ( # `  ( S repeatS  N )
) )  <->  -.  x  e.  ( 0..^ N ) ) )
37 oveq2 6296 . . . . . . . . . . . 12  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( x  -  ( # `  ( S repeatS  N ) ) )  =  ( x  -  N ) )
3837eleq1d 2512 . . . . . . . . . . 11  |-  ( (
# `  ( S repeatS  N ) )  =  N  ->  ( ( x  -  ( # `  ( S repeatS  N ) ) )  e.  ( 0..^ M )  <->  ( x  -  N )  e.  ( 0..^ M ) ) )
3938adantr 467 . . . . . . . . . 10  |-  ( ( ( # `  ( S repeatS  N ) )  =  N  /\  ( # `  ( S repeatS  M )
)  =  M )  ->  ( ( x  -  ( # `  ( S repeatS  N ) ) )  e.  ( 0..^ M )  <->  ( x  -  N )  e.  ( 0..^ M ) ) )
4036, 39imbi12d 322 . . . . . . . . 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 322 . . . . . . . 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 225 . . . . . . 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 37 . . . . . 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 434 . . . . 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 12872 . . . . 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 1267 . . . 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 3913 . . 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 4479 . 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 6316 . . . 4  |-  ( S repeatS  N )  e.  _V
50 ovex 6316 . . . 4  |-  ( S repeatS  M )  e.  _V
5149, 50pm3.2i 457 . . 3  |-  ( ( S repeatS  N )  e.  _V  /\  ( S repeatS  M )  e.  _V )
52 ccatfval 12716 . . 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 13 . 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 10902 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  -> 
( N  +  M
)  e.  NN0 )
55543adant1 1025 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  ( N  +  M )  e.  NN0 )
56 reps 12868 . . 3  |-  ( ( S  e.  V  /\  ( N  +  M
)  e.  NN0 )  ->  ( S repeatS  ( N  +  M ) )  =  ( x  e.  ( 0..^ ( N  +  M ) )  |->  S ) )
577, 55, 56syl2anc 666 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  ( S repeatS  ( N  +  M
) )  =  ( x  e.  ( 0..^ ( N  +  M
) )  |->  S ) )
5848, 53, 573eqtr4d 2494 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 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   _Vcvv 3044   ifcif 3880    |-> cmpt 4460   ` cfv 5581  (class class class)co 6288   0cc0 9536    + caddc 9539    - cmin 9857   NN0cn0 10866   ZZcz 10934  ..^cfzo 11912   #chash 12512   ++ cconcat 12655   repeatS creps 12660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-fzo 11913  df-hash 12513  df-concat 12663  df-reps 12668
This theorem is referenced by:  repswcshw  12906  repsw2  13018  repsw3  13019
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