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Theorem swrdccatin12lem2c 12829
Description: Lemma for swrdccatin12lem2 12830 and swrdccatin12lem3 12831. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
Hypothesis
Ref Expression
swrdccatin12.l  |-  L  =  ( # `  A
)
Assertion
Ref Expression
swrdccatin12lem2c  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( ( A ++  B )  e. Word  V  /\  M  e.  (
0 ... N )  /\  N  e.  ( 0 ... ( # `  ( A ++  B ) ) ) ) )

Proof of Theorem swrdccatin12lem2c
StepHypRef Expression
1 ccatcl 12707 . . 3  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( A ++  B )  e. Word  V )
21adantr 466 . 2  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( A ++  B )  e. Word  V
)
3 elfz0fzfz0 11893 . . 3  |-  ( ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) )  ->  M  e.  ( 0 ... N ) )
43adantl 467 . 2  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  M  e.  ( 0 ... N
) )
5 elfzuz2 11802 . . . . . . 7  |-  ( M  e.  ( 0 ... L )  ->  L  e.  ( ZZ>= `  0 )
)
65adantl 467 . . . . . 6  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... L ) )  ->  L  e.  ( ZZ>= `  0 )
)
7 fzss1 11835 . . . . . 6  |-  ( L  e.  ( ZZ>= `  0
)  ->  ( L ... ( L  +  (
# `  B )
) )  C_  (
0 ... ( L  +  ( # `  B ) ) ) )
86, 7syl 17 . . . . 5  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... L ) )  ->  ( L ... ( L  +  (
# `  B )
) )  C_  (
0 ... ( L  +  ( # `  B ) ) ) )
98sseld 3469 . . . 4  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... L ) )  ->  ( N  e.  ( L ... ( L  +  ( # `  B
) ) )  ->  N  e.  ( 0 ... ( L  +  ( # `  B ) ) ) ) )
109impr 623 . . 3  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  N  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )
11 ccatlen 12708 . . . . . . 7  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( # `  ( A ++  B ) )  =  ( ( # `  A
)  +  ( # `  B ) ) )
12 swrdccatin12.l . . . . . . . . . 10  |-  L  =  ( # `  A
)
1312eqcomi 2442 . . . . . . . . 9  |-  ( # `  A )  =  L
1413a1i 11 . . . . . . . 8  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( # `  A
)  =  L )
1514oveq1d 6320 . . . . . . 7  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( ( # `  A
)  +  ( # `  B ) )  =  ( L  +  (
# `  B )
) )
1611, 15eqtrd 2470 . . . . . 6  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( # `  ( A ++  B ) )  =  ( L  +  (
# `  B )
) )
1716oveq2d 6321 . . . . 5  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( 0 ... ( # `
 ( A ++  B
) ) )  =  ( 0 ... ( L  +  ( # `  B
) ) ) )
1817eleq2d 2499 . . . 4  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( N  e.  ( 0 ... ( # `  ( A ++  B ) ) )  <->  N  e.  ( 0 ... ( L  +  ( # `  B
) ) ) ) )
1918adantr 466 . . 3  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( N  e.  ( 0 ... ( # `
 ( A ++  B
) ) )  <->  N  e.  ( 0 ... ( L  +  ( # `  B
) ) ) ) )
2010, 19mpbird 235 . 2  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  N  e.  ( 0 ... ( # `
 ( A ++  B
) ) ) )
212, 4, 203jca 1185 1  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( ( A ++  B )  e. Word  V  /\  M  e.  (
0 ... N )  /\  N  e.  ( 0 ... ( # `  ( A ++  B ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    C_ wss 3442   ` cfv 5601  (class class class)co 6305   0cc0 9538    + caddc 9541   ZZ>=cuz 11159   ...cfz 11782   #chash 12512  Word cword 12643   ++ cconcat 12645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651  df-concat 12653
This theorem is referenced by:  swrdccatin12lem2  12830  swrdccatin12lem3  12831  pfxccatin12lem2  38354
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