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Theorem swrdccatin12d 12390
Description: The subword of a concatenation of two words within both of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by Mario Carneiro/AV, 21-Oct-2018.)
Hypotheses
Ref Expression
swrdccatind.l  |-  ( ph  ->  ( # `  A
)  =  L )
swrdccatind.w  |-  ( ph  ->  ( A  e. Word  V  /\  B  e. Word  V ) )
swrdccatin12d.1  |-  ( ph  ->  M  e.  ( 0 ... L ) )
swrdccatin12d.2  |-  ( ph  ->  N  e.  ( L ... ( L  +  ( # `  B ) ) ) )
Assertion
Ref Expression
swrdccatin12d  |-  ( ph  ->  ( ( A concat  B
) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <. 0 ,  ( N  -  L ) >. )
) )

Proof of Theorem swrdccatin12d
StepHypRef Expression
1 swrdccatind.l . 2  |-  ( ph  ->  ( # `  A
)  =  L )
2 swrdccatind.w . . . . . 6  |-  ( ph  ->  ( A  e. Word  V  /\  B  e. Word  V ) )
32adantl 466 . . . . 5  |-  ( ( ( # `  A
)  =  L  /\  ph )  ->  ( A  e. Word  V  /\  B  e. Word  V ) )
4 swrdccatin12d.1 . . . . . . . 8  |-  ( ph  ->  M  e.  ( 0 ... L ) )
5 swrdccatin12d.2 . . . . . . . 8  |-  ( ph  ->  N  e.  ( L ... ( L  +  ( # `  B ) ) ) )
64, 5jca 532 . . . . . . 7  |-  ( ph  ->  ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )
76adantl 466 . . . . . 6  |-  ( ( ( # `  A
)  =  L  /\  ph )  ->  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )
8 oveq2 6097 . . . . . . . . 9  |-  ( (
# `  A )  =  L  ->  ( 0 ... ( # `  A
) )  =  ( 0 ... L ) )
98eleq2d 2508 . . . . . . . 8  |-  ( (
# `  A )  =  L  ->  ( M  e.  ( 0 ... ( # `  A
) )  <->  M  e.  ( 0 ... L
) ) )
10 id 22 . . . . . . . . . 10  |-  ( (
# `  A )  =  L  ->  ( # `  A )  =  L )
11 oveq1 6096 . . . . . . . . . 10  |-  ( (
# `  A )  =  L  ->  ( (
# `  A )  +  ( # `  B
) )  =  ( L  +  ( # `  B ) ) )
1210, 11oveq12d 6107 . . . . . . . . 9  |-  ( (
# `  A )  =  L  ->  ( (
# `  A ) ... ( ( # `  A
)  +  ( # `  B ) ) )  =  ( L ... ( L  +  ( # `
 B ) ) ) )
1312eleq2d 2508 . . . . . . . 8  |-  ( (
# `  A )  =  L  ->  ( N  e.  ( ( # `  A ) ... (
( # `  A )  +  ( # `  B
) ) )  <->  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )
149, 13anbi12d 710 . . . . . . 7  |-  ( (
# `  A )  =  L  ->  ( ( M  e.  ( 0 ... ( # `  A
) )  /\  N  e.  ( ( # `  A
) ... ( ( # `  A )  +  (
# `  B )
) ) )  <->  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) ) )
1514adantr 465 . . . . . 6  |-  ( ( ( # `  A
)  =  L  /\  ph )  ->  ( ( M  e.  ( 0 ... ( # `  A
) )  /\  N  e.  ( ( # `  A
) ... ( ( # `  A )  +  (
# `  B )
) ) )  <->  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) ) )
167, 15mpbird 232 . . . . 5  |-  ( ( ( # `  A
)  =  L  /\  ph )  ->  ( M  e.  ( 0 ... ( # `
 A ) )  /\  N  e.  ( ( # `  A
) ... ( ( # `  A )  +  (
# `  B )
) ) ) )
17 eqid 2441 . . . . . 6  |-  ( # `  A )  =  (
# `  A )
1817swrdccatin12 12380 . . . . 5  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ... ( # `
 A ) )  /\  N  e.  ( ( # `  A
) ... ( ( # `  A )  +  (
# `  B )
) ) )  -> 
( ( A concat  B
) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  (
# `  A ) >. ) concat  ( B substr  <. 0 ,  ( N  -  ( # `  A ) ) >. ) ) ) )
193, 16, 18sylc 60 . . . 4  |-  ( ( ( # `  A
)  =  L  /\  ph )  ->  ( ( A concat  B ) substr  <. M ,  N >. )  =  ( ( A substr  <. M , 
( # `  A )
>. ) concat  ( B substr  <. 0 ,  ( N  -  ( # `  A ) ) >. ) ) )
2019ex 434 . . 3  |-  ( (
# `  A )  =  L  ->  ( ph  ->  ( ( A concat  B
) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  (
# `  A ) >. ) concat  ( B substr  <. 0 ,  ( N  -  ( # `  A ) ) >. ) ) ) )
21 opeq2 4058 . . . . . 6  |-  ( (
# `  A )  =  L  ->  <. M , 
( # `  A )
>.  =  <. M ,  L >. )
2221oveq2d 6105 . . . . 5  |-  ( (
# `  A )  =  L  ->  ( A substr  <. M ,  ( # `  A ) >. )  =  ( A substr  <. M ,  L >. ) )
23 oveq2 6097 . . . . . . 7  |-  ( (
# `  A )  =  L  ->  ( N  -  ( # `  A
) )  =  ( N  -  L ) )
2423opeq2d 4064 . . . . . 6  |-  ( (
# `  A )  =  L  ->  <. 0 ,  ( N  -  ( # `  A ) ) >.  =  <. 0 ,  ( N  -  L ) >. )
2524oveq2d 6105 . . . . 5  |-  ( (
# `  A )  =  L  ->  ( B substr  <. 0 ,  ( N  -  ( # `  A
) ) >. )  =  ( B substr  <. 0 ,  ( N  -  L ) >. )
)
2622, 25oveq12d 6107 . . . 4  |-  ( (
# `  A )  =  L  ->  ( ( A substr  <. M ,  (
# `  A ) >. ) concat  ( B substr  <. 0 ,  ( N  -  ( # `  A ) ) >. ) )  =  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <.
0 ,  ( N  -  L ) >.
) ) )
2726eqeq2d 2452 . . 3  |-  ( (
# `  A )  =  L  ->  ( ( ( A concat  B ) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  ( # `  A ) >. ) concat  ( B substr  <. 0 ,  ( N  -  ( # `  A ) ) >.
) )  <->  ( ( A concat  B ) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <.
0 ,  ( N  -  L ) >.
) ) ) )
2820, 27sylibd 214 . 2  |-  ( (
# `  A )  =  L  ->  ( ph  ->  ( ( A concat  B
) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <. 0 ,  ( N  -  L ) >. )
) ) )
291, 28mpcom 36 1  |-  ( ph  ->  ( ( A concat  B
) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <. 0 ,  ( N  -  L ) >. )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   <.cop 3881   ` cfv 5416  (class class class)co 6089   0cc0 9280    + caddc 9283    - cmin 9593   ...cfz 11435   #chash 12101  Word cword 12219   concat cconcat 12221   substr csubstr 12223
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-hash 12102  df-word 12227  df-concat 12229  df-substr 12231
This theorem is referenced by: (None)
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