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Theorem ccatopth 12675
Description: An opth 4727-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
Assertion
Ref Expression
ccatopth  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( A concat  B
)  =  ( C concat  D )  <->  ( A  =  C  /\  B  =  D ) ) )

Proof of Theorem ccatopth
StepHypRef Expression
1 oveq1 6302 . . . 4  |-  ( ( A concat  B )  =  ( C concat  D )  ->  ( ( A concat  B ) substr  <. 0 ,  ( # `  A
) >. )  =  ( ( C concat  D ) substr  <. 0 ,  ( # `  A ) >. )
)
2 swrdccat1 12662 . . . . . 6  |-  ( ( A  e. Word  X  /\  B  e. Word  X )  ->  ( ( A concat  B
) substr  <. 0 ,  (
# `  A ) >. )  =  A )
323ad2ant1 1017 . . . . 5  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( A concat  B
) substr  <. 0 ,  (
# `  A ) >. )  =  A )
4 simp3 998 . . . . . . . 8  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( # `  A )  =  ( # `  C
) )
54opeq2d 4226 . . . . . . 7  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  ->  <. 0 ,  ( # `  A ) >.  =  <. 0 ,  ( # `  C
) >. )
65oveq2d 6311 . . . . . 6  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( C concat  D
) substr  <. 0 ,  (
# `  A ) >. )  =  ( ( C concat  D ) substr  <. 0 ,  ( # `  C
) >. ) )
7 swrdccat1 12662 . . . . . . 7  |-  ( ( C  e. Word  X  /\  D  e. Word  X )  ->  ( ( C concat  D
) substr  <. 0 ,  (
# `  C ) >. )  =  C )
873ad2ant2 1018 . . . . . 6  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( C concat  D
) substr  <. 0 ,  (
# `  C ) >. )  =  C )
96, 8eqtrd 2508 . . . . 5  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( C concat  D
) substr  <. 0 ,  (
# `  A ) >. )  =  C )
103, 9eqeq12d 2489 . . . 4  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( ( A concat  B ) substr  <. 0 ,  ( # `  A
) >. )  =  ( ( C concat  D ) substr  <. 0 ,  ( # `  A ) >. )  <->  A  =  C ) )
111, 10syl5ib 219 . . 3  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( A concat  B
)  =  ( C concat  D )  ->  A  =  C ) )
12 simpr 461 . . . . . 6  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A concat  B )  =  ( C concat  D ) )  ->  ( A concat  B
)  =  ( C concat  D ) )
13 simpl3 1001 . . . . . . 7  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A concat  B )  =  ( C concat  D ) )  ->  ( # `  A
)  =  ( # `  C ) )
1412fveq2d 5876 . . . . . . . 8  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A concat  B )  =  ( C concat  D ) )  ->  ( # `  ( A concat  B ) )  =  ( # `  ( C concat  D ) ) )
15 simpl1 999 . . . . . . . . 9  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A concat  B )  =  ( C concat  D ) )  ->  ( A  e. Word  X  /\  B  e. Word  X
) )
16 ccatlen 12574 . . . . . . . . 9  |-  ( ( A  e. Word  X  /\  B  e. Word  X )  ->  ( # `  ( A concat  B ) )  =  ( ( # `  A
)  +  ( # `  B ) ) )
1715, 16syl 16 . . . . . . . 8  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A concat  B )  =  ( C concat  D ) )  ->  ( # `  ( A concat  B ) )  =  ( ( # `  A
)  +  ( # `  B ) ) )
18 simpl2 1000 . . . . . . . . 9  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A concat  B )  =  ( C concat  D ) )  ->  ( C  e. Word  X  /\  D  e. Word  X
) )
19 ccatlen 12574 . . . . . . . . 9  |-  ( ( C  e. Word  X  /\  D  e. Word  X )  ->  ( # `  ( C concat  D ) )  =  ( ( # `  C
)  +  ( # `  D ) ) )
2018, 19syl 16 . . . . . . . 8  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A concat  B )  =  ( C concat  D ) )  ->  ( # `  ( C concat  D ) )  =  ( ( # `  C
)  +  ( # `  D ) ) )
2114, 17, 203eqtr3d 2516 . . . . . . 7  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A concat  B )  =  ( C concat  D ) )  ->  ( ( # `  A )  +  (
# `  B )
)  =  ( (
# `  C )  +  ( # `  D
) ) )
2213, 21opeq12d 4227 . . . . . 6  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A concat  B )  =  ( C concat  D ) )  ->  <. ( # `  A
) ,  ( (
# `  A )  +  ( # `  B
) ) >.  =  <. (
# `  C ) ,  ( ( # `  C )  +  (
# `  D )
) >. )
2312, 22oveq12d 6313 . . . . 5  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A concat  B )  =  ( C concat  D ) )  ->  ( ( A concat  B ) substr  <. ( # `  A ) ,  ( ( # `  A
)  +  ( # `  B ) ) >.
)  =  ( ( C concat  D ) substr  <. (
# `  C ) ,  ( ( # `  C )  +  (
# `  D )
) >. ) )
24 swrdccat2 12663 . . . . . 6  |-  ( ( A  e. Word  X  /\  B  e. Word  X )  ->  ( ( A concat  B
) substr  <. ( # `  A
) ,  ( (
# `  A )  +  ( # `  B
) ) >. )  =  B )
2515, 24syl 16 . . . . 5  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A concat  B )  =  ( C concat  D ) )  ->  ( ( A concat  B ) substr  <. ( # `  A ) ,  ( ( # `  A
)  +  ( # `  B ) ) >.
)  =  B )
26 swrdccat2 12663 . . . . . 6  |-  ( ( C  e. Word  X  /\  D  e. Word  X )  ->  ( ( C concat  D
) substr  <. ( # `  C
) ,  ( (
# `  C )  +  ( # `  D
) ) >. )  =  D )
2718, 26syl 16 . . . . 5  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A concat  B )  =  ( C concat  D ) )  ->  ( ( C concat  D ) substr  <. ( # `  C ) ,  ( ( # `  C
)  +  ( # `  D ) ) >.
)  =  D )
2823, 25, 273eqtr3d 2516 . . . 4  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A concat  B )  =  ( C concat  D ) )  ->  B  =  D )
2928ex 434 . . 3  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( A concat  B
)  =  ( C concat  D )  ->  B  =  D ) )
3011, 29jcad 533 . 2  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( A concat  B
)  =  ( C concat  D )  ->  ( A  =  C  /\  B  =  D )
) )
31 oveq12 6304 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A concat  B )  =  ( C concat  D
) )
3230, 31impbid1 203 1  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( A concat  B
)  =  ( C concat  D )  <->  ( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   <.cop 4039   ` cfv 5594  (class class class)co 6295   0cc0 9504    + caddc 9507   #chash 12385  Word cword 12515   concat cconcat 12517   substr csubstr 12519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-concat 12525  df-substr 12527
This theorem is referenced by:  ccatopth2  12676  ccatlcan  12677  splval2  12713  s2eq2s1eq  12861  efgredleme  16634  efgredlemc  16636
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