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Theorem ccatopth 11776
Description: An opth 4435-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 6088 . . . 4  |-  ( ( A concat  B )  =  ( C concat  D )  ->  ( ( A concat  B ) substr  <. 0 ,  ( # `  A
) >. )  =  ( ( C concat  D ) substr  <. 0 ,  ( # `  A ) >. )
)
2 swrdccat1 11774 . . . . . 6  |-  ( ( A  e. Word  X  /\  B  e. Word  X )  ->  ( ( A concat  B
) substr  <. 0 ,  (
# `  A ) >. )  =  A )
323ad2ant1 978 . . . . 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 959 . . . . . . . 8  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( # `  A )  =  ( # `  C
) )
54opeq2d 3991 . . . . . . 7  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  ->  <. 0 ,  ( # `  A ) >.  =  <. 0 ,  ( # `  C
) >. )
65oveq2d 6097 . . . . . 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 11774 . . . . . . 7  |-  ( ( C  e. Word  X  /\  D  e. Word  X )  ->  ( ( C concat  D
) substr  <. 0 ,  (
# `  C ) >. )  =  C )
873ad2ant2 979 . . . . . 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 2468 . . . . 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 2450 . . . 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 211 . . 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 448 . . . . . 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 962 . . . . . . 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 5732 . . . . . . . 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 960 . . . . . . . . 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 11744 . . . . . . . . 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 961 . . . . . . . . 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 11744 . . . . . . . . 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 2476 . . . . . . 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 3992 . . . . . 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 6099 . . . . 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 11775 . . . . . 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 11775 . . . . . 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 2476 . . . 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 424 . . 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 520 . 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 6090 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A concat  B )  =  ( C concat  D
) )
3230, 31impbid1 195 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   <.cop 3817   ` cfv 5454  (class class class)co 6081   0cc0 8990    + caddc 8993   #chash 11618  Word cword 11717   concat cconcat 11718   substr csubstr 11720
This theorem is referenced by:  ccatopth2  11777  ccatlcan  11778  splval2  11786  efgredleme  15375  efgredlemc  15377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-concat 11724  df-substr 11726
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