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Theorem ccatopth 12686
Description: An opth 4711-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 ++  B
)  =  ( C ++  D )  <->  ( A  =  C  /\  B  =  D ) ) )

Proof of Theorem ccatopth
StepHypRef Expression
1 oveq1 6277 . . . 4  |-  ( ( A ++  B )  =  ( C ++  D )  ->  ( ( A ++  B ) substr  <. 0 ,  ( # `  A
) >. )  =  ( ( C ++  D ) substr  <. 0 ,  ( # `  A ) >. )
)
2 swrdccat1 12673 . . . . . 6  |-  ( ( A  e. Word  X  /\  B  e. Word  X )  ->  ( ( A ++  B
) substr  <. 0 ,  (
# `  A ) >. )  =  A )
323ad2ant1 1015 . . . . 5  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( A ++  B
) substr  <. 0 ,  (
# `  A ) >. )  =  A )
4 simp3 996 . . . . . . . 8  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( # `  A )  =  ( # `  C
) )
54opeq2d 4210 . . . . . . 7  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  ->  <. 0 ,  ( # `  A ) >.  =  <. 0 ,  ( # `  C
) >. )
65oveq2d 6286 . . . . . 6  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( C ++  D
) substr  <. 0 ,  (
# `  A ) >. )  =  ( ( C ++  D ) substr  <. 0 ,  ( # `  C
) >. ) )
7 swrdccat1 12673 . . . . . . 7  |-  ( ( C  e. Word  X  /\  D  e. Word  X )  ->  ( ( C ++  D
) substr  <. 0 ,  (
# `  C ) >. )  =  C )
873ad2ant2 1016 . . . . . 6  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( C ++  D
) substr  <. 0 ,  (
# `  C ) >. )  =  C )
96, 8eqtrd 2495 . . . . 5  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( C ++  D
) substr  <. 0 ,  (
# `  A ) >. )  =  C )
103, 9eqeq12d 2476 . . . 4  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( ( A ++  B ) substr  <. 0 ,  ( # `  A
) >. )  =  ( ( C ++  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 ++  B
)  =  ( C ++  D )  ->  A  =  C ) )
12 simpr 459 . . . . . 6  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A ++  B )  =  ( C ++  D ) )  ->  ( A ++  B
)  =  ( C ++  D ) )
13 simpl3 999 . . . . . . 7  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A ++  B )  =  ( C ++  D ) )  ->  ( # `  A
)  =  ( # `  C ) )
1412fveq2d 5852 . . . . . . . 8  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A ++  B )  =  ( C ++  D ) )  ->  ( # `  ( A ++  B ) )  =  ( # `  ( C ++  D ) ) )
15 simpl1 997 . . . . . . . . 9  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A ++  B )  =  ( C ++  D ) )  ->  ( A  e. Word  X  /\  B  e. Word  X
) )
16 ccatlen 12583 . . . . . . . . 9  |-  ( ( A  e. Word  X  /\  B  e. Word  X )  ->  ( # `  ( A ++  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 ++  B )  =  ( C ++  D ) )  ->  ( # `  ( A ++  B ) )  =  ( ( # `  A
)  +  ( # `  B ) ) )
18 simpl2 998 . . . . . . . . 9  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A ++  B )  =  ( C ++  D ) )  ->  ( C  e. Word  X  /\  D  e. Word  X
) )
19 ccatlen 12583 . . . . . . . . 9  |-  ( ( C  e. Word  X  /\  D  e. Word  X )  ->  ( # `  ( C ++  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 ++  B )  =  ( C ++  D ) )  ->  ( # `  ( C ++  D ) )  =  ( ( # `  C
)  +  ( # `  D ) ) )
2114, 17, 203eqtr3d 2503 . . . . . . 7  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A ++  B )  =  ( C ++  D ) )  ->  ( ( # `  A )  +  (
# `  B )
)  =  ( (
# `  C )  +  ( # `  D
) ) )
2213, 21opeq12d 4211 . . . . . 6  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A ++  B )  =  ( C ++  D ) )  ->  <. ( # `  A
) ,  ( (
# `  A )  +  ( # `  B
) ) >.  =  <. (
# `  C ) ,  ( ( # `  C )  +  (
# `  D )
) >. )
2312, 22oveq12d 6288 . . . . 5  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A ++  B )  =  ( C ++  D ) )  ->  ( ( A ++  B ) substr  <. ( # `
 A ) ,  ( ( # `  A
)  +  ( # `  B ) ) >.
)  =  ( ( C ++  D ) substr  <. (
# `  C ) ,  ( ( # `  C )  +  (
# `  D )
) >. ) )
24 swrdccat2 12674 . . . . . 6  |-  ( ( A  e. Word  X  /\  B  e. Word  X )  ->  ( ( A ++  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 ++  B )  =  ( C ++  D ) )  ->  ( ( A ++  B ) substr  <. ( # `
 A ) ,  ( ( # `  A
)  +  ( # `  B ) ) >.
)  =  B )
26 swrdccat2 12674 . . . . . 6  |-  ( ( C  e. Word  X  /\  D  e. Word  X )  ->  ( ( C ++  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 ++  B )  =  ( C ++  D ) )  ->  ( ( C ++  D ) substr  <. ( # `
 C ) ,  ( ( # `  C
)  +  ( # `  D ) ) >.
)  =  D )
2823, 25, 273eqtr3d 2503 . . . 4  |-  ( ( ( ( A  e. Word  X  /\  B  e. Word  X
)  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `
 A )  =  ( # `  C
) )  /\  ( A ++  B )  =  ( C ++  D ) )  ->  B  =  D )
2928ex 432 . . 3  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( A ++  B
)  =  ( C ++  D )  ->  B  =  D ) )
3011, 29jcad 531 . 2  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( A ++  B
)  =  ( C ++  D )  ->  ( A  =  C  /\  B  =  D )
) )
31 oveq12 6279 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A ++  B )  =  ( C ++  D
) )
3230, 31impbid1 203 1  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  A
)  =  ( # `  C ) )  -> 
( ( A ++  B
)  =  ( C ++  D )  <->  ( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   <.cop 4022   ` cfv 5570  (class class class)co 6270   0cc0 9481    + caddc 9484   #chash 12387  Word cword 12518   ++ cconcat 12520   substr csubstr 12522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-concat 12528  df-substr 12530
This theorem is referenced by:  ccatopth2  12687  ccatlcan  12688  splval2  12724  s2eq2s1eq  12872  efgredleme  16960  efgredlemc  16962
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