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Theorem ccatopth2 12469
Description: An opth 4666-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
Assertion
Ref Expression
ccatopth2  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( ( A concat  B
)  =  ( C concat  D )  <->  ( A  =  C  /\  B  =  D ) ) )

Proof of Theorem ccatopth2
StepHypRef Expression
1 fveq2 5791 . . . 4  |-  ( ( A concat  B )  =  ( C concat  D )  ->  ( # `  ( A concat  B ) )  =  ( # `  ( C concat  D ) ) )
2 ccatlen 12379 . . . . . . . 8  |-  ( ( A  e. Word  X  /\  B  e. Word  X )  ->  ( # `  ( A concat  B ) )  =  ( ( # `  A
)  +  ( # `  B ) ) )
323ad2ant1 1009 . . . . . . 7  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  ( A concat  B ) )  =  ( ( # `  A
)  +  ( # `  B ) ) )
4 simp3 990 . . . . . . . 8  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  B )  =  ( # `  D
) )
54oveq2d 6208 . . . . . . 7  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( ( # `  A
)  +  ( # `  B ) )  =  ( ( # `  A
)  +  ( # `  D ) ) )
63, 5eqtrd 2492 . . . . . 6  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  ( A concat  B ) )  =  ( ( # `  A
)  +  ( # `  D ) ) )
7 ccatlen 12379 . . . . . . 7  |-  ( ( C  e. Word  X  /\  D  e. Word  X )  ->  ( # `  ( C concat  D ) )  =  ( ( # `  C
)  +  ( # `  D ) ) )
873ad2ant2 1010 . . . . . 6  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  ( C concat  D ) )  =  ( ( # `  C
)  +  ( # `  D ) ) )
96, 8eqeq12d 2473 . . . . 5  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( ( # `  ( A concat  B ) )  =  ( # `  ( C concat  D ) )  <->  ( ( # `
 A )  +  ( # `  D
) )  =  ( ( # `  C
)  +  ( # `  D ) ) ) )
10 simp1l 1012 . . . . . . . 8  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  ->  A  e. Word  X )
11 lencl 12353 . . . . . . . 8  |-  ( A  e. Word  X  ->  ( # `
 A )  e. 
NN0 )
1210, 11syl 16 . . . . . . 7  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  A )  e.  NN0 )
1312nn0cnd 10741 . . . . . 6  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  A )  e.  CC )
14 simp2l 1014 . . . . . . . 8  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  ->  C  e. Word  X )
15 lencl 12353 . . . . . . . 8  |-  ( C  e. Word  X  ->  ( # `
 C )  e. 
NN0 )
1614, 15syl 16 . . . . . . 7  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  C )  e.  NN0 )
1716nn0cnd 10741 . . . . . 6  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  C )  e.  CC )
18 simp2r 1015 . . . . . . . 8  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  ->  D  e. Word  X )
19 lencl 12353 . . . . . . . 8  |-  ( D  e. Word  X  ->  ( # `
 D )  e. 
NN0 )
2018, 19syl 16 . . . . . . 7  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  D )  e.  NN0 )
2120nn0cnd 10741 . . . . . 6  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  D )  e.  CC )
2213, 17, 21addcan2d 9676 . . . . 5  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( ( ( # `  A )  +  (
# `  D )
)  =  ( (
# `  C )  +  ( # `  D
) )  <->  ( # `  A
)  =  ( # `  C ) ) )
239, 22bitrd 253 . . . 4  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( ( # `  ( A concat  B ) )  =  ( # `  ( C concat  D ) )  <->  ( # `  A
)  =  ( # `  C ) ) )
241, 23syl5ib 219 . . 3  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( ( A concat  B
)  =  ( C concat  D )  ->  ( # `
 A )  =  ( # `  C
) ) )
25 ccatopth 12468 . . . . . . 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  /\  B  =  D ) ) )
2625biimpd 207 . . . . . 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  =  C  /\  B  =  D )
) )
27263expia 1190 . . . . 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  =  C  /\  B  =  D ) ) ) )
2827com23 78 . . . 4  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
) )  ->  (
( A concat  B )  =  ( C concat  D
)  ->  ( ( # `
 A )  =  ( # `  C
)  ->  ( A  =  C  /\  B  =  D ) ) ) )
29283adant3 1008 . . 3  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( ( A concat  B
)  =  ( C concat  D )  ->  (
( # `  A )  =  ( # `  C
)  ->  ( A  =  C  /\  B  =  D ) ) ) )
3024, 29mpdd 40 . 2  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( ( A concat  B
)  =  ( C concat  D )  ->  ( A  =  C  /\  B  =  D )
) )
31 oveq12 6201 . 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
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( ( 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 965    = wceq 1370    e. wcel 1758   ` cfv 5518  (class class class)co 6192    + caddc 9388   NN0cn0 10682   #chash 12206  Word cword 12325   concat cconcat 12327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-fzo 11652  df-hash 12207  df-word 12333  df-concat 12335  df-substr 12337
This theorem is referenced by:  ccatrcan  12471
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