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Theorem ccatopth2 11463
Description: An opth 4245-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 5525 . . . 4  |-  ( ( A concat  B )  =  ( C concat  D )  ->  ( # `  ( A concat  B ) )  =  ( # `  ( C concat  D ) ) )
2 ccatlen 11430 . . . . . . . 8  |-  ( ( A  e. Word  X  /\  B  e. Word  X )  ->  ( # `  ( A concat  B ) )  =  ( ( # `  A
)  +  ( # `  B ) ) )
323ad2ant1 976 . . . . . . 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 957 . . . . . . . 8  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  B )  =  ( # `  D
) )
54oveq2d 5874 . . . . . . 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 2315 . . . . . 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 11430 . . . . . . 7  |-  ( ( C  e. Word  X  /\  D  e. Word  X )  ->  ( # `  ( C concat  D ) )  =  ( ( # `  C
)  +  ( # `  D ) ) )
873ad2ant2 977 . . . . . 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 2297 . . . . 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 979 . . . . . . . 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 11421 . . . . . . . 8  |-  ( A  e. Word  X  ->  ( # `
 A )  e. 
NN0 )
1210, 11syl 15 . . . . . . 7  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  A )  e.  NN0 )
1312nn0cnd 10020 . . . . . 6  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  A )  e.  CC )
14 simp2l 981 . . . . . . . 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 11421 . . . . . . . 8  |-  ( C  e. Word  X  ->  ( # `
 C )  e. 
NN0 )
1614, 15syl 15 . . . . . . 7  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  C )  e.  NN0 )
1716nn0cnd 10020 . . . . . 6  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  C )  e.  CC )
18 simp2r 982 . . . . . . . 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 11421 . . . . . . . 8  |-  ( D  e. Word  X  ->  ( # `
 D )  e. 
NN0 )
2018, 19syl 15 . . . . . . 7  |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X
)  /\  ( # `  B
)  =  ( # `  D ) )  -> 
( # `  D )  e.  NN0 )
2120nn0cnd 10020 . . . . . 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 9016 . . . . 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 244 . . . 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 210 . . 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 11462 . . . . . . 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 198 . . . . . 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 1153 . . . . 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 72 . . . 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 975 . . 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 36 . 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 5867 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A concat  B )  =  ( C concat  D
) )
3230, 31impbid1 194 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858    + caddc 8740   NN0cn0 9965   #chash 11337  Word cword 11403   concat cconcat 11404
This theorem is referenced by:  ccatrcan  11465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-concat 11410  df-substr 11412
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