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Theorem ccatopth2 11235
Description: An opth 4322-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 ++ B ) = ( C ++ D ) <-> ( A = C /\ B = D ) ) )
Proof of Theorem ccatopth2
StepHypRef Expression
1 fveq2 5623 . . . 4 |- ( ( A ++ B ) = ( C ++ D ) -> (♯ ` ( A ++ B ) ) = (♯ ` ( C ++ D ) ) )
2 ccatlen 11116 . . . . . . . 8 |- ( ( A e. Word X /\ B e. Word X ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
323ad2ant1 1042 . . . . . . 7 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
4 simp3 1023 . . . . . . . 8 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` B ) = (♯ ` D ) )
54oveq2d 6010 . . . . . . 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 2262 . . . . . 6 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` D ) ) )
7 ccatlen 11116 . . . . . . 7 |- ( ( C e. Word X /\ D e. Word X ) -> (♯ ` ( C ++ D ) ) = ( (♯ ` C ) + (♯ ` D ) ) )
873ad2ant2 1043 . . . . . 6 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` ( C ++ D ) ) = ( (♯ ` C ) + (♯ ` D ) ) )
96, 8eqeq12d 2244 . . . . 5 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( (♯ ` ( A ++ B ) ) = (♯ ` ( C ++ D ) ) <-> ( (♯ ` A ) + (♯ ` D ) ) = ( (♯ ` C ) + (♯ ` D ) ) ) )
10 simp1l 1045 . . . . . . . 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 11062 . . . . . . . 8 |- ( A e. Word X -> (♯ ` A ) e. NN0 )
1210, 11syl 14 . . . . . . 7 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` A ) e. NN0 )
1312nn0cnd 9412 . . . . . 6 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` A ) e. CC )
14 simp2l 1047 . . . . . . . 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 11062 . . . . . . . 8 |- ( C e. Word X -> (♯ ` C ) e. NN0 )
1614, 15syl 14 . . . . . . 7 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` C ) e. NN0 )
1716nn0cnd 9412 . . . . . 6 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` C ) e. CC )
18 simp2r 1048 . . . . . . . 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 11062 . . . . . . . 8 |- ( D e. Word X -> (♯ ` D ) e. NN0 )
2018, 19syl 14 . . . . . . 7 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` D ) e. NN0 )
2120nn0cnd 9412 . . . . . 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 8319 . . . . 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 188 . . . 4 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( (♯ ` ( A ++ B ) ) = (♯ ` ( C ++ D ) ) <-> (♯ ` A ) = (♯ ` C ) ) )
241, 23imbitrid 154 . . 3 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( ( A ++ B ) = ( C ++ D ) -> (♯ ` A ) = (♯ ` C ) ) )
25 ccatopth 11234 . . . . . . 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 /\ B = D ) ) )
2625biimpd 144 . . . . . 6 |- ( ( ( 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 ) ) )
27263expia 1229 . . . . 5 |- ( ( ( 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 ) ) ) )
2827com23 78 . . . 4 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) ) -> ( ( A ++ B ) = ( C ++ D ) -> ( (♯ ` A ) = (♯ ` C ) -> ( A = C /\ B = D ) ) ) )
29283adant3 1041 . . 3 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( ( A ++ B ) = ( C ++ D ) -> ( (♯ ` A ) = (♯ ` C ) -> ( A = C /\ B = D ) ) ) )
3024, 29mpdd 41 . 2 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( ( A ++ B ) = ( C ++ D ) -> ( A = C /\ B = D ) ) )
31 oveq12 6003 . 2 |- ( ( A = C /\ B = D ) -> ( A ++ B ) = ( C ++ D ) )
3230, 31impbid1 142 1 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( ( A ++ B ) = ( C ++ D ) <-> ( A = C /\ B = D ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   ` cfv 5314  (class class class)co 5994   + caddc 7990   NN0cn0 9357  ♯chash 10984  Word cword 11058   ++ cconcat 11111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-0id 8095  ax-rnegex 8096  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-1o 6552  df-er 6670  df-en 6878  df-dom 6879  df-fin 6880  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-inn 9099  df-n0 9358  df-z 9435  df-uz 9711  df-fz 10193  df-fzo 10327  df-ihash 10985  df-word 11059  df-concat 11112  df-substr 11164  df-pfx 11191
This theorem is referenced by:  ccatrcan  11237
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