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Theorem ccats1pfxeq 11402
Description: The last symbol of a word concatenated with the word with the last symbol removed results in the word itself. (Contributed by Alexander van der Vekens, 24-Oct-2018.) (Revised by AV, 9-May-2020.)
Assertion
Ref Expression
ccats1pfxeq |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> U = ( W ++ <" (lastS ` U ) "> ) ) )
Proof of Theorem ccats1pfxeq
StepHypRef Expression
1 oveq1 6056 . . . 4 |- ( W = ( U prefix (♯ ` W ) ) -> ( W ++ <" (lastS ` U ) "> ) = ( ( U prefix (♯ ` W ) ) ++ <" (lastS ` U ) "> ) )
21adantl 277 . . 3 |- ( ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) /\ W = ( U prefix (♯ ` W ) ) ) -> ( W ++ <" (lastS ` U ) "> ) = ( ( U prefix (♯ ` W ) ) ++ <" (lastS ` U ) "> ) )
3 lencl 11224 . . . . . . . . . . . 12 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
43nn0cnd 9554 . . . . . . . . . . 11 |- ( W e. Word V -> (♯ ` W ) e. CC )
5 pncan1 8649 . . . . . . . . . . 11 |- ( (♯ ` W ) e. CC -> ( ( (♯ ` W ) + 1 ) - 1 ) = (♯ ` W ) )
64, 5syl 14 . . . . . . . . . 10 |- ( W e. Word V -> ( ( (♯ ` W ) + 1 ) - 1 ) = (♯ ` W ) )
76eqcomd 2238 . . . . . . . . 9 |- ( W e. Word V -> (♯ ` W ) = ( ( (♯ ` W ) + 1 ) - 1 ) )
873ad2ant1 1045 . . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (♯ ` W ) = ( ( (♯ ` W ) + 1 ) - 1 ) )
9 oveq1 6056 . . . . . . . . . 10 |- ( (♯ ` U ) = ( (♯ ` W ) + 1 ) -> ( (♯ ` U ) - 1 ) = ( ( (♯ ` W ) + 1 ) - 1 ) )
109eqcomd 2238 . . . . . . . . 9 |- ( (♯ ` U ) = ( (♯ ` W ) + 1 ) -> ( ( (♯ ` W ) + 1 ) - 1 ) = ( (♯ ` U ) - 1 ) )
11103ad2ant3 1047 . . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( ( (♯ ` W ) + 1 ) - 1 ) = ( (♯ ` U ) - 1 ) )
128, 11eqtrd 2265 . . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (♯ ` W ) = ( (♯ ` U ) - 1 ) )
1312oveq2d 6065 . . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( U prefix (♯ ` W ) ) = ( U prefix ( (♯ ` U ) - 1 ) ) )
1413oveq1d 6064 . . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( ( U prefix (♯ ` W ) ) ++ <" (lastS ` U ) "> ) = ( ( U prefix ( (♯ ` U ) - 1 ) ) ++ <" (lastS ` U ) "> ) )
15 simp2 1025 . . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> U e. Word V )
16 nn0p1gt0 9524 . . . . . . . . . 10 |- ( (♯ ` W ) e. NN0 -> 0 < ( (♯ ` W ) + 1 ) )
173, 16syl 14 . . . . . . . . 9 |- ( W e. Word V -> 0 < ( (♯ ` W ) + 1 ) )
18173ad2ant1 1045 . . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> 0 < ( (♯ ` W ) + 1 ) )
19 breq2 4112 . . . . . . . . 9 |- ( (♯ ` U ) = ( (♯ ` W ) + 1 ) -> ( 0 < (♯ ` U ) <-> 0 < ( (♯ ` W ) + 1 ) ) )
20193ad2ant3 1047 . . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( 0 < (♯ ` U ) <-> 0 < ( (♯ ` W ) + 1 ) ) )
2118, 20mpbird 167 . . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> 0 < (♯ ` U ) )
22 wrdfin 11239 . . . . . . . . 9 |- ( U e. Word V -> U e. Fin )
23 fihashneq0 11155 . . . . . . . . 9 |- ( U e. Fin -> ( 0 < (♯ ` U ) <-> U =/= (/) ) )
2422, 23syl 14 . . . . . . . 8 |- ( U e. Word V -> ( 0 < (♯ ` U ) <-> U =/= (/) ) )
25243ad2ant2 1046 . . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( 0 < (♯ ` U ) <-> U =/= (/) ) )
2621, 25mpbid 147 . . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> U =/= (/) )
27 pfxlswccat 11401 . . . . . 6 |- ( ( U e. Word V /\ U =/= (/) ) -> ( ( U prefix ( (♯ ` U ) - 1 ) ) ++ <" (lastS ` U ) "> ) = U )
2815, 26, 27syl2anc 411 . . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( ( U prefix ( (♯ ` U ) - 1 ) ) ++ <" (lastS ` U ) "> ) = U )
2914, 28eqtrd 2265 . . . 4 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( ( U prefix (♯ ` W ) ) ++ <" (lastS ` U ) "> ) = U )
3029adantr 276 . . 3 |- ( ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) /\ W = ( U prefix (♯ ` W ) ) ) -> ( ( U prefix (♯ ` W ) ) ++ <" (lastS ` U ) "> ) = U )
312, 30eqtr2d 2266 . 2 |- ( ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) /\ W = ( U prefix (♯ ` W ) ) ) -> U = ( W ++ <" (lastS ` U ) "> ) )
3231ex 115 1 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> U = ( W ++ <" (lastS ` U ) "> ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   (/)c0 3507   class class class wbr 4108   ` cfv 5351  (class class class)co 6049   Fincfn 6974   CCcc 8124   0cc0 8126   1c1 8127   + caddc 8129   < clt 8307   - cmin 8443   NN0cn0 9495  ♯chash 11136  Word cword 11220  lastSclsw 11265   ++ cconcat 11274   <"cs1 11299   prefix cpfx 11360
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-1o 6646  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-inn 9237  df-n0 9496  df-z 9577  df-uz 9853  df-fz 10342  df-fzo 10476  df-ihash 11137  df-word 11221  df-lsw 11266  df-concat 11275  df-s1 11300  df-substr 11334  df-pfx 11361
This theorem is referenced by:  ccats1pfxeqrex  11403  ccats1pfxeqbi  11430
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