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Theorem ccats1pfxeq 11254
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 6014 . . . 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 11083 . . . . . . . . . . . 12 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
43nn0cnd 9432 . . . . . . . . . . 11 |- ( W e. Word V -> (♯ ` W ) e. CC )
5 pncan1 8531 . . . . . . . . . . 11 |- ( (♯ ` W ) e. CC -> ( ( (♯ ` W ) + 1 ) - 1 ) = (♯ ` W ) )
64, 5syl 14 . . . . . . . . . 10 |- ( W e. Word V -> ( ( (♯ ` W ) + 1 ) - 1 ) = (♯ ` W ) )
76eqcomd 2235 . . . . . . . . 9 |- ( W e. Word V -> (♯ ` W ) = ( ( (♯ ` W ) + 1 ) - 1 ) )
873ad2ant1 1042 . . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (♯ ` W ) = ( ( (♯ ` W ) + 1 ) - 1 ) )
9 oveq1 6014 . . . . . . . . . 10 |- ( (♯ ` U ) = ( (♯ ` W ) + 1 ) -> ( (♯ ` U ) - 1 ) = ( ( (♯ ` W ) + 1 ) - 1 ) )
109eqcomd 2235 . . . . . . . . 9 |- ( (♯ ` U ) = ( (♯ ` W ) + 1 ) -> ( ( (♯ ` W ) + 1 ) - 1 ) = ( (♯ ` U ) - 1 ) )
11103ad2ant3 1044 . . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( ( (♯ ` W ) + 1 ) - 1 ) = ( (♯ ` U ) - 1 ) )
128, 11eqtrd 2262 . . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (♯ ` W ) = ( (♯ ` U ) - 1 ) )
1312oveq2d 6023 . . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( U prefix (♯ ` W ) ) = ( U prefix ( (♯ ` U ) - 1 ) ) )
1413oveq1d 6022 . . . . 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 1022 . . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> U e. Word V )
16 nn0p1gt0 9406 . . . . . . . . . 10 |- ( (♯ ` W ) e. NN0 -> 0 < ( (♯ ` W ) + 1 ) )
173, 16syl 14 . . . . . . . . 9 |- ( W e. Word V -> 0 < ( (♯ ` W ) + 1 ) )
18173ad2ant1 1042 . . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> 0 < ( (♯ ` W ) + 1 ) )
19 breq2 4087 . . . . . . . . 9 |- ( (♯ ` U ) = ( (♯ ` W ) + 1 ) -> ( 0 < (♯ ` U ) <-> 0 < ( (♯ ` W ) + 1 ) ) )
20193ad2ant3 1044 . . . . . . . 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 11098 . . . . . . . . 9 |- ( U e. Word V -> U e. Fin )
23 fihashneq0 11024 . . . . . . . . 9 |- ( U e. Fin -> ( 0 < (♯ ` U ) <-> U =/= (/) ) )
2422, 23syl 14 . . . . . . . 8 |- ( U e. Word V -> ( 0 < (♯ ` U ) <-> U =/= (/) ) )
25243ad2ant2 1043 . . . . . . 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 11253 . . . . . 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 2262 . . . 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 2263 . 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 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   (/)c0 3491   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   Fincfn 6895   CCcc 8005   0cc0 8007   1c1 8008   + caddc 8010   < clt 8189   - cmin 8325   NN0cn0 9377  ♯chash 11005  Word cword 11079  lastSclsw 11124   ++ cconcat 11133   <"cs1 11156   prefix cpfx 11212
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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124
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 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-inn 9119  df-n0 9378  df-z 9455  df-uz 9731  df-fz 10213  df-fzo 10347  df-ihash 11006  df-word 11080  df-lsw 11125  df-concat 11134  df-s1 11157  df-substr 11186  df-pfx 11213
This theorem is referenced by:  ccats1pfxeqrex  11255  ccats1pfxeqbi  11282
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