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Theorem ccats1pfxeq 11267
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 6017 . . . 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 11093 . . . . . . . . . . . 12 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
43nn0cnd 9440 . . . . . . . . . . 11 |- ( W e. Word V -> (♯ ` W ) e. CC )
5 pncan1 8539 . . . . . . . . . . 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 6017 . . . . . . . . . 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 6026 . . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( U prefix (♯ ` W ) ) = ( U prefix ( (♯ ` U ) - 1 ) ) )
1413oveq1d 6025 . . . . 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 9414 . . . . . . . . . 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 11108 . . . . . . . . 9 |- ( U e. Word V -> U e. Fin )
23 fihashneq0 11033 . . . . . . . . 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 11266 . . . . . 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 5321  (class class class)co 6010   Fincfn 6900   CCcc 8013   0cc0 8015   1c1 8016   + caddc 8018   < clt 8197   - cmin 8333   NN0cn0 9385  ♯chash 11014  Word cword 11089  lastSclsw 11134   ++ cconcat 11143   <"cs1 11168   prefix cpfx 11225
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 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132
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 4385  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-1o 6573  df-er 6693  df-en 6901  df-dom 6902  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-inn 9127  df-n0 9386  df-z 9463  df-uz 9739  df-fz 10222  df-fzo 10356  df-ihash 11015  df-word 11090  df-lsw 11135  df-concat 11144  df-s1 11169  df-substr 11199  df-pfx 11226
This theorem is referenced by:  ccats1pfxeqrex  11268  ccats1pfxeqbi  11295
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