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Theorem ccats1pfxeq 11288
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 6020 . . . 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 11110 . . . . . . . . . . . 12 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
43nn0cnd 9450 . . . . . . . . . . 11 |- ( W e. Word V -> (♯ ` W ) e. CC )
5 pncan1 8549 . . . . . . . . . . 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 6020 . . . . . . . . . 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 6029 . . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( U prefix (♯ ` W ) ) = ( U prefix ( (♯ ` U ) - 1 ) ) )
1413oveq1d 6028 . . . . 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 9424 . . . . . . . . . 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 4090 . . . . . . . . 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 11125 . . . . . . . . 9 |- ( U e. Word V -> U e. Fin )
23 fihashneq0 11049 . . . . . . . . 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 11287 . . . . . 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 3492   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   Fincfn 6904   CCcc 8023   0cc0 8025   1c1 8026   + caddc 8028   < clt 8207   - cmin 8343   NN0cn0 9395  ♯chash 11030  Word cword 11106  lastSclsw 11151   ++ cconcat 11160   <"cs1 11185   prefix cpfx 11246
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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142
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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-inn 9137  df-n0 9396  df-z 9473  df-uz 9749  df-fz 10237  df-fzo 10371  df-ihash 11031  df-word 11107  df-lsw 11152  df-concat 11161  df-s1 11186  df-substr 11220  df-pfx 11247
This theorem is referenced by:  ccats1pfxeqrex  11289  ccats1pfxeqbi  11316
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