Theorem ccats1pfxeqbi 11269

Description: A word is a prefix of a word with length greater by 1 than the first word iff the second word is the first word concatenated with the last symbol of the second word. (Contributed by AV, 24-Oct-2018.) (Revised by AV, 10-May-2020.)

Assertion

Ref	Expression
ccats1pfxeqbi	|- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) <-> U = ( W ++ <" (lastS ` U ) "> ) ) )

Proof of Theorem ccats1pfxeqbi

Step	Hyp	Ref	Expression
1		ccats1pfxeq 11241	. 2 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> U = ( W ++ <" (lastS ` U ) "> ) ) )
2		simp1 1021	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> W e. Word V )
3		lencl 11070	. . . . . . . . 9 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
4		nn0p1nn 9404	. . . . . . . . 9 |- ( (♯ ` W ) e. NN0 -> ( (♯ ` W ) + 1 ) e. NN )
5	3, 4	syl 14	. . . . . . . 8 |- ( W e. Word V -> ( (♯ ` W ) + 1 ) e. NN )
6	5	3ad2ant1 1042	. . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( (♯ ` W ) + 1 ) e. NN )
7		3simpc 1020	. . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) )
8		lswlgt0cl 11119	. . . . . . 7 |- ( ( ( (♯ ` W ) + 1 ) e. NN /\ ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) ) -> (lastS ` U ) e. V )
9	6, 7, 8	syl2anc 411	. . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (lastS ` U ) e. V )
10	9	s1cld 11150	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> <" (lastS ` U ) "> e. Word V )
11		eqidd 2230	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (♯ ` W ) = (♯ ` W ) )
12		pfxccatid 11268	. . . . . 6 |- ( ( W e. Word V /\ <" (lastS ` U ) "> e. Word V /\ (♯ ` W ) = (♯ ` W ) ) -> ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) = W )
13	12	eqcomd 2235	. . . . 5 |- ( ( W e. Word V /\ <" (lastS ` U ) "> e. Word V /\ (♯ ` W ) = (♯ ` W ) ) -> W = ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) )
14	2, 10, 11, 13	syl3anc 1271	. . . 4 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> W = ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) )
15		oveq1 6007	. . . . 5 |- ( U = ( W ++ <" (lastS ` U ) "> ) -> ( U prefix (♯ ` W ) ) = ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) )
16	15	eqcomd 2235	. . . 4 |- ( U = ( W ++ <" (lastS ` U ) "> ) -> ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) = ( U prefix (♯ ` W ) ) )
17	14, 16	sylan9eq 2282	. . 3 |- ( ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) /\ U = ( W ++ <" (lastS ` U ) "> ) ) -> W = ( U prefix (♯ ` W ) ) )
18	17	ex 115	. 2 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( U = ( W ++ <" (lastS ` U ) "> ) -> W = ( U prefix (♯ ` W ) ) ) )
19	1, 18	impbid 129	1 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) <-> U = ( W ++ <" (lastS ` U ) "> ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   ` cfv 5317  (class class class)co 6000   1c1 7996   + caddc 7998   NNcn 9106   NN0cn0 9365  ♯chash 10992  Word cword 11066  lastSclsw 11111   ++ cconcat 11120   <"cs1 11143   prefix cpfx 11199

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 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112

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 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-1o 6560  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-fzo 10335  df-ihash 10993  df-word 11067  df-lsw 11112  df-concat 11121  df-s1 11144  df-substr 11173  df-pfx 11200

This theorem is referenced by: (None)