Theorem ccats1pfxeqrex 11233

Description: There exists a symbol such that its concatenation after the prefix obtained by deleting the last symbol of a nonempty word results in the word itself. (Contributed by AV, 5-Oct-2018.) (Revised by AV, 9-May-2020.)

Assertion

Ref	Expression
ccats1pfxeqrex	|- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> E. s e. V U = ( W ++ <" s "> ) ) )

Distinct variable groups:   U, s   V, s   W, s

Proof of Theorem ccats1pfxeqrex

Step	Hyp	Ref	Expression
1		simp2 1022	. . 3 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> U e. Word V )
2		lencl 11062	. . . . . . 7 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
3	2	3ad2ant1 1042	. . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (♯ ` W ) e. NN0 )
4		nn0p1nn 9396	. . . . . 6 |- ( (♯ ` W ) e. NN0 -> ( (♯ ` W ) + 1 ) e. NN )
5		nngt0 9123	. . . . . 6 |- ( ( (♯ ` W ) + 1 ) e. NN -> 0 < ( (♯ ` W ) + 1 ) )
6	3, 4, 5	3syl 17	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> 0 < ( (♯ ` W ) + 1 ) )
7		breq2 4086	. . . . . 6 |- ( (♯ ` U ) = ( (♯ ` W ) + 1 ) -> ( 0 < (♯ ` U ) <-> 0 < ( (♯ ` W ) + 1 ) ) )
8	7	3ad2ant3 1044	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( 0 < (♯ ` U ) <-> 0 < ( (♯ ` W ) + 1 ) ) )
9	6, 8	mpbird 167	. . . 4 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> 0 < (♯ ` U ) )
10		wrdfin 11077	. . . . . 6 |- ( U e. Word V -> U e. Fin )
11		fihashneq0 11003	. . . . . 6 |- ( U e. Fin -> ( 0 < (♯ ` U ) <-> U =/= (/) ) )
12	10, 11	syl 14	. . . . 5 |- ( U e. Word V -> ( 0 < (♯ ` U ) <-> U =/= (/) ) )
13	12	biimpa 296	. . . 4 |- ( ( U e. Word V /\ 0 < (♯ ` U ) ) -> U =/= (/) )
14	1, 9, 13	syl2anc 411	. . 3 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> U =/= (/) )
15		lswcl 11108	. . 3 |- ( ( U e. Word V /\ U =/= (/) ) -> (lastS ` U ) e. V )
16	1, 14, 15	syl2anc 411	. 2 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (lastS ` U ) e. V )
17		ccats1pfxeq 11232	. 2 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> U = ( W ++ <" (lastS ` U ) "> ) ) )
18		s1eq 11138	. . . 4 |- ( s = (lastS ` U ) -> <" s "> = <" (lastS ` U ) "> )
19	18	oveq2d 6010	. . 3 |- ( s = (lastS ` U ) -> ( W ++ <" s "> ) = ( W ++ <" (lastS ` U ) "> ) )
20	19	rspceeqv 2925	. 2 |- ( ( (lastS ` U ) e. V /\ U = ( W ++ <" (lastS ` U ) "> ) ) -> E. s e. V U = ( W ++ <" s "> ) )
21	16, 17, 20	syl6an 1476	1 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> E. s e. V U = ( W ++ <" s "> ) ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   E.wrex 2509   (/)c0 3491   class class class wbr 4082   ` cfv 5314  (class class class)co 5994   Fincfn 6877   0cc0 7987   1c1 7988   + caddc 7990   < clt 8169   NNcn 9098   NN0cn0 9357  ♯chash 10984  Word cword 11058  lastSclsw 11102   ++ cconcat 11111   <"cs1 11134   prefix cpfx 11190

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 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104

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 4381  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-1o 6552  df-er 6670  df-en 6878  df-dom 6879  df-fin 6880  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-inn 9099  df-n0 9358  df-z 9435  df-uz 9711  df-fz 10193  df-fzo 10327  df-ihash 10985  df-word 11059  df-lsw 11103  df-concat 11112  df-s1 11135  df-substr 11164  df-pfx 11191

This theorem is referenced by:  reuccatpfxs1lem  11264