Theorem ccats1pfxeqrex 11432

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 1025	. . 3 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> U e. Word V )
2		lencl 11253	. . . . . . 7 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
3	2	3ad2ant1 1045	. . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (♯ ` W ) e. NN0 )
4		nn0p1nn 9552	. . . . . 6 |- ( (♯ ` W ) e. NN0 -> ( (♯ ` W ) + 1 ) e. NN )
5		nngt0 9279	. . . . . 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 4118	. . . . . 6 |- ( (♯ ` U ) = ( (♯ ` W ) + 1 ) -> ( 0 < (♯ ` U ) <-> 0 < ( (♯ ` W ) + 1 ) ) )
8	7	3ad2ant3 1047	. . . . 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 11268	. . . . . 6 |- ( U e. Word V -> U e. Fin )
11		fihashneq0 11182	. . . . . 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 11300	. . 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 11431	. 2 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> U = ( W ++ <" (lastS ` U ) "> ) ) )
18		s1eq 11332	. . . 4 |- ( s = (lastS ` U ) -> <" s "> = <" (lastS ` U ) "> )
19	18	oveq2d 6074	. . 3 |- ( s = (lastS ` U ) -> ( W ++ <" s "> ) = ( W ++ <" (lastS ` U ) "> ) )
20	19	rspceeqv 2942	. 2 |- ( ( (lastS ` U ) e. V /\ U = ( W ++ <" (lastS ` U ) "> ) ) -> E. s e. V U = ( W ++ <" s "> ) )
21	16, 17, 20	syl6an 1479	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 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   E.wrex 2523   (/)c0 3512   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   Fincfn 6988   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   NNcn 9254   NN0cn0 9513  ♯chash 11163  Word cword 11249  lastSclsw 11294   ++ cconcat 11303   <"cs1 11328   prefix cpfx 11389

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-lsw 11295  df-concat 11304  df-s1 11329  df-substr 11363  df-pfx 11390

This theorem is referenced by:  reuccatpfxs1lem  11463