Theorem ccats1pfxeqrex 11343

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 11164	. . . . . . 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 9484	. . . . . 6 |- ( (♯ ` W ) e. NN0 -> ( (♯ ` W ) + 1 ) e. NN )
5		nngt0 9211	. . . . . 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 4097	. . . . . 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 11179	. . . . . 6 |- ( U e. Word V -> U e. Fin )
11		fihashneq0 11100	. . . . . 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 11211	. . 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 11342	. 2 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> U = ( W ++ <" (lastS ` U ) "> ) ) )
18		s1eq 11243	. . . 4 |- ( s = (lastS ` U ) -> <" s "> = <" (lastS ` U ) "> )
19	18	oveq2d 6044	. . 3 |- ( s = (lastS ` U ) -> ( W ++ <" s "> ) = ( W ++ <" (lastS ` U ) "> ) )
20	19	rspceeqv 2929	. 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 2202   =/= wne 2403   E.wrex 2512   (/)c0 3496   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   Fincfn 6952   0cc0 8075   1c1 8076   + caddc 8078   < clt 8257   NNcn 9186   NN0cn0 9445  ♯chash 11081  Word cword 11160  lastSclsw 11205   ++ cconcat 11214   <"cs1 11239   prefix cpfx 11300

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-inn 9187  df-n0 9446  df-z 9523  df-uz 9799  df-fz 10287  df-fzo 10421  df-ihash 11082  df-word 11161  df-lsw 11206  df-concat 11215  df-s1 11240  df-substr 11274  df-pfx 11301

This theorem is referenced by:  reuccatpfxs1lem  11374