Theorem ccats1pfxeqrex 11300

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 1024	. . 3 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> U e. Word V )
2		lencl 11121	. . . . . . 7 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
3	2	3ad2ant1 1044	. . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (♯ ` W ) e. NN0 )
4		nn0p1nn 9441	. . . . . 6 |- ( (♯ ` W ) e. NN0 -> ( (♯ ` W ) + 1 ) e. NN )
5		nngt0 9168	. . . . . 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 4092	. . . . . 6 |- ( (♯ ` U ) = ( (♯ ` W ) + 1 ) -> ( 0 < (♯ ` U ) <-> 0 < ( (♯ ` W ) + 1 ) ) )
8	7	3ad2ant3 1046	. . . . 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 11136	. . . . . 6 |- ( U e. Word V -> U e. Fin )
11		fihashneq0 11057	. . . . . 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 11168	. . 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 11299	. 2 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> U = ( W ++ <" (lastS ` U ) "> ) ) )
18		s1eq 11200	. . . 4 |- ( s = (lastS ` U ) -> <" s "> = <" (lastS ` U ) "> )
19	18	oveq2d 6034	. . 3 |- ( s = (lastS ` U ) -> ( W ++ <" s "> ) = ( W ++ <" (lastS ` U ) "> ) )
20	19	rspceeqv 2928	. 2 |- ( ( (lastS ` U ) e. V /\ U = ( W ++ <" (lastS ` U ) "> ) ) -> E. s e. V U = ( W ++ <" s "> ) )
21	16, 17, 20	syl6an 1478	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 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   E.wrex 2511   (/)c0 3494   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   Fincfn 6909   0cc0 8032   1c1 8033   + caddc 8035   < clt 8214   NNcn 9143   NN0cn0 9402  ♯chash 11038  Word cword 11117  lastSclsw 11162   ++ cconcat 11171   <"cs1 11196   prefix cpfx 11257

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-fzo 10378  df-ihash 11039  df-word 11118  df-lsw 11163  df-concat 11172  df-s1 11197  df-substr 11231  df-pfx 11258

This theorem is referenced by:  reuccatpfxs1lem  11331