Theorem ccats1pfxeqrex 11470

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 1029	. . 3 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> U e. Word V )
2		lencl 11291	. . . . . . 7 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
3	2	3ad2ant1 1049	. . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (♯ ` W ) e. NN0 )
4		nn0p1nn 9585	. . . . . 6 |- ( (♯ ` W ) e. NN0 -> ( (♯ ` W ) + 1 ) e. NN )
5		nngt0 9312	. . . . . 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 4132	. . . . . 6 |- ( (♯ ` U ) = ( (♯ ` W ) + 1 ) -> ( 0 < (♯ ` U ) <-> 0 < ( (♯ ` W ) + 1 ) ) )
8	7	3ad2ant3 1051	. . . . 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 11306	. . . . . 6 |- ( U e. Word V -> U e. Fin )
11		fihashneq0 11216	. . . . . 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 415	. . 3 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> U =/= (/) )
15		lswcl 11338	. . 3 |- ( ( U e. Word V /\ U =/= (/) ) -> (lastS ` U ) e. V )
16	1, 14, 15	syl2anc 415	. 2 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (lastS ` U ) e. V )
17		ccats1pfxeq 11469	. 2 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> U = ( W ++ <" (lastS ` U ) "> ) ) )
18		s1eq 11370	. . . 4 |- ( s = (lastS ` U ) -> <" s "> = <" (lastS ` U ) "> )
19	18	oveq2d 6095	. . 3 |- ( s = (lastS ` U ) -> ( W ++ <" s "> ) = ( W ++ <" (lastS ` U ) "> ) )
20	19	rspceeqv 2948	. 2 |- ( ( (lastS ` U ) e. V /\ U = ( W ++ <" (lastS ` U ) "> ) ) -> E. s e. V U = ( W ++ <" s "> ) )
21	16, 17, 20	syl6an 1483	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 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   E.wrex 2529   (/)c0 3520   class class class wbr 4128   ` cfv 5375  (class class class)co 6079   Fincfn 7016   0cc0 8173   1c1 8174   + caddc 8176   < clt 8354   NNcn 9287   NN0cn0 9546  ♯chash 11197  Word cword 11287  lastSclsw 11332   ++ cconcat 11341   <"cs1 11366   prefix cpfx 11427

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-1o 6681  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-fz 10395  df-fzo 10533  df-ihash 11198  df-word 11288  df-lsw 11333  df-concat 11342  df-s1 11367  df-substr 11401  df-pfx 11428

This theorem is referenced by:  reuccatpfxs1lem  11501