Theorem ccats1pfxeqbi 11514

Description: A word is a prefix of a word with length greater by 1 than the first word iff the second word is the first word concatenated with the last symbol of the second word. (Contributed by AV, 24-Oct-2018.) (Revised by AV, 10-May-2020.)

Assertion

Ref	Expression
ccats1pfxeqbi	|- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) <-> U = ( W ++ <" (lastS ` U ) "> ) ) )

Proof of Theorem ccats1pfxeqbi

Step	Hyp	Ref	Expression
1		ccats1pfxeq 11486	. 2 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> U = ( W ++ <" (lastS ` U ) "> ) ) )
2		simp1 1028	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> W e. Word V )
3		lencl 11308	. . . . . . . . 9 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
4		nn0p1nn 9602	. . . . . . . . 9 |- ( (♯ ` W ) e. NN0 -> ( (♯ ` W ) + 1 ) e. NN )
5	3, 4	syl 14	. . . . . . . 8 |- ( W e. Word V -> ( (♯ ` W ) + 1 ) e. NN )
6	5	3ad2ant1 1049	. . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( (♯ ` W ) + 1 ) e. NN )
7		3simpc 1027	. . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) )
8		lswlgt0cl 11357	. . . . . . 7 |- ( ( ( (♯ ` W ) + 1 ) e. NN /\ ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) ) -> (lastS ` U ) e. V )
9	6, 7, 8	syl2anc 415	. . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (lastS ` U ) e. V )
10	9	s1cld 11390	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> <" (lastS ` U ) "> e. Word V )
11		eqidd 2239	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (♯ ` W ) = (♯ ` W ) )
12		pfxccatid 11513	. . . . . 6 |- ( ( W e. Word V /\ <" (lastS ` U ) "> e. Word V /\ (♯ ` W ) = (♯ ` W ) ) -> ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) = W )
13	12	eqcomd 2244	. . . . 5 |- ( ( W e. Word V /\ <" (lastS ` U ) "> e. Word V /\ (♯ ` W ) = (♯ ` W ) ) -> W = ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) )
14	2, 10, 11, 13	syl3anc 1278	. . . 4 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> W = ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) )
15		oveq1 6092	. . . . 5 |- ( U = ( W ++ <" (lastS ` U ) "> ) -> ( U prefix (♯ ` W ) ) = ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) )
16	15	eqcomd 2244	. . . 4 |- ( U = ( W ++ <" (lastS ` U ) "> ) -> ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) = ( U prefix (♯ ` W ) ) )
17	14, 16	sylan9eq 2291	. . 3 |- ( ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) /\ U = ( W ++ <" (lastS ` U ) "> ) ) -> W = ( U prefix (♯ ` W ) ) )
18	17	ex 115	. 2 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( U = ( W ++ <" (lastS ` U ) "> ) -> W = ( U prefix (♯ ` W ) ) ) )
19	1, 18	impbid 129	1 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) <-> U = ( W ++ <" (lastS ` U ) "> ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   1c1 8180   + caddc 8182   NNcn 9304   NN0cn0 9563  ♯chash 11214  Word cword 11304  lastSclsw 11349   ++ cconcat 11358   <"cs1 11383   prefix cpfx 11444

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-lsw 11350  df-concat 11359  df-s1 11384  df-substr 11418  df-pfx 11445

This theorem is used by: (None)