Theorem ccats1pfxeqbi 11389

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 11361	. 2 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> U = ( W ++ <" (lastS ` U ) "> ) ) )
2		simp1 1024	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> W e. Word V )
3		lencl 11183	. . . . . . . . 9 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
4		nn0p1nn 9500	. . . . . . . . 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 1045	. . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( (♯ ` W ) + 1 ) e. NN )
7		3simpc 1023	. . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) )
8		lswlgt0cl 11232	. . . . . . 7 |- ( ( ( (♯ ` W ) + 1 ) e. NN /\ ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) ) -> (lastS ` U ) e. V )
9	6, 7, 8	syl2anc 411	. . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (lastS ` U ) e. V )
10	9	s1cld 11265	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> <" (lastS ` U ) "> e. Word V )
11		eqidd 2232	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (♯ ` W ) = (♯ ` W ) )
12		pfxccatid 11388	. . . . . 6 |- ( ( W e. Word V /\ <" (lastS ` U ) "> e. Word V /\ (♯ ` W ) = (♯ ` W ) ) -> ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) = W )
13	12	eqcomd 2237	. . . . 5 |- ( ( W e. Word V /\ <" (lastS ` U ) "> e. Word V /\ (♯ ` W ) = (♯ ` W ) ) -> W = ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) )
14	2, 10, 11, 13	syl3anc 1274	. . . 4 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> W = ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) )
15		oveq1 6035	. . . . 5 |- ( U = ( W ++ <" (lastS ` U ) "> ) -> ( U prefix (♯ ` W ) ) = ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) )
16	15	eqcomd 2237	. . . 4 |- ( U = ( W ++ <" (lastS ` U ) "> ) -> ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) = ( U prefix (♯ ` W ) ) )
17	14, 16	sylan9eq 2284	. . 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 ) "> ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   ` cfv 5333  (class class class)co 6028   1c1 8093   + caddc 8095   NNcn 9202   NN0cn0 9461  ♯chash 11100  Word cword 11179  lastSclsw 11224   ++ cconcat 11233   <"cs1 11258   prefix cpfx 11319

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 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209

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 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-fz 10306  df-fzo 10440  df-ihash 11101  df-word 11180  df-lsw 11225  df-concat 11234  df-s1 11259  df-substr 11293  df-pfx 11320

This theorem is referenced by: (None)