Theorem ccats1pfxeqbi 11233

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 11205	. 2 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> U = ( W ++ <" (lastS ` U ) "> ) ) )
2		simp1 1000	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> W e. Word V )
3		lencl 11035	. . . . . . . . 9 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
4		nn0p1nn 9369	. . . . . . . . 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 1021	. . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( (♯ ` W ) + 1 ) e. NN )
7		3simpc 999	. . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) )
8		lswlgt0cl 11083	. . . . . . 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 11114	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> <" (lastS ` U ) "> e. Word V )
11		eqidd 2208	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> (♯ ` W ) = (♯ ` W ) )
12		pfxccatid 11232	. . . . . 6 |- ( ( W e. Word V /\ <" (lastS ` U ) "> e. Word V /\ (♯ ` W ) = (♯ ` W ) ) -> ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) = W )
13	12	eqcomd 2213	. . . . 5 |- ( ( W e. Word V /\ <" (lastS ` U ) "> e. Word V /\ (♯ ` W ) = (♯ ` W ) ) -> W = ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) )
14	2, 10, 11, 13	syl3anc 1250	. . . 4 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> W = ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) )
15		oveq1 5974	. . . . 5 |- ( U = ( W ++ <" (lastS ` U ) "> ) -> ( U prefix (♯ ` W ) ) = ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) )
16	15	eqcomd 2213	. . . 4 |- ( U = ( W ++ <" (lastS ` U ) "> ) -> ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) = ( U prefix (♯ ` W ) ) )
17	14, 16	sylan9eq 2260	. . 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 981   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   1c1 7961   + caddc 7963   NNcn 9071   NN0cn0 9330  ♯chash 10957  Word cword 11031  lastSclsw 11075   ++ cconcat 11084   <"cs1 11107   prefix cpfx 11163

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-er 6643  df-en 6851  df-dom 6852  df-fin 6853  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166  df-fzo 10300  df-ihash 10958  df-word 11032  df-lsw 11076  df-concat 11085  df-s1 11108  df-substr 11137  df-pfx 11164

This theorem is referenced by: (None)