Theorem ccats1pfxeqbi 11322

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 11294	. 2 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> U = ( W ++ <" (lastS ` U ) "> ) ) )
2		simp1 1023	. . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> W e. Word V )
3		lencl 11116	. . . . . . . . 9 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
4		nn0p1nn 9440	. . . . . . . . 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 1044	. . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( (♯ ` W ) + 1 ) e. NN )
7		3simpc 1022	. . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) )
8		lswlgt0cl 11165	. . . . . . 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 11198	. . . . 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 11321	. . . . . 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 1273	. . . 4 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> W = ( ( W ++ <" (lastS ` U ) "> ) prefix (♯ ` W ) ) )
15		oveq1 6024	. . . . 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 1004   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   1c1 8032   + caddc 8034   NNcn 9142   NN0cn0 9401  ♯chash 11036  Word cword 11112  lastSclsw 11157   ++ cconcat 11166   <"cs1 11191   prefix cpfx 11252

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 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

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 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-lsw 11158  df-concat 11167  df-s1 11192  df-substr 11226  df-pfx 11253

This theorem is referenced by: (None)