Theorem pfxsuff1eqwrdeq 11284

Description: Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Revised by AV, 6-May-2020.)

Assertion

Ref	Expression
pfxsuff1eqwrdeq	|- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) -> ( W = U <-> ( (♯ ` W ) = (♯ ` U ) /\ ( ( W prefix ( (♯ ` W ) - 1 ) ) = ( U prefix ( (♯ ` W ) - 1 ) ) /\ (lastS ` W ) = (lastS ` U ) ) ) ) )

Proof of Theorem pfxsuff1eqwrdeq

Step	Hyp	Ref	Expression
1		wrdfin 11136	. . . . . . . 8 |- ( W e. Word V -> W e. Fin )
2		fihashneq0 11057	. . . . . . . 8 |- ( W e. Fin -> ( 0 < (♯ ` W ) <-> W =/= (/) ) )
3	1, 2	syl 14	. . . . . . 7 |- ( W e. Word V -> ( 0 < (♯ ` W ) <-> W =/= (/) ) )
4	3	biimpa 296	. . . . . 6 |- ( ( W e. Word V /\ 0 < (♯ ` W ) ) -> W =/= (/) )
5		lennncl 11137	. . . . . 6 |- ( ( W e. Word V /\ W =/= (/) ) -> (♯ ` W ) e. NN )
6	4, 5	syldan 282	. . . . 5 |- ( ( W e. Word V /\ 0 < (♯ ` W ) ) -> (♯ ` W ) e. NN )
7	6	3adant2 1042	. . . 4 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) -> (♯ ` W ) e. NN )
8		fzo0end 10469	. . . 4 |- ( (♯ ` W ) e. NN -> ( (♯ ` W ) - 1 ) e. ( 0..^ (♯ ` W ) ) )
9	7, 8	syl 14	. . 3 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) -> ( (♯ ` W ) - 1 ) e. ( 0..^ (♯ ` W ) ) )
10		pfxsuffeqwrdeq 11283	. . 3 |- ( ( W e. Word V /\ U e. Word V /\ ( (♯ ` W ) - 1 ) e. ( 0..^ (♯ ` W ) ) ) -> ( W = U <-> ( (♯ ` W ) = (♯ ` U ) /\ ( ( W prefix ( (♯ ` W ) - 1 ) ) = ( U prefix ( (♯ ` W ) - 1 ) ) /\ ( W substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = ( U substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) ) ) ) )
11	9, 10	syld3an3 1318	. 2 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) -> ( W = U <-> ( (♯ ` W ) = (♯ ` U ) /\ ( ( W prefix ( (♯ ` W ) - 1 ) ) = ( U prefix ( (♯ ` W ) - 1 ) ) /\ ( W substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = ( U substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) ) ) ) )
12	3	biimpd 144	. . . . . . . . . 10 |- ( W e. Word V -> ( 0 < (♯ ` W ) -> W =/= (/) ) )
13	12	imdistani 445	. . . . . . . . 9 |- ( ( W e. Word V /\ 0 < (♯ ` W ) ) -> ( W e. Word V /\ W =/= (/) ) )
14	13	3adant2 1042	. . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) -> ( W e. Word V /\ W =/= (/) ) )
15	14	adantr 276	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) /\ (♯ ` W ) = (♯ ` U ) ) -> ( W e. Word V /\ W =/= (/) ) )
16		swrdlsw 11254	. . . . . . 7 |- ( ( W e. Word V /\ W =/= (/) ) -> ( W substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = <" (lastS ` W ) "> )
17	15, 16	syl 14	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) /\ (♯ ` W ) = (♯ ` U ) ) -> ( W substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = <" (lastS ` W ) "> )
18		breq2 4092	. . . . . . . . . 10 |- ( (♯ ` W ) = (♯ ` U ) -> ( 0 < (♯ ` W ) <-> 0 < (♯ ` U ) ) )
19	18	3anbi3d 1354	. . . . . . . . 9 |- ( (♯ ` W ) = (♯ ` U ) -> ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) <-> ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` U ) ) ) )
20		wrdfin 11136	. . . . . . . . . . . . . 14 |- ( U e. Word V -> U e. Fin )
21		fihashneq0 11057	. . . . . . . . . . . . . 14 |- ( U e. Fin -> ( 0 < (♯ ` U ) <-> U =/= (/) ) )
22	20, 21	syl 14	. . . . . . . . . . . . 13 |- ( U e. Word V -> ( 0 < (♯ ` U ) <-> U =/= (/) ) )
23	22	biimpd 144	. . . . . . . . . . . 12 |- ( U e. Word V -> ( 0 < (♯ ` U ) -> U =/= (/) ) )
24	23	imdistani 445	. . . . . . . . . . 11 |- ( ( U e. Word V /\ 0 < (♯ ` U ) ) -> ( U e. Word V /\ U =/= (/) ) )
25	24	3adant1 1041	. . . . . . . . . 10 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` U ) ) -> ( U e. Word V /\ U =/= (/) ) )
26		swrdlsw 11254	. . . . . . . . . 10 |- ( ( U e. Word V /\ U =/= (/) ) -> ( U substr <. ( (♯ ` U ) - 1 ) , (♯ ` U ) >. ) = <" (lastS ` U ) "> )
27	25, 26	syl 14	. . . . . . . . 9 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` U ) ) -> ( U substr <. ( (♯ ` U ) - 1 ) , (♯ ` U ) >. ) = <" (lastS ` U ) "> )
28	19, 27	biimtrdi 163	. . . . . . . 8 |- ( (♯ ` W ) = (♯ ` U ) -> ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) -> ( U substr <. ( (♯ ` U ) - 1 ) , (♯ ` U ) >. ) = <" (lastS ` U ) "> ) )
29	28	impcom 125	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) /\ (♯ ` W ) = (♯ ` U ) ) -> ( U substr <. ( (♯ ` U ) - 1 ) , (♯ ` U ) >. ) = <" (lastS ` U ) "> )
30		oveq1 6025	. . . . . . . . . . 11 |- ( (♯ ` W ) = (♯ ` U ) -> ( (♯ ` W ) - 1 ) = ( (♯ ` U ) - 1 ) )
31		id 19	. . . . . . . . . . 11 |- ( (♯ ` W ) = (♯ ` U ) -> (♯ ` W ) = (♯ ` U ) )
32	30, 31	opeq12d 3870	. . . . . . . . . 10 |- ( (♯ ` W ) = (♯ ` U ) -> <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. = <. ( (♯ ` U ) - 1 ) , (♯ ` U ) >. )
33	32	oveq2d 6034	. . . . . . . . 9 |- ( (♯ ` W ) = (♯ ` U ) -> ( U substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = ( U substr <. ( (♯ ` U ) - 1 ) , (♯ ` U ) >. ) )
34	33	eqeq1d 2240	. . . . . . . 8 |- ( (♯ ` W ) = (♯ ` U ) -> ( ( U substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = <" (lastS ` U ) "> <-> ( U substr <. ( (♯ ` U ) - 1 ) , (♯ ` U ) >. ) = <" (lastS ` U ) "> ) )
35	34	adantl 277	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) /\ (♯ ` W ) = (♯ ` U ) ) -> ( ( U substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = <" (lastS ` U ) "> <-> ( U substr <. ( (♯ ` U ) - 1 ) , (♯ ` U ) >. ) = <" (lastS ` U ) "> ) )
36	29, 35	mpbird 167	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) /\ (♯ ` W ) = (♯ ` U ) ) -> ( U substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = <" (lastS ` U ) "> )
37	17, 36	eqeq12d 2246	. . . . 5 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) /\ (♯ ` W ) = (♯ ` U ) ) -> ( ( W substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = ( U substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) <-> <" (lastS ` W ) "> = <" (lastS ` U ) "> ) )
38		lswex 11169	. . . . . . 7 |- ( W e. Word V -> (lastS ` W ) e. _V )
39	38	3ad2ant1 1044	. . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) -> (lastS ` W ) e. _V )
40		lswex 11169	. . . . . . . 8 |- ( U e. Word V -> (lastS ` U ) e. _V )
41	40	3ad2ant2 1045	. . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) -> (lastS ` U ) e. _V )
42	41	adantr 276	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) /\ (♯ ` W ) = (♯ ` U ) ) -> (lastS ` U ) e. _V )
43		s111 11212	. . . . . 6 |- ( ( (lastS ` W ) e. _V /\ (lastS ` U ) e. _V ) -> ( <" (lastS ` W ) "> = <" (lastS ` U ) "> <-> (lastS ` W ) = (lastS ` U ) ) )
44	39, 42, 43	syl2an2r 599	. . . . 5 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) /\ (♯ ` W ) = (♯ ` U ) ) -> ( <" (lastS ` W ) "> = <" (lastS ` U ) "> <-> (lastS ` W ) = (lastS ` U ) ) )
45	37, 44	bitrd 188	. . . 4 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) /\ (♯ ` W ) = (♯ ` U ) ) -> ( ( W substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = ( U substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) <-> (lastS ` W ) = (lastS ` U ) ) )
46	45	anbi2d 464	. . 3 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) /\ (♯ ` W ) = (♯ ` U ) ) -> ( ( ( W prefix ( (♯ ` W ) - 1 ) ) = ( U prefix ( (♯ ` W ) - 1 ) ) /\ ( W substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = ( U substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) ) <-> ( ( W prefix ( (♯ ` W ) - 1 ) ) = ( U prefix ( (♯ ` W ) - 1 ) ) /\ (lastS ` W ) = (lastS ` U ) ) ) )
47	46	pm5.32da 452	. 2 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) -> ( ( (♯ ` W ) = (♯ ` U ) /\ ( ( W prefix ( (♯ ` W ) - 1 ) ) = ( U prefix ( (♯ ` W ) - 1 ) ) /\ ( W substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = ( U substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) ) ) <-> ( (♯ ` W ) = (♯ ` U ) /\ ( ( W prefix ( (♯ ` W ) - 1 ) ) = ( U prefix ( (♯ ` W ) - 1 ) ) /\ (lastS ` W ) = (lastS ` U ) ) ) ) )
48	11, 47	bitrd 188	1 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) -> ( W = U <-> ( (♯ ` W ) = (♯ ` U ) /\ ( ( W prefix ( (♯ ` W ) - 1 ) ) = ( U prefix ( (♯ ` W ) - 1 ) ) /\ (lastS ` W ) = (lastS ` U ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   _Vcvv 2802   (/)c0 3494   <.cop 3672   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   Fincfn 6909   0cc0 8032   1c1 8033   < clt 8214   - cmin 8350   NNcn 9143  ..^cfzo 10377  ♯chash 11038  Word cword 11117  lastSclsw 11162   <"cs1 11196   substr csubstr 11230   prefix cpfx 11257

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 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149

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 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-fzo 10378  df-ihash 11039  df-word 11118  df-lsw 11163  df-s1 11197  df-substr 11231  df-pfx 11258

This theorem is referenced by: (None)