Theorem pfxsuff1eqwrdeq 11190

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 11050	. . . . . . . 8 |- ( W e. Word V -> W e. Fin )
2		fihashneq0 10976	. . . . . . . 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 11051	. . . . . 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 1019	. . . 4 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) -> (♯ ` W ) e. NN )
8		fzo0end 10389	. . . 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 11189	. . 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 1295	. 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 1019	. . . . . . . 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 11160	. . . . . . 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 4063	. . . . . . . . . 10 |- ( (♯ ` W ) = (♯ ` U ) -> ( 0 < (♯ ` W ) <-> 0 < (♯ ` U ) ) )
19	18	3anbi3d 1331	. . . . . . . . 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 11050	. . . . . . . . . . . . . 14 |- ( U e. Word V -> U e. Fin )
21		fihashneq0 10976	. . . . . . . . . . . . . 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 1018	. . . . . . . . . 10 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` U ) ) -> ( U e. Word V /\ U =/= (/) ) )
26		swrdlsw 11160	. . . . . . . . . 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 5974	. . . . . . . . . . 11 |- ( (♯ ` W ) = (♯ ` U ) -> ( (♯ ` W ) - 1 ) = ( (♯ ` U ) - 1 ) )
31		id 19	. . . . . . . . . . 11 |- ( (♯ ` W ) = (♯ ` U ) -> (♯ ` W ) = (♯ ` U ) )
32	30, 31	opeq12d 3841	. . . . . . . . . 10 |- ( (♯ ` W ) = (♯ ` U ) -> <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. = <. ( (♯ ` U ) - 1 ) , (♯ ` U ) >. )
33	32	oveq2d 5983	. . . . . . . . 9 |- ( (♯ ` W ) = (♯ ` U ) -> ( U substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = ( U substr <. ( (♯ ` U ) - 1 ) , (♯ ` U ) >. ) )
34	33	eqeq1d 2216	. . . . . . . 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 2222	. . . . 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 11082	. . . . . . 7 |- ( W e. Word V -> (lastS ` W ) e. _V )
39	38	3ad2ant1 1021	. . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) -> (lastS ` W ) e. _V )
40		lswex 11082	. . . . . . . 8 |- ( U e. Word V -> (lastS ` U ) e. _V )
41	40	3ad2ant2 1022	. . . . . . 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 11123	. . . . . 6 |- ( ( (lastS ` W ) e. _V /\ (lastS ` U ) e. _V ) -> ( <" (lastS ` W ) "> = <" (lastS ` U ) "> <-> (lastS ` W ) = (lastS ` U ) ) )
44	39, 42, 43	syl2an2r 595	. . . . 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 981   = wceq 1373   e. wcel 2178   =/= wne 2378   _Vcvv 2776   (/)c0 3468   <.cop 3646   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   Fincfn 6850   0cc0 7960   1c1 7961   < clt 8142   - cmin 8278   NNcn 9071  ..^cfzo 10299  ♯chash 10957  Word cword 11031  lastSclsw 11075   <"cs1 11107   substr csubstr 11136   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-s1 11108  df-substr 11137  df-pfx 11164

This theorem is referenced by: (None)