Theorem pfxsuff1eqwrdeq 11231

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 11090	. . . . . . . 8 |- ( W e. Word V -> W e. Fin )
2		fihashneq0 11016	. . . . . . . 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 11091	. . . . . 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 1040	. . . 4 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) -> (♯ ` W ) e. NN )
8		fzo0end 10429	. . . 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 11230	. . 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 1316	. 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 1040	. . . . . . . 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 11201	. . . . . . 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 4087	. . . . . . . . . 10 |- ( (♯ ` W ) = (♯ ` U ) -> ( 0 < (♯ ` W ) <-> 0 < (♯ ` U ) ) )
19	18	3anbi3d 1352	. . . . . . . . 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 11090	. . . . . . . . . . . . . 14 |- ( U e. Word V -> U e. Fin )
21		fihashneq0 11016	. . . . . . . . . . . . . 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 1039	. . . . . . . . . 10 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` U ) ) -> ( U e. Word V /\ U =/= (/) ) )
26		swrdlsw 11201	. . . . . . . . . 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 6008	. . . . . . . . . . 11 |- ( (♯ ` W ) = (♯ ` U ) -> ( (♯ ` W ) - 1 ) = ( (♯ ` U ) - 1 ) )
31		id 19	. . . . . . . . . . 11 |- ( (♯ ` W ) = (♯ ` U ) -> (♯ ` W ) = (♯ ` U ) )
32	30, 31	opeq12d 3865	. . . . . . . . . 10 |- ( (♯ ` W ) = (♯ ` U ) -> <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. = <. ( (♯ ` U ) - 1 ) , (♯ ` U ) >. )
33	32	oveq2d 6017	. . . . . . . . 9 |- ( (♯ ` W ) = (♯ ` U ) -> ( U substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = ( U substr <. ( (♯ ` U ) - 1 ) , (♯ ` U ) >. ) )
34	33	eqeq1d 2238	. . . . . . . 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 2244	. . . . 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 11123	. . . . . . 7 |- ( W e. Word V -> (lastS ` W ) e. _V )
39	38	3ad2ant1 1042	. . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ 0 < (♯ ` W ) ) -> (lastS ` W ) e. _V )
40		lswex 11123	. . . . . . . 8 |- ( U e. Word V -> (lastS ` U ) e. _V )
41	40	3ad2ant2 1043	. . . . . . 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 11164	. . . . . 6 |- ( ( (lastS ` W ) e. _V /\ (lastS ` U ) e. _V ) -> ( <" (lastS ` W ) "> = <" (lastS ` U ) "> <-> (lastS ` W ) = (lastS ` U ) ) )
44	39, 42, 43	syl2an2r 597	. . . . 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 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   _Vcvv 2799   (/)c0 3491   <.cop 3669   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   Fincfn 6887   0cc0 7999   1c1 8000   < clt 8181   - cmin 8317   NNcn 9110  ..^cfzo 10338  ♯chash 10997  Word cword 11071  lastSclsw 11116   <"cs1 11148   substr csubstr 11177   prefix cpfx 11204

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723  df-fz 10205  df-fzo 10339  df-ihash 10998  df-word 11072  df-lsw 11117  df-s1 11149  df-substr 11178  df-pfx 11205

This theorem is referenced by: (None)