Theorem pfxsuff1eqwrdeq 11153

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 11015	. . . . . . . 8 |- ( W e. Word V -> W e. Fin )
2		fihashneq0 10941	. . . . . . . 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 11016	. . . . . 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 10354	. . . 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 11152	. . 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 11125	. . . . . . 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 4049	. . . . . . . . . 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 11015	. . . . . . . . . . . . . 14 |- ( U e. Word V -> U e. Fin )
21		fihashneq0 10941	. . . . . . . . . . . . . 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 11125	. . . . . . . . . 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 5953	. . . . . . . . . . 11 |- ( (♯ ` W ) = (♯ ` U ) -> ( (♯ ` W ) - 1 ) = ( (♯ ` U ) - 1 ) )
31		id 19	. . . . . . . . . . 11 |- ( (♯ ` W ) = (♯ ` U ) -> (♯ ` W ) = (♯ ` U ) )
32	30, 31	opeq12d 3827	. . . . . . . . . 10 |- ( (♯ ` W ) = (♯ ` U ) -> <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. = <. ( (♯ ` U ) - 1 ) , (♯ ` U ) >. )
33	32	oveq2d 5962	. . . . . . . . 9 |- ( (♯ ` W ) = (♯ ` U ) -> ( U substr <. ( (♯ ` W ) - 1 ) , (♯ ` W ) >. ) = ( U substr <. ( (♯ ` U ) - 1 ) , (♯ ` U ) >. ) )
34	33	eqeq1d 2214	. . . . . . . 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 2220	. . . . 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 11047	. . . . . . 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 11047	. . . . . . . 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 11088	. . . . . 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 2176   =/= wne 2376   _Vcvv 2772   (/)c0 3460   <.cop 3636   class class class wbr 4045   ` cfv 5272  (class class class)co 5946   Fincfn 6829   0cc0 7927   1c1 7928   < clt 8109   - cmin 8245   NNcn 9038  ..^cfzo 10266  ♯chash 10922  Word cword 10996  lastSclsw 11040   <"cs1 11072   substr csubstr 11101   prefix cpfx 11128

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-1o 6504  df-er 6622  df-en 6830  df-dom 6831  df-fin 6832  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-inn 9039  df-n0 9298  df-z 9375  df-uz 9651  df-fz 10133  df-fzo 10267  df-ihash 10923  df-word 10997  df-lsw 11041  df-s1 11073  df-substr 11102  df-pfx 11129

This theorem is referenced by: (None)