Theorem 2swrd1eqwrdeq 13454

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 Mario Carneiro/AV, 23-Oct-2018.)

Assertion

Ref	Expression
2swrd1eqwrdeq	|- ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) -> ( W = U <-> ( ( # ` W ) = ( # ` U ) /\ ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 1 ) >. ) /\ ( lastS ` W ) = ( lastS ` U ) ) ) ) )

Proof of Theorem 2swrd1eqwrdeq

Step	Hyp	Ref	Expression
1		lencl 13324	. . . . . . 7 |- ( W e. Word V -> ( # ` W ) e. NN0 )
2		nn0z 11400	. . . . . . 7 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. ZZ )
3		elnnz 11387	. . . . . . . 8 |- ( ( # ` W ) e. NN <-> ( ( # ` W ) e. ZZ /\ 0 < ( # ` W ) ) )
4	3	simplbi2 655	. . . . . . 7 |- ( ( # ` W ) e. ZZ -> ( 0 < ( # ` W ) -> ( # ` W ) e. NN ) )
5	1, 2, 4	3syl 18	. . . . . 6 |- ( W e. Word V -> ( 0 < ( # ` W ) -> ( # ` W ) e. NN ) )
6	5	a1d 25	. . . . 5 |- ( W e. Word V -> ( U e. Word V -> ( 0 < ( # ` W ) -> ( # ` W ) e. NN ) ) )
7	6	3imp 1256	. . . 4 |- ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) -> ( # ` W ) e. NN )
8		fzo0end 12560	. . . 4 |- ( ( # ` W ) e. NN -> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) )
9	7, 8	syl 17	. . 3 |- ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) -> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) )
10		2swrdeqwrdeq 13453	. . 3 |- ( ( W e. Word V /\ U e. Word V /\ ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( W = U <-> ( ( # ` W ) = ( # ` U ) /\ ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 1 ) >. ) /\ ( W substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) = ( U substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) ) ) ) )
11	9, 10	syld3an3 1371	. 2 |- ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) -> ( W = U <-> ( ( # ` W ) = ( # ` U ) /\ ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 1 ) >. ) /\ ( W substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) = ( U substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) ) ) ) )
12		hashneq0 13155	. . . . . . . . . . 11 |- ( W e. Word V -> ( 0 < ( # ` W ) <-> W =/= (/) ) )
13	12	biimpd 219	. . . . . . . . . 10 |- ( W e. Word V -> ( 0 < ( # ` W ) -> W =/= (/) ) )
14	13	imdistani 726	. . . . . . . . 9 |- ( ( W e. Word V /\ 0 < ( # ` W ) ) -> ( W e. Word V /\ W =/= (/) ) )
15	14	3adant2 1080	. . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) -> ( W e. Word V /\ W =/= (/) ) )
16	15	adantr 481	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( W e. Word V /\ W =/= (/) ) )
17		swrdlsw 13452	. . . . . . 7 |- ( ( W e. Word V /\ W =/= (/) ) -> ( W substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) = <" ( lastS ` W ) "> )
18	16, 17	syl 17	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( W substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) = <" ( lastS ` W ) "> )
19		breq2 4657	. . . . . . . . . 10 |- ( ( # ` W ) = ( # ` U ) -> ( 0 < ( # ` W ) <-> 0 < ( # ` U ) ) )
20	19	3anbi3d 1405	. . . . . . . . 9 |- ( ( # ` W ) = ( # ` U ) -> ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) <-> ( W e. Word V /\ U e. Word V /\ 0 < ( # ` U ) ) ) )
21		hashneq0 13155	. . . . . . . . . . . . 13 |- ( U e. Word V -> ( 0 < ( # ` U ) <-> U =/= (/) ) )
22	21	biimpd 219	. . . . . . . . . . . 12 |- ( U e. Word V -> ( 0 < ( # ` U ) -> U =/= (/) ) )
23	22	imdistani 726	. . . . . . . . . . 11 |- ( ( U e. Word V /\ 0 < ( # ` U ) ) -> ( U e. Word V /\ U =/= (/) ) )
24	23	3adant1 1079	. . . . . . . . . 10 |- ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` U ) ) -> ( U e. Word V /\ U =/= (/) ) )
25		swrdlsw 13452	. . . . . . . . . 10 |- ( ( U e. Word V /\ U =/= (/) ) -> ( U substr <. ( ( # ` U ) - 1 ) , ( # ` U ) >. ) = <" ( lastS ` U ) "> )
26	24, 25	syl 17	. . . . . . . . 9 |- ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` U ) ) -> ( U substr <. ( ( # ` U ) - 1 ) , ( # ` U ) >. ) = <" ( lastS ` U ) "> )
27	20, 26	syl6bi 243	. . . . . . . 8 |- ( ( # ` W ) = ( # ` U ) -> ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) -> ( U substr <. ( ( # ` U ) - 1 ) , ( # ` U ) >. ) = <" ( lastS ` U ) "> ) )
28	27	impcom 446	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( U substr <. ( ( # ` U ) - 1 ) , ( # ` U ) >. ) = <" ( lastS ` U ) "> )
29		oveq1 6657	. . . . . . . . . . 11 |- ( ( # ` W ) = ( # ` U ) -> ( ( # ` W ) - 1 ) = ( ( # ` U ) - 1 ) )
30		id 22	. . . . . . . . . . 11 |- ( ( # ` W ) = ( # ` U ) -> ( # ` W ) = ( # ` U ) )
31	29, 30	opeq12d 4410	. . . . . . . . . 10 |- ( ( # ` W ) = ( # ` U ) -> <. ( ( # ` W ) - 1 ) , ( # ` W ) >. = <. ( ( # ` U ) - 1 ) , ( # ` U ) >. )
32	31	oveq2d 6666	. . . . . . . . 9 |- ( ( # ` W ) = ( # ` U ) -> ( U substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) = ( U substr <. ( ( # ` U ) - 1 ) , ( # ` U ) >. ) )
33	32	eqeq1d 2624	. . . . . . . 8 |- ( ( # ` W ) = ( # ` U ) -> ( ( U substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) = <" ( lastS ` U ) "> <-> ( U substr <. ( ( # ` U ) - 1 ) , ( # ` U ) >. ) = <" ( lastS ` U ) "> ) )
34	33	adantl 482	. . . . . . 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 ) "> ) )
35	28, 34	mpbird 247	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( U substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) = <" ( lastS ` U ) "> )
36	18, 35	eqeq12d 2637	. . . . 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 ) "> ) )
37		hashgt0n0 13156	. . . . . . . . 9 |- ( ( W e. Word V /\ 0 < ( # ` W ) ) -> W =/= (/) )
38		lswcl 13355	. . . . . . . . 9 |- ( ( W e. Word V /\ W =/= (/) ) -> ( lastS ` W ) e. V )
39	37, 38	syldan 487	. . . . . . . 8 |- ( ( W e. Word V /\ 0 < ( # ` W ) ) -> ( lastS ` W ) e. V )
40	39	3adant2 1080	. . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) -> ( lastS ` W ) e. V )
41	40	adantr 481	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( lastS ` W ) e. V )
42		hashgt0n0 13156	. . . . . . . . . 10 |- ( ( U e. Word V /\ 0 < ( # ` U ) ) -> U =/= (/) )
43		lswcl 13355	. . . . . . . . . 10 |- ( ( U e. Word V /\ U =/= (/) ) -> ( lastS ` U ) e. V )
44	42, 43	syldan 487	. . . . . . . . 9 |- ( ( U e. Word V /\ 0 < ( # ` U ) ) -> ( lastS ` U ) e. V )
45	44	3adant1 1079	. . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` U ) ) -> ( lastS ` U ) e. V )
46	20, 45	syl6bi 243	. . . . . . 7 |- ( ( # ` W ) = ( # ` U ) -> ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) -> ( lastS ` U ) e. V ) )
47	46	impcom 446	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( lastS ` U ) e. V )
48		s111 13395	. . . . . 6 |- ( ( ( lastS ` W ) e. V /\ ( lastS ` U ) e. V ) -> ( <" ( lastS ` W ) "> = <" ( lastS ` U ) "> <-> ( lastS ` W ) = ( lastS ` U ) ) )
49	41, 47, 48	syl2anc 693	. . . . 5 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( <" ( lastS ` W ) "> = <" ( lastS ` U ) "> <-> ( lastS ` W ) = ( lastS ` U ) ) )
50	36, 49	bitrd 268	. . . 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 ) ) )
51	50	anbi2d 740	. . 3 |- ( ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 1 ) >. ) /\ ( W substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) = ( U substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) ) <-> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 1 ) >. ) /\ ( lastS ` W ) = ( lastS ` U ) ) ) )
52	51	pm5.32da 673	. 2 |- ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) -> ( ( ( # ` W ) = ( # ` U ) /\ ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 1 ) >. ) /\ ( W substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) = ( U substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) ) ) <-> ( ( # ` W ) = ( # ` U ) /\ ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 1 ) >. ) /\ ( lastS ` W ) = ( lastS ` U ) ) ) ) )
53	11, 52	bitrd 268	1 |- ( ( W e. Word V /\ U e. Word V /\ 0 < ( # ` W ) ) -> ( W = U <-> ( ( # ` W ) = ( # ` U ) /\ ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 1 ) >. ) /\ ( lastS ` W ) = ( lastS ` U ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   < clt 10074   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   <"cs1 13294   substr csubstr 13295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-s1 13302  df-substr 13303

This theorem is referenced by:  wwlksnextinj  26794  clwwlksf1  26917