Theorem swrdeq 13444

Description: Two subwords of words are equal iff they have the same length and the same symbols at each position. (Contributed by Alexander van der Vekens, 7-Aug-2018.)

Assertion

Ref	Expression
swrdeq	|- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> ( ( W substr <. 0 , M >. ) = ( U substr <. 0 , N >. ) <-> ( M = N /\ A. i e. ( 0..^ M ) ( W ` i ) = ( U ` i ) ) ) )

Distinct variable groups:   i, M   i, N   U, i   i, V   i, W

Proof of Theorem swrdeq

Step	Hyp	Ref	Expression
1		swrdcl 13419	. . . . 5 |- ( W e. Word V -> ( W substr <. 0 , M >. ) e. Word V )
2		swrdcl 13419	. . . . 5 |- ( U e. Word V -> ( U substr <. 0 , N >. ) e. Word V )
3	1, 2	anim12i 590	. . . 4 |- ( ( W e. Word V /\ U e. Word V ) -> ( ( W substr <. 0 , M >. ) e. Word V /\ ( U substr <. 0 , N >. ) e. Word V ) )
4	3	3ad2ant1 1082	. . 3 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> ( ( W substr <. 0 , M >. ) e. Word V /\ ( U substr <. 0 , N >. ) e. Word V ) )
5		eqwrd 13346	. . 3 |- ( ( ( W substr <. 0 , M >. ) e. Word V /\ ( U substr <. 0 , N >. ) e. Word V ) -> ( ( W substr <. 0 , M >. ) = ( U substr <. 0 , N >. ) <-> ( ( # ` ( W substr <. 0 , M >. ) ) = ( # ` ( U substr <. 0 , N >. ) ) /\ A. i e. ( 0..^ ( # ` ( W substr <. 0 , M >. ) ) ) ( ( W substr <. 0 , M >. ) ` i ) = ( ( U substr <. 0 , N >. ) ` i ) ) ) )
6	4, 5	syl 17	. 2 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> ( ( W substr <. 0 , M >. ) = ( U substr <. 0 , N >. ) <-> ( ( # ` ( W substr <. 0 , M >. ) ) = ( # ` ( U substr <. 0 , N >. ) ) /\ A. i e. ( 0..^ ( # ` ( W substr <. 0 , M >. ) ) ) ( ( W substr <. 0 , M >. ) ` i ) = ( ( U substr <. 0 , N >. ) ` i ) ) ) )
7		simpl 473	. . . . . 6 |- ( ( W e. Word V /\ U e. Word V ) -> W e. Word V )
8	7	3ad2ant1 1082	. . . . 5 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> W e. Word V )
9		simpl 473	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> M e. NN0 )
10	9	3ad2ant2 1083	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> M e. NN0 )
11		lencl 13324	. . . . . . . 8 |- ( W e. Word V -> ( # ` W ) e. NN0 )
12	11	adantr 481	. . . . . . 7 |- ( ( W e. Word V /\ U e. Word V ) -> ( # ` W ) e. NN0 )
13	12	3ad2ant1 1082	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> ( # ` W ) e. NN0 )
14		simpl 473	. . . . . . 7 |- ( ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) -> M <_ ( # ` W ) )
15	14	3ad2ant3 1084	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> M <_ ( # ` W ) )
16		elfz2nn0 12431	. . . . . 6 |- ( M e. ( 0 ... ( # ` W ) ) <-> ( M e. NN0 /\ ( # ` W ) e. NN0 /\ M <_ ( # ` W ) ) )
17	10, 13, 15, 16	syl3anbrc 1246	. . . . 5 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> M e. ( 0 ... ( # ` W ) ) )
18		swrd0len 13422	. . . . 5 |- ( ( W e. Word V /\ M e. ( 0 ... ( # ` W ) ) ) -> ( # ` ( W substr <. 0 , M >. ) ) = M )
19	8, 17, 18	syl2anc 693	. . . 4 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> ( # ` ( W substr <. 0 , M >. ) ) = M )
20		simpr 477	. . . . . 6 |- ( ( W e. Word V /\ U e. Word V ) -> U e. Word V )
21	20	3ad2ant1 1082	. . . . 5 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> U e. Word V )
22		simpr 477	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> N e. NN0 )
23	22	3ad2ant2 1083	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> N e. NN0 )
24		lencl 13324	. . . . . . . 8 |- ( U e. Word V -> ( # ` U ) e. NN0 )
25	24	adantl 482	. . . . . . 7 |- ( ( W e. Word V /\ U e. Word V ) -> ( # ` U ) e. NN0 )
26	25	3ad2ant1 1082	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> ( # ` U ) e. NN0 )
27		simpr 477	. . . . . . 7 |- ( ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) -> N <_ ( # ` U ) )
28	27	3ad2ant3 1084	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> N <_ ( # ` U ) )
29		elfz2nn0 12431	. . . . . 6 |- ( N e. ( 0 ... ( # ` U ) ) <-> ( N e. NN0 /\ ( # ` U ) e. NN0 /\ N <_ ( # ` U ) ) )
30	23, 26, 28, 29	syl3anbrc 1246	. . . . 5 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> N e. ( 0 ... ( # ` U ) ) )
31		swrd0len 13422	. . . . 5 |- ( ( U e. Word V /\ N e. ( 0 ... ( # ` U ) ) ) -> ( # ` ( U substr <. 0 , N >. ) ) = N )
32	21, 30, 31	syl2anc 693	. . . 4 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> ( # ` ( U substr <. 0 , N >. ) ) = N )
33	19, 32	eqeq12d 2637	. . 3 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> ( ( # ` ( W substr <. 0 , M >. ) ) = ( # ` ( U substr <. 0 , N >. ) ) <-> M = N ) )
34	33	anbi1d 741	. 2 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> ( ( ( # ` ( W substr <. 0 , M >. ) ) = ( # ` ( U substr <. 0 , N >. ) ) /\ A. i e. ( 0..^ ( # ` ( W substr <. 0 , M >. ) ) ) ( ( W substr <. 0 , M >. ) ` i ) = ( ( U substr <. 0 , N >. ) ` i ) ) <-> ( M = N /\ A. i e. ( 0..^ ( # ` ( W substr <. 0 , M >. ) ) ) ( ( W substr <. 0 , M >. ) ` i ) = ( ( U substr <. 0 , N >. ) ` i ) ) ) )
35	8	adantr 481	. . . . . . 7 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) -> W e. Word V )
36	17	adantr 481	. . . . . . 7 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) -> M e. ( 0 ... ( # ` W ) ) )
37	35, 36, 18	syl2anc 693	. . . . . 6 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) -> ( # ` ( W substr <. 0 , M >. ) ) = M )
38	37	oveq2d 6666	. . . . 5 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) -> ( 0..^ ( # ` ( W substr <. 0 , M >. ) ) ) = ( 0..^ M ) )
39	38	raleqdv 3144	. . . 4 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) -> ( A. i e. ( 0..^ ( # ` ( W substr <. 0 , M >. ) ) ) ( ( W substr <. 0 , M >. ) ` i ) = ( ( U substr <. 0 , N >. ) ` i ) <-> A. i e. ( 0..^ M ) ( ( W substr <. 0 , M >. ) ` i ) = ( ( U substr <. 0 , N >. ) ` i ) ) )
40	35	adantr 481	. . . . . . 7 |- ( ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> W e. Word V )
41	36	adantr 481	. . . . . . 7 |- ( ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> M e. ( 0 ... ( # ` W ) ) )
42		simpr 477	. . . . . . 7 |- ( ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> i e. ( 0..^ M ) )
43		swrd0fv 13439	. . . . . . 7 |- ( ( W e. Word V /\ M e. ( 0 ... ( # ` W ) ) /\ i e. ( 0..^ M ) ) -> ( ( W substr <. 0 , M >. ) ` i ) = ( W ` i ) )
44	40, 41, 42, 43	syl3anc 1326	. . . . . 6 |- ( ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> ( ( W substr <. 0 , M >. ) ` i ) = ( W ` i ) )
45		oveq2 6658	. . . . . . . . . . 11 |- ( M = N -> ( 0..^ M ) = ( 0..^ N ) )
46	45	eleq2d 2687	. . . . . . . . . 10 |- ( M = N -> ( i e. ( 0..^ M ) <-> i e. ( 0..^ N ) ) )
47	46	adantl 482	. . . . . . . . 9 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) -> ( i e. ( 0..^ M ) <-> i e. ( 0..^ N ) ) )
48	21	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ i e. ( 0..^ N ) ) -> U e. Word V )
49	30	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ i e. ( 0..^ N ) ) -> N e. ( 0 ... ( # ` U ) ) )
50		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ i e. ( 0..^ N ) ) -> i e. ( 0..^ N ) )
51	48, 49, 50	3jca 1242	. . . . . . . . . . 11 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ i e. ( 0..^ N ) ) -> ( U e. Word V /\ N e. ( 0 ... ( # ` U ) ) /\ i e. ( 0..^ N ) ) )
52	51	ex 450	. . . . . . . . . 10 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> ( i e. ( 0..^ N ) -> ( U e. Word V /\ N e. ( 0 ... ( # ` U ) ) /\ i e. ( 0..^ N ) ) ) )
53	52	adantr 481	. . . . . . . . 9 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) -> ( i e. ( 0..^ N ) -> ( U e. Word V /\ N e. ( 0 ... ( # ` U ) ) /\ i e. ( 0..^ N ) ) ) )
54	47, 53	sylbid 230	. . . . . . . 8 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) -> ( i e. ( 0..^ M ) -> ( U e. Word V /\ N e. ( 0 ... ( # ` U ) ) /\ i e. ( 0..^ N ) ) ) )
55	54	imp 445	. . . . . . 7 |- ( ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> ( U e. Word V /\ N e. ( 0 ... ( # ` U ) ) /\ i e. ( 0..^ N ) ) )
56		swrd0fv 13439	. . . . . . 7 |- ( ( U e. Word V /\ N e. ( 0 ... ( # ` U ) ) /\ i e. ( 0..^ N ) ) -> ( ( U substr <. 0 , N >. ) ` i ) = ( U ` i ) )
57	55, 56	syl 17	. . . . . 6 |- ( ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> ( ( U substr <. 0 , N >. ) ` i ) = ( U ` i ) )
58	44, 57	eqeq12d 2637	. . . . 5 |- ( ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> ( ( ( W substr <. 0 , M >. ) ` i ) = ( ( U substr <. 0 , N >. ) ` i ) <-> ( W ` i ) = ( U ` i ) ) )
59	58	ralbidva 2985	. . . 4 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) -> ( A. i e. ( 0..^ M ) ( ( W substr <. 0 , M >. ) ` i ) = ( ( U substr <. 0 , N >. ) ` i ) <-> A. i e. ( 0..^ M ) ( W ` i ) = ( U ` i ) ) )
60	39, 59	bitrd 268	. . 3 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) /\ M = N ) -> ( A. i e. ( 0..^ ( # ` ( W substr <. 0 , M >. ) ) ) ( ( W substr <. 0 , M >. ) ` i ) = ( ( U substr <. 0 , N >. ) ` i ) <-> A. i e. ( 0..^ M ) ( W ` i ) = ( U ` i ) ) )
61	60	pm5.32da 673	. 2 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> ( ( M = N /\ A. i e. ( 0..^ ( # ` ( W substr <. 0 , M >. ) ) ) ( ( W substr <. 0 , M >. ) ` i ) = ( ( U substr <. 0 , N >. ) ` i ) ) <-> ( M = N /\ A. i e. ( 0..^ M ) ( W ` i ) = ( U ` i ) ) ) )
62	6, 34, 61	3bitrd 294	1 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ ( # ` W ) /\ N <_ ( # ` U ) ) ) -> ( ( W substr <. 0 , M >. ) = ( U substr <. 0 , N >. ) <-> ( M = N /\ A. i e. ( 0..^ M ) ( W ` i ) = ( U ` i ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   <_ cle 10075   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   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-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-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-substr 13303

This theorem is referenced by:  clwlksf1clwwlklem  26968