Theorem signstfvc 30651

Description: Zero-skipping sign in a word compared to a shorter word. (Contributed by Thierry Arnoux, 11-Oct-2018.)

Hypotheses

Ref	Expression
signsv.p	|- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
signsv.w	|- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
signsv.t	|- T = ( f e. Word RR |-> ( n e. ( 0..^ ( # ` f ) ) |-> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( f ` i ) ) ) ) ) )
signsv.v	|- V = ( f e. Word RR |-> sum_ j e. ( 1..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) )

Assertion

Ref	Expression
signstfvc	|- ( ( F e. Word RR /\ G e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ G ) ) ` N ) = ( ( T ` F ) ` N ) )

Distinct variable groups:   a, b, .+^   f, i, n, F   f, W, i, n   i, N, n

Allowed substitution hints:   .+^ ( f, i, j, n)   T( f, i, j, n, a, b)   F( j, a, b)   G( f, i, j, n, a, b)   N( f, j, a, b)   V( f, i, j, n, a, b)   W( j, a, b)

Proof of Theorem signstfvc

Dummy variables e g k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . . . 8 |- ( g = (/) -> ( F ++ g ) = ( F ++ (/) ) )
2	1	fveq2d 6195	. . . . . . 7 |- ( g = (/) -> ( T ` ( F ++ g ) ) = ( T ` ( F ++ (/) ) ) )
3	2	fveq1d 6193	. . . . . 6 |- ( g = (/) -> ( ( T ` ( F ++ g ) ) ` N ) = ( ( T ` ( F ++ (/) ) ) ` N ) )
4	3	eqeq1d 2624	. . . . 5 |- ( g = (/) -> ( ( ( T ` ( F ++ g ) ) ` N ) = ( ( T ` F ) ` N ) <-> ( ( T ` ( F ++ (/) ) ) ` N ) = ( ( T ` F ) ` N ) ) )
5	4	imbi2d 330	. . . 4 |- ( g = (/) -> ( ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ g ) ) ` N ) = ( ( T ` F ) ` N ) ) <-> ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ (/) ) ) ` N ) = ( ( T ` F ) ` N ) ) ) )
6		oveq2 6658	. . . . . . . 8 |- ( g = e -> ( F ++ g ) = ( F ++ e ) )
7	6	fveq2d 6195	. . . . . . 7 |- ( g = e -> ( T ` ( F ++ g ) ) = ( T ` ( F ++ e ) ) )
8	7	fveq1d 6193	. . . . . 6 |- ( g = e -> ( ( T ` ( F ++ g ) ) ` N ) = ( ( T ` ( F ++ e ) ) ` N ) )
9	8	eqeq1d 2624	. . . . 5 |- ( g = e -> ( ( ( T ` ( F ++ g ) ) ` N ) = ( ( T ` F ) ` N ) <-> ( ( T ` ( F ++ e ) ) ` N ) = ( ( T ` F ) ` N ) ) )
10	9	imbi2d 330	. . . 4 |- ( g = e -> ( ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ g ) ) ` N ) = ( ( T ` F ) ` N ) ) <-> ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ e ) ) ` N ) = ( ( T ` F ) ` N ) ) ) )
11		oveq2 6658	. . . . . . . 8 |- ( g = ( e ++ <" k "> ) -> ( F ++ g ) = ( F ++ ( e ++ <" k "> ) ) )
12	11	fveq2d 6195	. . . . . . 7 |- ( g = ( e ++ <" k "> ) -> ( T ` ( F ++ g ) ) = ( T ` ( F ++ ( e ++ <" k "> ) ) ) )
13	12	fveq1d 6193	. . . . . 6 |- ( g = ( e ++ <" k "> ) -> ( ( T ` ( F ++ g ) ) ` N ) = ( ( T ` ( F ++ ( e ++ <" k "> ) ) ) ` N ) )
14	13	eqeq1d 2624	. . . . 5 |- ( g = ( e ++ <" k "> ) -> ( ( ( T ` ( F ++ g ) ) ` N ) = ( ( T ` F ) ` N ) <-> ( ( T ` ( F ++ ( e ++ <" k "> ) ) ) ` N ) = ( ( T ` F ) ` N ) ) )
15	14	imbi2d 330	. . . 4 |- ( g = ( e ++ <" k "> ) -> ( ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ g ) ) ` N ) = ( ( T ` F ) ` N ) ) <-> ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ ( e ++ <" k "> ) ) ) ` N ) = ( ( T ` F ) ` N ) ) ) )
16		oveq2 6658	. . . . . . . 8 |- ( g = G -> ( F ++ g ) = ( F ++ G ) )
17	16	fveq2d 6195	. . . . . . 7 |- ( g = G -> ( T ` ( F ++ g ) ) = ( T ` ( F ++ G ) ) )
18	17	fveq1d 6193	. . . . . 6 |- ( g = G -> ( ( T ` ( F ++ g ) ) ` N ) = ( ( T ` ( F ++ G ) ) ` N ) )
19	18	eqeq1d 2624	. . . . 5 |- ( g = G -> ( ( ( T ` ( F ++ g ) ) ` N ) = ( ( T ` F ) ` N ) <-> ( ( T ` ( F ++ G ) ) ` N ) = ( ( T ` F ) ` N ) ) )
20	19	imbi2d 330	. . . 4 |- ( g = G -> ( ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ g ) ) ` N ) = ( ( T ` F ) ` N ) ) <-> ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ G ) ) ` N ) = ( ( T ` F ) ` N ) ) ) )
21		ccatrid 13370	. . . . . . 7 |- ( F e. Word RR -> ( F ++ (/) ) = F )
22	21	fveq2d 6195	. . . . . 6 |- ( F e. Word RR -> ( T ` ( F ++ (/) ) ) = ( T ` F ) )
23	22	fveq1d 6193	. . . . 5 |- ( F e. Word RR -> ( ( T ` ( F ++ (/) ) ) ` N ) = ( ( T ` F ) ` N ) )
24	23	adantr 481	. . . 4 |- ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ (/) ) ) ` N ) = ( ( T ` F ) ` N ) )
25		simprl 794	. . . . . . . . . . . 12 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> F e. Word RR )
26		simpll 790	. . . . . . . . . . . 12 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> e e. Word RR )
27		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> k e. RR )
28	27	s1cld 13383	. . . . . . . . . . . 12 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> <" k "> e. Word RR )
29		ccatass 13371	. . . . . . . . . . . 12 |- ( ( F e. Word RR /\ e e. Word RR /\ <" k "> e. Word RR ) -> ( ( F ++ e ) ++ <" k "> ) = ( F ++ ( e ++ <" k "> ) ) )
30	25, 26, 28, 29	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( ( F ++ e ) ++ <" k "> ) = ( F ++ ( e ++ <" k "> ) ) )
31	30	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( T ` ( ( F ++ e ) ++ <" k "> ) ) = ( T ` ( F ++ ( e ++ <" k "> ) ) ) )
32	31	fveq1d 6193	. . . . . . . . 9 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( ( T ` ( ( F ++ e ) ++ <" k "> ) ) ` N ) = ( ( T ` ( F ++ ( e ++ <" k "> ) ) ) ` N ) )
33		ccatcl 13359	. . . . . . . . . . 11 |- ( ( F e. Word RR /\ e e. Word RR ) -> ( F ++ e ) e. Word RR )
34	25, 26, 33	syl2anc 693	. . . . . . . . . 10 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( F ++ e ) e. Word RR )
35		lencl 13324	. . . . . . . . . . . . . . 15 |- ( F e. Word RR -> ( # ` F ) e. NN0 )
36	25, 35	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( # ` F ) e. NN0 )
37	36	nn0zd 11480	. . . . . . . . . . . . 13 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( # ` F ) e. ZZ )
38		lencl 13324	. . . . . . . . . . . . . . 15 |- ( ( F ++ e ) e. Word RR -> ( # ` ( F ++ e ) ) e. NN0 )
39	34, 38	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( # ` ( F ++ e ) ) e. NN0 )
40	39	nn0zd 11480	. . . . . . . . . . . . 13 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( # ` ( F ++ e ) ) e. ZZ )
41	36	nn0red 11352	. . . . . . . . . . . . . . 15 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( # ` F ) e. RR )
42		lencl 13324	. . . . . . . . . . . . . . . 16 |- ( e e. Word RR -> ( # ` e ) e. NN0 )
43	26, 42	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( # ` e ) e. NN0 )
44		nn0addge1 11339	. . . . . . . . . . . . . . 15 |- ( ( ( # ` F ) e. RR /\ ( # ` e ) e. NN0 ) -> ( # ` F ) <_ ( ( # ` F ) + ( # ` e ) ) )
45	41, 43, 44	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( # ` F ) <_ ( ( # ` F ) + ( # ` e ) ) )
46		ccatlen 13360	. . . . . . . . . . . . . . 15 |- ( ( F e. Word RR /\ e e. Word RR ) -> ( # ` ( F ++ e ) ) = ( ( # ` F ) + ( # ` e ) ) )
47	25, 26, 46	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( # ` ( F ++ e ) ) = ( ( # ` F ) + ( # ` e ) ) )
48	45, 47	breqtrrd 4681	. . . . . . . . . . . . 13 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( # ` F ) <_ ( # ` ( F ++ e ) ) )
49		eluz2 11693	. . . . . . . . . . . . 13 |- ( ( # ` ( F ++ e ) ) e. ( ZZ>= ` ( # ` F ) ) <-> ( ( # ` F ) e. ZZ /\ ( # ` ( F ++ e ) ) e. ZZ /\ ( # ` F ) <_ ( # ` ( F ++ e ) ) ) )
50	37, 40, 48, 49	syl3anbrc 1246	. . . . . . . . . . . 12 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( # ` ( F ++ e ) ) e. ( ZZ>= ` ( # ` F ) ) )
51		fzoss2 12496	. . . . . . . . . . . 12 |- ( ( # ` ( F ++ e ) ) e. ( ZZ>= ` ( # ` F ) ) -> ( 0..^ ( # ` F ) ) C_ ( 0..^ ( # ` ( F ++ e ) ) ) )
52	50, 51	syl 17	. . . . . . . . . . 11 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( 0..^ ( # ` F ) ) C_ ( 0..^ ( # ` ( F ++ e ) ) ) )
53		simprr 796	. . . . . . . . . . 11 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> N e. ( 0..^ ( # ` F ) ) )
54	52, 53	sseldd 3604	. . . . . . . . . 10 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> N e. ( 0..^ ( # ` ( F ++ e ) ) ) )
55		signsv.p	. . . . . . . . . . 11 |- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
56		signsv.w	. . . . . . . . . . 11 |- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
57		signsv.t	. . . . . . . . . . 11 |- T = ( f e. Word RR |-> ( n e. ( 0..^ ( # ` f ) ) |-> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( f ` i ) ) ) ) ) )
58		signsv.v	. . . . . . . . . . 11 |- V = ( f e. Word RR |-> sum_ j e. ( 1..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) )
59	55, 56, 57, 58	signstfvp 30648	. . . . . . . . . 10 |- ( ( ( F ++ e ) e. Word RR /\ k e. RR /\ N e. ( 0..^ ( # ` ( F ++ e ) ) ) ) -> ( ( T ` ( ( F ++ e ) ++ <" k "> ) ) ` N ) = ( ( T ` ( F ++ e ) ) ` N ) )
60	34, 27, 54, 59	syl3anc 1326	. . . . . . . . 9 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( ( T ` ( ( F ++ e ) ++ <" k "> ) ) ` N ) = ( ( T ` ( F ++ e ) ) ` N ) )
61	32, 60	eqtr3d 2658	. . . . . . . 8 |- ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) -> ( ( T ` ( F ++ ( e ++ <" k "> ) ) ) ` N ) = ( ( T ` ( F ++ e ) ) ` N ) )
62	61	adantr 481	. . . . . . 7 |- ( ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) /\ ( ( T ` ( F ++ e ) ) ` N ) = ( ( T ` F ) ` N ) ) -> ( ( T ` ( F ++ ( e ++ <" k "> ) ) ) ` N ) = ( ( T ` ( F ++ e ) ) ` N ) )
63		simpr 477	. . . . . . 7 |- ( ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) /\ ( ( T ` ( F ++ e ) ) ` N ) = ( ( T ` F ) ` N ) ) -> ( ( T ` ( F ++ e ) ) ` N ) = ( ( T ` F ) ` N ) )
64	62, 63	eqtrd 2656	. . . . . 6 |- ( ( ( ( e e. Word RR /\ k e. RR ) /\ ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) ) /\ ( ( T ` ( F ++ e ) ) ` N ) = ( ( T ` F ) ` N ) ) -> ( ( T ` ( F ++ ( e ++ <" k "> ) ) ) ` N ) = ( ( T ` F ) ` N ) )
65	64	exp31 630	. . . . 5 |- ( ( e e. Word RR /\ k e. RR ) -> ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( ( T ` ( F ++ e ) ) ` N ) = ( ( T ` F ) ` N ) -> ( ( T ` ( F ++ ( e ++ <" k "> ) ) ) ` N ) = ( ( T ` F ) ` N ) ) ) )
66	65	a2d 29	. . . 4 |- ( ( e e. Word RR /\ k e. RR ) -> ( ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ e ) ) ` N ) = ( ( T ` F ) ` N ) ) -> ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ ( e ++ <" k "> ) ) ) ` N ) = ( ( T ` F ) ` N ) ) ) )
67	5, 10, 15, 20, 24, 66	wrdind 13476	. . 3 |- ( G e. Word RR -> ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ G ) ) ` N ) = ( ( T ` F ) ` N ) ) )
68	67	3impib 1262	. 2 |- ( ( G e. Word RR /\ F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ G ) ) ` N ) = ( ( T ` F ) ` N ) )
69	68	3com12 1269	1 |- ( ( F e. Word RR /\ G e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ G ) ) ` N ) = ( ( T ` F ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   ifcif 4086   {cpr 4179   {ctp 4181   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   -ucneg 10267   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294  sgncsgn 13826   sum_csu 14416   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   gsum_g cgsu 16101

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-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303

This theorem is referenced by:  signstres  30652