Theorem swrdccat3a 13494

Description: A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by Alexander van der Vekens, 29-May-2018.)

Hypothesis

Ref	Expression
swrdccatin12.l	|- L = ( # ` A )

Assertion

Ref	Expression
swrdccat3a	|- ( ( A e. Word V /\ B e. Word V ) -> ( N e. ( 0 ... ( L + ( # ` B ) ) ) -> ( ( A ++ B ) substr <. 0 , N >. ) = if ( N <_ L , ( A substr <. 0 , N >. ) , ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) ) ) )

Proof of Theorem swrdccat3a

Step	Hyp	Ref	Expression
1		elfznn0 12433	. . . . . 6 |- ( N e. ( 0 ... ( L + ( # ` B ) ) ) -> N e. NN0 )
2		0elfz 12436	. . . . . 6 |- ( N e. NN0 -> 0 e. ( 0 ... N ) )
3	1, 2	syl 17	. . . . 5 |- ( N e. ( 0 ... ( L + ( # ` B ) ) ) -> 0 e. ( 0 ... N ) )
4	3	ancri 575	. . . 4 |- ( N e. ( 0 ... ( L + ( # ` B ) ) ) -> ( 0 e. ( 0 ... N ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) )
5		swrdccatin12.l	. . . . . 6 |- L = ( # ` A )
6	5	swrdccat3 13492	. . . . 5 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( 0 e. ( 0 ... N ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) -> ( ( A ++ B ) substr <. 0 , N >. ) = if ( N <_ L , ( A substr <. 0 , N >. ) , if ( L <_ 0 , ( B substr <. ( 0 - L ) , ( N - L ) >. ) , ( ( A substr <. 0 , L >. ) ++ ( B substr <. 0 , ( N - L ) >. ) ) ) ) ) )
7	6	imp 445	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( 0 e. ( 0 ... N ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) ) -> ( ( A ++ B ) substr <. 0 , N >. ) = if ( N <_ L , ( A substr <. 0 , N >. ) , if ( L <_ 0 , ( B substr <. ( 0 - L ) , ( N - L ) >. ) , ( ( A substr <. 0 , L >. ) ++ ( B substr <. 0 , ( N - L ) >. ) ) ) ) )
8	4, 7	sylan2 491	. . 3 |- ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) -> ( ( A ++ B ) substr <. 0 , N >. ) = if ( N <_ L , ( A substr <. 0 , N >. ) , if ( L <_ 0 , ( B substr <. ( 0 - L ) , ( N - L ) >. ) , ( ( A substr <. 0 , L >. ) ++ ( B substr <. 0 , ( N - L ) >. ) ) ) ) )
9		iftrue 4092	. . . . 5 |- ( N <_ L -> if ( N <_ L , ( A substr <. 0 , N >. ) , ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) ) = ( A substr <. 0 , N >. ) )
10	9	adantl 482	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ N <_ L ) -> if ( N <_ L , ( A substr <. 0 , N >. ) , ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) ) = ( A substr <. 0 , N >. ) )
11		iffalse 4095	. . . . . 6 |- ( -. N <_ L -> if ( N <_ L , ( A substr <. 0 , N >. ) , ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) ) = ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) )
12	11	3ad2ant2 1083	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. N <_ L /\ L <_ 0 ) -> if ( N <_ L , ( A substr <. 0 , N >. ) , ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) ) = ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) )
13		lencl 13324	. . . . . . . . . . . . 13 |- ( A e. Word V -> ( # ` A ) e. NN0 )
14	5, 13	syl5eqel 2705	. . . . . . . . . . . 12 |- ( A e. Word V -> L e. NN0 )
15		nn0le0eq0 11321	. . . . . . . . . . . 12 |- ( L e. NN0 -> ( L <_ 0 <-> L = 0 ) )
16	14, 15	syl 17	. . . . . . . . . . 11 |- ( A e. Word V -> ( L <_ 0 <-> L = 0 ) )
17	16	biimpd 219	. . . . . . . . . 10 |- ( A e. Word V -> ( L <_ 0 -> L = 0 ) )
18	17	adantr 481	. . . . . . . . 9 |- ( ( A e. Word V /\ B e. Word V ) -> ( L <_ 0 -> L = 0 ) )
19	5	eqeq1i 2627	. . . . . . . . . . . . . . . 16 |- ( L = 0 <-> ( # ` A ) = 0 )
20	19	biimpi 206	. . . . . . . . . . . . . . 15 |- ( L = 0 -> ( # ` A ) = 0 )
21		hasheq0 13154	. . . . . . . . . . . . . . 15 |- ( A e. Word V -> ( ( # ` A ) = 0 <-> A = (/) ) )
22	20, 21	syl5ib 234	. . . . . . . . . . . . . 14 |- ( A e. Word V -> ( L = 0 -> A = (/) ) )
23	22	adantr 481	. . . . . . . . . . . . 13 |- ( ( A e. Word V /\ B e. Word V ) -> ( L = 0 -> A = (/) ) )
24	23	imp 445	. . . . . . . . . . . 12 |- ( ( ( A e. Word V /\ B e. Word V ) /\ L = 0 ) -> A = (/) )
25		0m0e0 11130	. . . . . . . . . . . . . . . 16 |- ( 0 - 0 ) = 0
26		oveq2 6658	. . . . . . . . . . . . . . . . 17 |- ( 0 = L -> ( 0 - 0 ) = ( 0 - L ) )
27	26	eqcoms 2630	. . . . . . . . . . . . . . . 16 |- ( L = 0 -> ( 0 - 0 ) = ( 0 - L ) )
28	25, 27	syl5eqr 2670	. . . . . . . . . . . . . . 15 |- ( L = 0 -> 0 = ( 0 - L ) )
29	28	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( A e. Word V /\ B e. Word V ) /\ L = 0 ) -> 0 = ( 0 - L ) )
30	29	opeq1d 4408	. . . . . . . . . . . . 13 |- ( ( ( A e. Word V /\ B e. Word V ) /\ L = 0 ) -> <. 0 , ( N - L ) >. = <. ( 0 - L ) , ( N - L ) >. )
31	30	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( ( A e. Word V /\ B e. Word V ) /\ L = 0 ) -> ( B substr <. 0 , ( N - L ) >. ) = ( B substr <. ( 0 - L ) , ( N - L ) >. ) )
32	24, 31	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( A e. Word V /\ B e. Word V ) /\ L = 0 ) -> ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) = ( (/) ++ ( B substr <. ( 0 - L ) , ( N - L ) >. ) ) )
33		swrdcl 13419	. . . . . . . . . . . . . 14 |- ( B e. Word V -> ( B substr <. ( 0 - L ) , ( N - L ) >. ) e. Word V )
34		ccatlid 13369	. . . . . . . . . . . . . 14 |- ( ( B substr <. ( 0 - L ) , ( N - L ) >. ) e. Word V -> ( (/) ++ ( B substr <. ( 0 - L ) , ( N - L ) >. ) ) = ( B substr <. ( 0 - L ) , ( N - L ) >. ) )
35	33, 34	syl 17	. . . . . . . . . . . . 13 |- ( B e. Word V -> ( (/) ++ ( B substr <. ( 0 - L ) , ( N - L ) >. ) ) = ( B substr <. ( 0 - L ) , ( N - L ) >. ) )
36	35	adantl 482	. . . . . . . . . . . 12 |- ( ( A e. Word V /\ B e. Word V ) -> ( (/) ++ ( B substr <. ( 0 - L ) , ( N - L ) >. ) ) = ( B substr <. ( 0 - L ) , ( N - L ) >. ) )
37	36	adantr 481	. . . . . . . . . . 11 |- ( ( ( A e. Word V /\ B e. Word V ) /\ L = 0 ) -> ( (/) ++ ( B substr <. ( 0 - L ) , ( N - L ) >. ) ) = ( B substr <. ( 0 - L ) , ( N - L ) >. ) )
38	32, 37	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( A e. Word V /\ B e. Word V ) /\ L = 0 ) -> ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) = ( B substr <. ( 0 - L ) , ( N - L ) >. ) )
39	38	ex 450	. . . . . . . . 9 |- ( ( A e. Word V /\ B e. Word V ) -> ( L = 0 -> ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) = ( B substr <. ( 0 - L ) , ( N - L ) >. ) ) )
40	18, 39	syld 47	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V ) -> ( L <_ 0 -> ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) = ( B substr <. ( 0 - L ) , ( N - L ) >. ) ) )
41	40	adantr 481	. . . . . . 7 |- ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) -> ( L <_ 0 -> ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) = ( B substr <. ( 0 - L ) , ( N - L ) >. ) ) )
42	41	imp 445	. . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ L <_ 0 ) -> ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) = ( B substr <. ( 0 - L ) , ( N - L ) >. ) )
43	42	3adant2 1080	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. N <_ L /\ L <_ 0 ) -> ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) = ( B substr <. ( 0 - L ) , ( N - L ) >. ) )
44	12, 43	eqtrd 2656	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. N <_ L /\ L <_ 0 ) -> if ( N <_ L , ( A substr <. 0 , N >. ) , ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) ) = ( B substr <. ( 0 - L ) , ( N - L ) >. ) )
45	11	3ad2ant2 1083	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. N <_ L /\ -. L <_ 0 ) -> if ( N <_ L , ( A substr <. 0 , N >. ) , ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) ) = ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) )
46	5	opeq2i 4406	. . . . . . . . . . 11 |- <. 0 , L >. = <. 0 , ( # ` A ) >.
47	46	oveq2i 6661	. . . . . . . . . 10 |- ( A substr <. 0 , L >. ) = ( A substr <. 0 , ( # ` A ) >. )
48		swrdid 13428	. . . . . . . . . 10 |- ( A e. Word V -> ( A substr <. 0 , ( # ` A ) >. ) = A )
49	47, 48	syl5req 2669	. . . . . . . . 9 |- ( A e. Word V -> A = ( A substr <. 0 , L >. ) )
50	49	adantr 481	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V ) -> A = ( A substr <. 0 , L >. ) )
51	50	adantr 481	. . . . . . 7 |- ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) -> A = ( A substr <. 0 , L >. ) )
52	51	3ad2ant1 1082	. . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. N <_ L /\ -. L <_ 0 ) -> A = ( A substr <. 0 , L >. ) )
53	52	oveq1d 6665	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. N <_ L /\ -. L <_ 0 ) -> ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) = ( ( A substr <. 0 , L >. ) ++ ( B substr <. 0 , ( N - L ) >. ) ) )
54	45, 53	eqtrd 2656	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. N <_ L /\ -. L <_ 0 ) -> if ( N <_ L , ( A substr <. 0 , N >. ) , ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) ) = ( ( A substr <. 0 , L >. ) ++ ( B substr <. 0 , ( N - L ) >. ) ) )
55	10, 44, 54	2if2 4136	. . 3 |- ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) -> if ( N <_ L , ( A substr <. 0 , N >. ) , ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) ) = if ( N <_ L , ( A substr <. 0 , N >. ) , if ( L <_ 0 , ( B substr <. ( 0 - L ) , ( N - L ) >. ) , ( ( A substr <. 0 , L >. ) ++ ( B substr <. 0 , ( N - L ) >. ) ) ) ) )
56	8, 55	eqtr4d 2659	. 2 |- ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... ( L + ( # ` B ) ) ) ) -> ( ( A ++ B ) substr <. 0 , N >. ) = if ( N <_ L , ( A substr <. 0 , N >. ) , ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) ) )
57	56	ex 450	1 |- ( ( A e. Word V /\ B e. Word V ) -> ( N e. ( 0 ... ( L + ( # ` B ) ) ) -> ( ( A ++ B ) substr <. 0 , N >. ) = if ( N <_ L , ( A substr <. 0 , N >. ) , ( A ++ ( B substr <. 0 , ( N - L ) >. ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   (/)c0 3915   ifcif 4086   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   + caddc 9939   <_ cle 10075   - cmin 10266   NN0cn0 11292   ...cfz 12326   #chash 13117  Word cword 13291   ++ cconcat 13293   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-concat 13301  df-substr 13303

This theorem is referenced by:  swrdccatid  13497