Theorem swrdccat3b 11287

Description: A suffix of a concatenation is either a suffix of the second concatenated word or a concatenation of a suffix of the first word with the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by Alexander van der Vekens, 30-May-2018.) (Proof shortened by AV, 14-Oct-2022.)

Hypothesis

Ref	Expression
swrdccatin2.l	|- L = (♯ ` A )

Assertion

Ref	Expression
swrdccat3b	|- ( ( A e. Word V /\ B e. Word V ) -> ( M e. ( 0 ... ( L + (♯ ` B ) ) ) -> ( ( A ++ B ) substr <. M , ( L + (♯ ` B ) ) >. ) = if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) ) )

Proof of Theorem swrdccat3b

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( A e. Word V /\ B e. Word V ) )
2		simpr 110	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> M e. ( 0 ... ( L + (♯ ` B ) ) ) )
3		elfzubelfz 10244	. . . . 5 |- ( M e. ( 0 ... ( L + (♯ ` B ) ) ) -> ( L + (♯ ` B ) ) e. ( 0 ... ( L + (♯ ` B ) ) ) )
4	3	adantl 277	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( L + (♯ ` B ) ) e. ( 0 ... ( L + (♯ ` B ) ) ) )
5		swrdccatin2.l	. . . . . 6 |- L = (♯ ` A )
6	5	pfxccat3 11281	. . . . 5 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( 0 ... ( L + (♯ ` B ) ) ) /\ ( L + (♯ ` B ) ) e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , ( L + (♯ ` B ) ) >. ) = if ( ( L + (♯ ` B ) ) <_ L , ( A substr <. M , ( L + (♯ ` B ) ) >. ) , if ( L <_ M , ( B substr <. ( M - L ) , ( ( L + (♯ ` B ) ) - L ) >. ) , ( ( A substr <. M , L >. ) ++ ( B prefix ( ( L + (♯ ` B ) ) - L ) ) ) ) ) ) )
7	6	imp 124	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... ( L + (♯ ` B ) ) ) /\ ( L + (♯ ` B ) ) e. ( 0 ... ( L + (♯ ` B ) ) ) ) ) -> ( ( A ++ B ) substr <. M , ( L + (♯ ` B ) ) >. ) = if ( ( L + (♯ ` B ) ) <_ L , ( A substr <. M , ( L + (♯ ` B ) ) >. ) , if ( L <_ M , ( B substr <. ( M - L ) , ( ( L + (♯ ` B ) ) - L ) >. ) , ( ( A substr <. M , L >. ) ++ ( B prefix ( ( L + (♯ ` B ) ) - L ) ) ) ) ) )
8	1, 2, 4, 7	syl12anc 1269	. . 3 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , ( L + (♯ ` B ) ) >. ) = if ( ( L + (♯ ` B ) ) <_ L , ( A substr <. M , ( L + (♯ ` B ) ) >. ) , if ( L <_ M , ( B substr <. ( M - L ) , ( ( L + (♯ ` B ) ) - L ) >. ) , ( ( A substr <. M , L >. ) ++ ( B prefix ( ( L + (♯ ` B ) ) - L ) ) ) ) ) )
9	5	swrdccat3blem 11286	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ ( L + (♯ ` B ) ) <_ L ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) )
10		iftrue 3607	. . . . . 6 |- ( L <_ M -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( B substr <. ( M - L ) , (♯ ` B ) >. ) )
11	10	3ad2ant3 1044	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L /\ L <_ M ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( B substr <. ( M - L ) , (♯ ` B ) >. ) )
12		lencl 11088	. . . . . . . . . . . 12 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
13	12	nn0cnd 9435	. . . . . . . . . . 11 |- ( A e. Word V -> (♯ ` A ) e. CC )
14		lencl 11088	. . . . . . . . . . . 12 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
15	14	nn0cnd 9435	. . . . . . . . . . 11 |- ( B e. Word V -> (♯ ` B ) e. CC )
16	5	eqcomi 2233	. . . . . . . . . . . . 13 |- (♯ ` A ) = L
17	16	eleq1i 2295	. . . . . . . . . . . 12 |- ( (♯ ` A ) e. CC <-> L e. CC )
18		pncan2 8364	. . . . . . . . . . . 12 |- ( ( L e. CC /\ (♯ ` B ) e. CC ) -> ( ( L + (♯ ` B ) ) - L ) = (♯ ` B ) )
19	17, 18	sylanb 284	. . . . . . . . . . 11 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( ( L + (♯ ` B ) ) - L ) = (♯ ` B ) )
20	13, 15, 19	syl2an 289	. . . . . . . . . 10 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( L + (♯ ` B ) ) - L ) = (♯ ` B ) )
21	20	eqcomd 2235	. . . . . . . . 9 |- ( ( A e. Word V /\ B e. Word V ) -> (♯ ` B ) = ( ( L + (♯ ` B ) ) - L ) )
22	21	adantr 276	. . . . . . . 8 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> (♯ ` B ) = ( ( L + (♯ ` B ) ) - L ) )
23	22	3ad2ant1 1042	. . . . . . 7 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L /\ L <_ M ) -> (♯ ` B ) = ( ( L + (♯ ` B ) ) - L ) )
24	23	opeq2d 3864	. . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L /\ L <_ M ) -> <. ( M - L ) , (♯ ` B ) >. = <. ( M - L ) , ( ( L + (♯ ` B ) ) - L ) >. )
25	24	oveq2d 6023	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L /\ L <_ M ) -> ( B substr <. ( M - L ) , (♯ ` B ) >. ) = ( B substr <. ( M - L ) , ( ( L + (♯ ` B ) ) - L ) >. ) )
26	11, 25	eqtrd 2262	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L /\ L <_ M ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( B substr <. ( M - L ) , ( ( L + (♯ ` B ) ) - L ) >. ) )
27		iffalse 3610	. . . . . 6 |- ( -. L <_ M -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( ( A substr <. M , L >. ) ++ B ) )
28	27	3ad2ant3 1044	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L /\ -. L <_ M ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( ( A substr <. M , L >. ) ++ B ) )
29	20	adantr 276	. . . . . . . . 9 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( ( L + (♯ ` B ) ) - L ) = (♯ ` B ) )
30	29	3ad2ant1 1042	. . . . . . . 8 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L /\ -. L <_ M ) -> ( ( L + (♯ ` B ) ) - L ) = (♯ ` B ) )
31	30	oveq2d 6023	. . . . . . 7 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L /\ -. L <_ M ) -> ( B prefix ( ( L + (♯ ` B ) ) - L ) ) = ( B prefix (♯ ` B ) ) )
32		pfxid 11233	. . . . . . . . 9 |- ( B e. Word V -> ( B prefix (♯ ` B ) ) = B )
33	32	ad2antlr 489	. . . . . . . 8 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( B prefix (♯ ` B ) ) = B )
34	33	3ad2ant1 1042	. . . . . . 7 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L /\ -. L <_ M ) -> ( B prefix (♯ ` B ) ) = B )
35	31, 34	eqtr2d 2263	. . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L /\ -. L <_ M ) -> B = ( B prefix ( ( L + (♯ ` B ) ) - L ) ) )
36	35	oveq2d 6023	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L /\ -. L <_ M ) -> ( ( A substr <. M , L >. ) ++ B ) = ( ( A substr <. M , L >. ) ++ ( B prefix ( ( L + (♯ ` B ) ) - L ) ) ) )
37	28, 36	eqtrd 2262	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L /\ -. L <_ M ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( ( A substr <. M , L >. ) ++ ( B prefix ( ( L + (♯ ` B ) ) - L ) ) ) )
38	4	elfzelzd 10234	. . . . 5 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( L + (♯ ` B ) ) e. ZZ )
39	5, 12	eqeltrid 2316	. . . . . . 7 |- ( A e. Word V -> L e. NN0 )
40	39	ad2antrr 488	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> L e. NN0 )
41	40	nn0zd 9578	. . . . 5 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> L e. ZZ )
42		zdcle 9534	. . . . 5 |- ( ( ( L + (♯ ` B ) ) e. ZZ /\ L e. ZZ ) -> DECID ( L + (♯ ` B ) ) <_ L )
43	38, 41, 42	syl2anc 411	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> DECID ( L + (♯ ` B ) ) <_ L )
44	41	adantr 276	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L ) -> L e. ZZ )
45		elfznn0 10322	. . . . . . 7 |- ( M e. ( 0 ... ( L + (♯ ` B ) ) ) -> M e. NN0 )
46	45	ad2antlr 489	. . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L ) -> M e. NN0 )
47	46	nn0zd 9578	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L ) -> M e. ZZ )
48		zdcle 9534	. . . . 5 |- ( ( L e. ZZ /\ M e. ZZ ) -> DECID L <_ M )
49	44, 47, 48	syl2anc 411	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L ) -> DECID L <_ M )
50	9, 26, 37, 43, 49	2if2dc 3642	. . 3 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = if ( ( L + (♯ ` B ) ) <_ L , ( A substr <. M , ( L + (♯ ` B ) ) >. ) , if ( L <_ M , ( B substr <. ( M - L ) , ( ( L + (♯ ` B ) ) - L ) >. ) , ( ( A substr <. M , L >. ) ++ ( B prefix ( ( L + (♯ ` B ) ) - L ) ) ) ) ) )
51	8, 50	eqtr4d 2265	. 2 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , ( L + (♯ ` B ) ) >. ) = if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) )
52	51	ex 115	1 |- ( ( A e. Word V /\ B e. Word V ) -> ( M e. ( 0 ... ( L + (♯ ` B ) ) ) -> ( ( A ++ B ) substr <. M , ( L + (♯ ` B ) ) >. ) = if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   ifcif 3602   <.cop 3669   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   CCcc 8008   0cc0 8010   + caddc 8013   <_ cle 8193   - cmin 8328   NN0cn0 9380   ZZcz 9457   ...cfz 10216  ♯chash 11009  Word cword 11084   ++ cconcat 11138   substr csubstr 11192   prefix cpfx 11219

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-fz 10217  df-fzo 10351  df-ihash 11010  df-word 11085  df-concat 11139  df-substr 11193  df-pfx 11220

This theorem is referenced by: (None)