Theorem swrdccat3b 11370

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 10316	. . . . 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 11364	. . . . 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 1272	. . 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 11369	. . . 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 3614	. . . . . 6 |- ( L <_ M -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( B substr <. ( M - L ) , (♯ ` B ) >. ) )
11	10	3ad2ant3 1047	. . . . 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 11166	. . . . . . . . . . . 12 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
13	12	nn0cnd 9501	. . . . . . . . . . 11 |- ( A e. Word V -> (♯ ` A ) e. CC )
14		lencl 11166	. . . . . . . . . . . 12 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
15	14	nn0cnd 9501	. . . . . . . . . . 11 |- ( B e. Word V -> (♯ ` B ) e. CC )
16	5	eqcomi 2235	. . . . . . . . . . . . 13 |- (♯ ` A ) = L
17	16	eleq1i 2297	. . . . . . . . . . . 12 |- ( (♯ ` A ) e. CC <-> L e. CC )
18		pncan2 8428	. . . . . . . . . . . 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 2237	. . . . . . . . 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 1045	. . . . . . 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 3874	. . . . . 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 6044	. . . . 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 2264	. . . 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 3617	. . . . . 6 |- ( -. L <_ M -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( ( A substr <. M , L >. ) ++ B ) )
28	27	3ad2ant3 1047	. . . . 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 1045	. . . . . . . 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 6044	. . . . . . 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 11316	. . . . . . . . 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 1045	. . . . . . 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 2265	. . . . . 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 6044	. . . . 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 2264	. . . 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 10306	. . . . 5 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( L + (♯ ` B ) ) e. ZZ )
39	5, 12	eqeltrid 2318	. . . . . . 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 9644	. . . . 5 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> L e. ZZ )
42		zdcle 9600	. . . . 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 10394	. . . . . . 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 9644	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L ) -> M e. ZZ )
48		zdcle 9600	. . . . 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 3649	. . 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 2267	. 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 842   /\ w3a 1005   = wceq 1398   e. wcel 2202   ifcif 3607   <.cop 3676   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8073   0cc0 8075   + caddc 8078   <_ cle 8257   - cmin 8392   NN0cn0 9444   ZZcz 9523   ...cfz 10288  ♯chash 11083  Word cword 11162   ++ cconcat 11216   substr csubstr 11275   prefix cpfx 11302

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-inn 9186  df-n0 9445  df-z 9524  df-uz 9800  df-fz 10289  df-fzo 10423  df-ihash 11084  df-word 11163  df-concat 11217  df-substr 11276  df-pfx 11303

This theorem is referenced by: (None)