Theorem swrdccat3b 11490

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 10419	. . . . 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 11484	. . . . 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 1276	. . 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 11489	. . . 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 3642	. . . . . 6 |- ( L <_ M -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( B substr <. ( M - L ) , (♯ ` B ) >. ) )
11	10	3ad2ant3 1051	. . . . 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 11286	. . . . . . . . . . . 12 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
13	12	nn0cnd 9601	. . . . . . . . . . 11 |- ( A e. Word V -> (♯ ` A ) e. CC )
14		lencl 11286	. . . . . . . . . . . 12 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
15	14	nn0cnd 9601	. . . . . . . . . . 11 |- ( B e. Word V -> (♯ ` B ) e. CC )
16	5	eqcomi 2242	. . . . . . . . . . . . 13 |- (♯ ` A ) = L
17	16	eleq1i 2304	. . . . . . . . . . . 12 |- ( (♯ ` A ) e. CC <-> L e. CC )
18		pncan2 8523	. . . . . . . . . . . 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 2244	. . . . . . . . 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 1049	. . . . . . 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 3906	. . . . . 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 6091	. . . . 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 2271	. . . 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 3645	. . . . . 6 |- ( -. L <_ M -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( ( A substr <. M , L >. ) ++ B ) )
28	27	3ad2ant3 1051	. . . . 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 1049	. . . . . . . 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 6091	. . . . . . 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 11436	. . . . . . . . 9 |- ( B e. Word V -> ( B prefix (♯ ` B ) ) = B )
33	32	ad2antlr 493	. . . . . . . 8 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( B prefix (♯ ` B ) ) = B )
34	33	3ad2ant1 1049	. . . . . . 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 2272	. . . . . 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 6091	. . . . 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 2271	. . . 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 10408	. . . . 5 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( L + (♯ ` B ) ) e. ZZ )
39	5, 12	eqeltrid 2325	. . . . . . 7 |- ( A e. Word V -> L e. NN0 )
40	39	ad2antrr 492	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> L e. NN0 )
41	40	nn0zd 9745	. . . . 5 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> L e. ZZ )
42		zdcle 9700	. . . . 5 |- ( ( ( L + (♯ ` B ) ) e. ZZ /\ L e. ZZ ) -> DECID ( L + (♯ ` B ) ) <_ L )
43	38, 41, 42	syl2anc 415	. . . 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 10499	. . . . . . 7 |- ( M e. ( 0 ... ( L + (♯ ` B ) ) ) -> M e. NN0 )
46	45	ad2antlr 493	. . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L ) -> M e. NN0 )
47	46	nn0zd 9745	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ -. ( L + (♯ ` B ) ) <_ L ) -> M e. ZZ )
48		zdcle 9700	. . . . 5 |- ( ( L e. ZZ /\ M e. ZZ ) -> DECID L <_ M )
49	44, 47, 48	syl2anc 415	. . . 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 3677	. . 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 2274	. 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 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   ifcif 3635   <.cop 3708   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   0cc0 8169   + caddc 8172   <_ cle 8351   - cmin 8487   NN0cn0 9542   ZZcz 9623   ...cfz 10390  ♯chash 11192  Word cword 11282   ++ cconcat 11336   substr csubstr 11395   prefix cpfx 11422

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-concat 11337  df-substr 11396  df-pfx 11423

This theorem is referenced by: (None)