Theorem pfxccatin12 11425

Description: The subword of a concatenation of two words within both of the concatenated words. (Contributed by Alexander van der Vekens, 5-Apr-2018.) (Revised by AV, 9-May-2020.)

Hypothesis

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

Assertion

Ref	Expression
pfxccatin12	|- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ) )

Proof of Theorem pfxccatin12

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		swrdccatin2.l	. . . . 5 |- L = (♯ ` A )
2	1	pfxccatin12lem2c 11422	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( A ++ B ) e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` ( A ++ B ) ) ) ) )
3		swrdvalfn 11348	. . . 4 |- ( ( ( A ++ B ) e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` ( A ++ B ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) )
4	2, 3	syl 14	. . 3 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) )
5		simpll 527	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> A e. Word V )
6		elfzelz 10359	. . . . . . 7 |- ( M e. ( 0 ... L ) -> M e. ZZ )
7	6	ad2antrl 490	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> M e. ZZ )
8		elfzel1 10358	. . . . . . 7 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> L e. ZZ )
9	8	ad2antll 491	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> L e. ZZ )
10		swrdclg 11342	. . . . . 6 |- ( ( A e. Word V /\ M e. ZZ /\ L e. ZZ ) -> ( A substr <. M , L >. ) e. Word V )
11	5, 7, 9, 10	syl3anc 1274	. . . . 5 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( A substr <. M , L >. ) e. Word V )
12		simplr 529	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> B e. Word V )
13		elfzle1 10361	. . . . . . . 8 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> L <_ N )
14		elfzelz 10359	. . . . . . . . 9 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> N e. ZZ )
15		znn0sub 9643	. . . . . . . . 9 |- ( ( L e. ZZ /\ N e. ZZ ) -> ( L <_ N <-> ( N - L ) e. NN0 ) )
16	8, 14, 15	syl2anc 411	. . . . . . . 8 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> ( L <_ N <-> ( N - L ) e. NN0 ) )
17	13, 16	mpbid 147	. . . . . . 7 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> ( N - L ) e. NN0 )
18	17	ad2antll 491	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( N - L ) e. NN0 )
19		pfxclg 11370	. . . . . 6 |- ( ( B e. Word V /\ ( N - L ) e. NN0 ) -> ( B prefix ( N - L ) ) e. Word V )
20	12, 18, 19	syl2anc 411	. . . . 5 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( B prefix ( N - L ) ) e. Word V )
21		ccatvalfn 11289	. . . . 5 |- ( ( ( A substr <. M , L >. ) e. Word V /\ ( B prefix ( N - L ) ) e. Word V ) -> ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) Fn ( 0..^ ( (♯ ` ( A substr <. M , L >. ) ) + (♯ ` ( B prefix ( N - L ) ) ) ) ) )
22	11, 20, 21	syl2anc 411	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) Fn ( 0..^ ( (♯ ` ( A substr <. M , L >. ) ) + (♯ ` ( B prefix ( N - L ) ) ) ) ) )
23		simprl 531	. . . . . . . . 9 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> M e. ( 0 ... L ) )
24		lencl 11228	. . . . . . . . . . . 12 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
25		nn0fz0 10453	. . . . . . . . . . . 12 |- ( (♯ ` A ) e. NN0 <-> (♯ ` A ) e. ( 0 ... (♯ ` A ) ) )
26	24, 25	sylib 122	. . . . . . . . . . 11 |- ( A e. Word V -> (♯ ` A ) e. ( 0 ... (♯ ` A ) ) )
27	1, 26	eqeltrid 2319	. . . . . . . . . 10 |- ( A e. Word V -> L e. ( 0 ... (♯ ` A ) ) )
28	27	ad2antrr 488	. . . . . . . . 9 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> L e. ( 0 ... (♯ ` A ) ) )
29		swrdlen 11344	. . . . . . . . 9 |- ( ( A e. Word V /\ M e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` A ) ) ) -> (♯ ` ( A substr <. M , L >. ) ) = ( L - M ) )
30	5, 23, 28, 29	syl3anc 1274	. . . . . . . 8 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> (♯ ` ( A substr <. M , L >. ) ) = ( L - M ) )
31		lencl 11228	. . . . . . . . . . . 12 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
32	31	nn0zd 9698	. . . . . . . . . . 11 |- ( B e. Word V -> (♯ ` B ) e. ZZ )
33		elfzmlbp 10466	. . . . . . . . . . 11 |- ( ( (♯ ` B ) e. ZZ /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( N - L ) e. ( 0 ... (♯ ` B ) ) )
34	32, 33	sylan 283	. . . . . . . . . 10 |- ( ( B e. Word V /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( N - L ) e. ( 0 ... (♯ ` B ) ) )
35		pfxlen 11377	. . . . . . . . . 10 |- ( ( B e. Word V /\ ( N - L ) e. ( 0 ... (♯ ` B ) ) ) -> (♯ ` ( B prefix ( N - L ) ) ) = ( N - L ) )
36	34, 35	syldan 282	. . . . . . . . 9 |- ( ( B e. Word V /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> (♯ ` ( B prefix ( N - L ) ) ) = ( N - L ) )
37	36	ad2ant2l 508	. . . . . . . 8 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> (♯ ` ( B prefix ( N - L ) ) ) = ( N - L ) )
38	30, 37	oveq12d 6068	. . . . . . 7 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( (♯ ` ( A substr <. M , L >. ) ) + (♯ ` ( B prefix ( N - L ) ) ) ) = ( ( L - M ) + ( N - L ) ) )
39		elfz2nn0 10446	. . . . . . . . . . 11 |- ( M e. ( 0 ... L ) <-> ( M e. NN0 /\ L e. NN0 /\ M <_ L ) )
40		nn0cn 9506	. . . . . . . . . . . . . . . 16 |- ( L e. NN0 -> L e. CC )
41	40	ad2antll 491	. . . . . . . . . . . . . . 15 |- ( ( N e. ZZ /\ ( M e. NN0 /\ L e. NN0 ) ) -> L e. CC )
42		nn0cn 9506	. . . . . . . . . . . . . . . 16 |- ( M e. NN0 -> M e. CC )
43	42	ad2antrl 490	. . . . . . . . . . . . . . 15 |- ( ( N e. ZZ /\ ( M e. NN0 /\ L e. NN0 ) ) -> M e. CC )
44		zcn 9582	. . . . . . . . . . . . . . . 16 |- ( N e. ZZ -> N e. CC )
45	44	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( N e. ZZ /\ ( M e. NN0 /\ L e. NN0 ) ) -> N e. CC )
46	41, 43, 45	3jca 1204	. . . . . . . . . . . . . 14 |- ( ( N e. ZZ /\ ( M e. NN0 /\ L e. NN0 ) ) -> ( L e. CC /\ M e. CC /\ N e. CC ) )
47	46	ex 115	. . . . . . . . . . . . 13 |- ( N e. ZZ -> ( ( M e. NN0 /\ L e. NN0 ) -> ( L e. CC /\ M e. CC /\ N e. CC ) ) )
48	47, 14	syl11 31	. . . . . . . . . . . 12 |- ( ( M e. NN0 /\ L e. NN0 ) -> ( N e. ( L ... ( L + (♯ ` B ) ) ) -> ( L e. CC /\ M e. CC /\ N e. CC ) ) )
49	48	3adant3 1044	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ L e. NN0 /\ M <_ L ) -> ( N e. ( L ... ( L + (♯ ` B ) ) ) -> ( L e. CC /\ M e. CC /\ N e. CC ) ) )
50	39, 49	sylbi 121	. . . . . . . . . 10 |- ( M e. ( 0 ... L ) -> ( N e. ( L ... ( L + (♯ ` B ) ) ) -> ( L e. CC /\ M e. CC /\ N e. CC ) ) )
51	50	imp 124	. . . . . . . . 9 |- ( ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( L e. CC /\ M e. CC /\ N e. CC ) )
52		npncan3 8511	. . . . . . . . 9 |- ( ( L e. CC /\ M e. CC /\ N e. CC ) -> ( ( L - M ) + ( N - L ) ) = ( N - M ) )
53	51, 52	syl 14	. . . . . . . 8 |- ( ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( ( L - M ) + ( N - L ) ) = ( N - M ) )
54	53	adantl 277	. . . . . . 7 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( L - M ) + ( N - L ) ) = ( N - M ) )
55	38, 54	eqtr2d 2266	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( N - M ) = ( (♯ ` ( A substr <. M , L >. ) ) + (♯ ` ( B prefix ( N - L ) ) ) ) )
56	55	oveq2d 6066	. . . . 5 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( 0..^ ( N - M ) ) = ( 0..^ ( (♯ ` ( A substr <. M , L >. ) ) + (♯ ` ( B prefix ( N - L ) ) ) ) ) )
57	56	fneq2d 5447	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) Fn ( 0..^ ( N - M ) ) <-> ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) Fn ( 0..^ ( (♯ ` ( A substr <. M , L >. ) ) + (♯ ` ( B prefix ( N - L ) ) ) ) ) ) )
58	22, 57	mpbird 167	. . 3 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) Fn ( 0..^ ( N - M ) ) )
59		elfzoelz 10481	. . . . . 6 |- ( k e. ( 0..^ ( N - M ) ) -> k e. ZZ )
60	59	adantl 277	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> k e. ZZ )
61		0zd 9589	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> 0 e. ZZ )
62	9, 7	zsubcld 9705	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( L - M ) e. ZZ )
63	62	adantr 276	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( L - M ) e. ZZ )
64		fzodcel 10487	. . . . 5 |- ( ( k e. ZZ /\ 0 e. ZZ /\ ( L - M ) e. ZZ ) -> DECID k e. ( 0..^ ( L - M ) ) )
65	60, 61, 63, 64	syl3anc 1274	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> DECID k e. ( 0..^ ( L - M ) ) )
66		exmiddc 844	. . . . 5 |- (DECID k e. ( 0..^ ( L - M ) ) -> ( k e. ( 0..^ ( L - M ) ) \/ -. k e. ( 0..^ ( L - M ) ) ) )
67		simprl 531	. . . . . . . . 9 |- ( ( k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) )
68		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> k e. ( 0..^ ( N - M ) ) )
69	68	anim2i 342	. . . . . . . . . 10 |- ( ( k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( k e. ( 0..^ ( L - M ) ) /\ k e. ( 0..^ ( N - M ) ) ) )
70	69	ancomd 267	. . . . . . . . 9 |- ( ( k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( k e. ( 0..^ ( N - M ) ) /\ k e. ( 0..^ ( L - M ) ) ) )
71	1	pfxccatin12lem3 11424	. . . . . . . . 9 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( k e. ( 0..^ ( N - M ) ) /\ k e. ( 0..^ ( L - M ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( A substr <. M , L >. ) ` k ) ) )
72	67, 70, 71	sylc 62	. . . . . . . 8 |- ( ( k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( A substr <. M , L >. ) ` k ) )
73	11, 20	jca 306	. . . . . . . . . . 11 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( A substr <. M , L >. ) e. Word V /\ ( B prefix ( N - L ) ) e. Word V ) )
74	73	ad2antrl 490	. . . . . . . . . 10 |- ( ( k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( ( A substr <. M , L >. ) e. Word V /\ ( B prefix ( N - L ) ) e. Word V ) )
75		simpl 109	. . . . . . . . . . 11 |- ( ( k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> k e. ( 0..^ ( L - M ) ) )
76	30	oveq2d 6066	. . . . . . . . . . . 12 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( 0..^ (♯ ` ( A substr <. M , L >. ) ) ) = ( 0..^ ( L - M ) ) )
77	76	ad2antrl 490	. . . . . . . . . . 11 |- ( ( k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( 0..^ (♯ ` ( A substr <. M , L >. ) ) ) = ( 0..^ ( L - M ) ) )
78	75, 77	eleqtrrd 2312	. . . . . . . . . 10 |- ( ( k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> k e. ( 0..^ (♯ ` ( A substr <. M , L >. ) ) ) )
79		df-3an 1007	. . . . . . . . . 10 |- ( ( ( A substr <. M , L >. ) e. Word V /\ ( B prefix ( N - L ) ) e. Word V /\ k e. ( 0..^ (♯ ` ( A substr <. M , L >. ) ) ) ) <-> ( ( ( A substr <. M , L >. ) e. Word V /\ ( B prefix ( N - L ) ) e. Word V ) /\ k e. ( 0..^ (♯ ` ( A substr <. M , L >. ) ) ) ) )
80	74, 78, 79	sylanbrc 417	. . . . . . . . 9 |- ( ( k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( ( A substr <. M , L >. ) e. Word V /\ ( B prefix ( N - L ) ) e. Word V /\ k e. ( 0..^ (♯ ` ( A substr <. M , L >. ) ) ) ) )
81		ccatval1 11285	. . . . . . . . 9 |- ( ( ( A substr <. M , L >. ) e. Word V /\ ( B prefix ( N - L ) ) e. Word V /\ k e. ( 0..^ (♯ ` ( A substr <. M , L >. ) ) ) ) -> ( ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ` k ) = ( ( A substr <. M , L >. ) ` k ) )
82	80, 81	syl 14	. . . . . . . 8 |- ( ( k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ` k ) = ( ( A substr <. M , L >. ) ` k ) )
83	72, 82	eqtr4d 2268	. . . . . . 7 |- ( ( k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ` k ) )
84	83	ex 115	. . . . . 6 |- ( k e. ( 0..^ ( L - M ) ) -> ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ` k ) ) )
85		simprl 531	. . . . . . . . 9 |- ( ( -. k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) )
86	68	anim2i 342	. . . . . . . . . 10 |- ( ( -. k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( -. k e. ( 0..^ ( L - M ) ) /\ k e. ( 0..^ ( N - M ) ) ) )
87	86	ancomd 267	. . . . . . . . 9 |- ( ( -. k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( k e. ( 0..^ ( N - M ) ) /\ -. k e. ( 0..^ ( L - M ) ) ) )
88	1	pfxccatin12lem2 11423	. . . . . . . . 9 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( k e. ( 0..^ ( N - M ) ) /\ -. k e. ( 0..^ ( L - M ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( B prefix ( N - L ) ) ` ( k - (♯ ` ( A substr <. M , L >. ) ) ) ) ) )
89	85, 87, 88	sylc 62	. . . . . . . 8 |- ( ( -. k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( B prefix ( N - L ) ) ` ( k - (♯ ` ( A substr <. M , L >. ) ) ) ) )
90	73	ad2antrl 490	. . . . . . . . . 10 |- ( ( -. k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( ( A substr <. M , L >. ) e. Word V /\ ( B prefix ( N - L ) ) e. Word V ) )
91		elfzuz 10355	. . . . . . . . . . . . . . . 16 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> N e. ( ZZ>= ` L ) )
92		eluzelz 9863	. . . . . . . . . . . . . . . . 17 |- ( N e. ( ZZ>= ` L ) -> N e. ZZ )
93		id 19	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( L e. NN0 /\ M e. NN0 /\ N e. ZZ ) -> ( L e. NN0 /\ M e. NN0 /\ N e. ZZ ) )
94	93	3expia 1232	. . . . . . . . . . . . . . . . . . . 20 |- ( ( L e. NN0 /\ M e. NN0 ) -> ( N e. ZZ -> ( L e. NN0 /\ M e. NN0 /\ N e. ZZ ) ) )
95	94	ancoms 268	. . . . . . . . . . . . . . . . . . 19 |- ( ( M e. NN0 /\ L e. NN0 ) -> ( N e. ZZ -> ( L e. NN0 /\ M e. NN0 /\ N e. ZZ ) ) )
96	95	3adant3 1044	. . . . . . . . . . . . . . . . . 18 |- ( ( M e. NN0 /\ L e. NN0 /\ M <_ L ) -> ( N e. ZZ -> ( L e. NN0 /\ M e. NN0 /\ N e. ZZ ) ) )
97	39, 96	sylbi 121	. . . . . . . . . . . . . . . . 17 |- ( M e. ( 0 ... L ) -> ( N e. ZZ -> ( L e. NN0 /\ M e. NN0 /\ N e. ZZ ) ) )
98	92, 97	syl5com 29	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` L ) -> ( M e. ( 0 ... L ) -> ( L e. NN0 /\ M e. NN0 /\ N e. ZZ ) ) )
99	91, 98	syl 14	. . . . . . . . . . . . . . 15 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> ( M e. ( 0 ... L ) -> ( L e. NN0 /\ M e. NN0 /\ N e. ZZ ) ) )
100	99	impcom 125	. . . . . . . . . . . . . 14 |- ( ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( L e. NN0 /\ M e. NN0 /\ N e. ZZ ) )
101	100	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( L e. NN0 /\ M e. NN0 /\ N e. ZZ ) )
102	101	ad2antrl 490	. . . . . . . . . . . 12 |- ( ( -. k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( L e. NN0 /\ M e. NN0 /\ N e. ZZ ) )
103		pfxccatin12lem4 11418	. . . . . . . . . . . 12 |- ( ( L e. NN0 /\ M e. NN0 /\ N e. ZZ ) -> ( ( k e. ( 0..^ ( N - M ) ) /\ -. k e. ( 0..^ ( L - M ) ) ) -> k e. ( ( L - M )..^ ( ( L - M ) + ( N - L ) ) ) ) )
104	102, 87, 103	sylc 62	. . . . . . . . . . 11 |- ( ( -. k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> k e. ( ( L - M )..^ ( ( L - M ) + ( N - L ) ) ) )
105	30, 38	oveq12d 6068	. . . . . . . . . . . 12 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( (♯ ` ( A substr <. M , L >. ) )..^ ( (♯ ` ( A substr <. M , L >. ) ) + (♯ ` ( B prefix ( N - L ) ) ) ) ) = ( ( L - M )..^ ( ( L - M ) + ( N - L ) ) ) )
106	105	ad2antrl 490	. . . . . . . . . . 11 |- ( ( -. k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( (♯ ` ( A substr <. M , L >. ) )..^ ( (♯ ` ( A substr <. M , L >. ) ) + (♯ ` ( B prefix ( N - L ) ) ) ) ) = ( ( L - M )..^ ( ( L - M ) + ( N - L ) ) ) )
107	104, 106	eleqtrrd 2312	. . . . . . . . . 10 |- ( ( -. k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> k e. ( (♯ ` ( A substr <. M , L >. ) )..^ ( (♯ ` ( A substr <. M , L >. ) ) + (♯ ` ( B prefix ( N - L ) ) ) ) ) )
108		df-3an 1007	. . . . . . . . . 10 |- ( ( ( A substr <. M , L >. ) e. Word V /\ ( B prefix ( N - L ) ) e. Word V /\ k e. ( (♯ ` ( A substr <. M , L >. ) )..^ ( (♯ ` ( A substr <. M , L >. ) ) + (♯ ` ( B prefix ( N - L ) ) ) ) ) ) <-> ( ( ( A substr <. M , L >. ) e. Word V /\ ( B prefix ( N - L ) ) e. Word V ) /\ k e. ( (♯ ` ( A substr <. M , L >. ) )..^ ( (♯ ` ( A substr <. M , L >. ) ) + (♯ ` ( B prefix ( N - L ) ) ) ) ) ) )
109	90, 107, 108	sylanbrc 417	. . . . . . . . 9 |- ( ( -. k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( ( A substr <. M , L >. ) e. Word V /\ ( B prefix ( N - L ) ) e. Word V /\ k e. ( (♯ ` ( A substr <. M , L >. ) )..^ ( (♯ ` ( A substr <. M , L >. ) ) + (♯ ` ( B prefix ( N - L ) ) ) ) ) ) )
110		ccatval2 11286	. . . . . . . . 9 |- ( ( ( A substr <. M , L >. ) e. Word V /\ ( B prefix ( N - L ) ) e. Word V /\ k e. ( (♯ ` ( A substr <. M , L >. ) )..^ ( (♯ ` ( A substr <. M , L >. ) ) + (♯ ` ( B prefix ( N - L ) ) ) ) ) ) -> ( ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ` k ) = ( ( B prefix ( N - L ) ) ` ( k - (♯ ` ( A substr <. M , L >. ) ) ) ) )
111	109, 110	syl 14	. . . . . . . 8 |- ( ( -. k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ` k ) = ( ( B prefix ( N - L ) ) ` ( k - (♯ ` ( A substr <. M , L >. ) ) ) ) )
112	89, 111	eqtr4d 2268	. . . . . . 7 |- ( ( -. k e. ( 0..^ ( L - M ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ` k ) )
113	112	ex 115	. . . . . 6 |- ( -. k e. ( 0..^ ( L - M ) ) -> ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ` k ) ) )
114	84, 113	jaoi 724	. . . . 5 |- ( ( k e. ( 0..^ ( L - M ) ) \/ -. k e. ( 0..^ ( L - M ) ) ) -> ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ` k ) ) )
115	66, 114	syl 14	. . . 4 |- (DECID k e. ( 0..^ ( L - M ) ) -> ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ` k ) ) )
116	65, 115	mpcom 36	. . 3 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ` k ) )
117	4, 58, 116	eqfnfvd 5778	. 2 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) )
118	117	ex 115	1 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2203   <.cop 3692   class class class wbr 4109   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   CCcc 8125   0cc0 8127   + caddc 8130   <_ cle 8309   - cmin 8444   NN0cn0 9496   ZZcz 9577   ZZ>=cuz 9853   ...cfz 10342  ..^cfzo 10476  ♯chash 11138  Word cword 11224   ++ cconcat 11278   substr csubstr 11337   prefix cpfx 11364

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-concat 11279  df-substr 11338  df-pfx 11365

This theorem is referenced by:  pfxccat3  11426  pfxccatpfx2  11429  pfxccatin12d  11437