Theorem pfxccatin12 11260

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 11257	. . . 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 11183	. . . 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 10217	. . . . . . 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 10216	. . . . . . 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 11177	. . . . . 6 |- ( ( A e. Word V /\ M e. ZZ /\ L e. ZZ ) -> ( A substr <. M , L >. ) e. Word V )
11	5, 7, 9, 10	syl3anc 1271	. . . . 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 528	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> B e. Word V )
13		elfzle1 10219	. . . . . . . 8 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> L <_ N )
14		elfzelz 10217	. . . . . . . . 9 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> N e. ZZ )
15		znn0sub 9508	. . . . . . . . 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 11205	. . . . . 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 11131	. . . . 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 529	. . . . . . . . 9 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> M e. ( 0 ... L ) )
24		lencl 11070	. . . . . . . . . . . 12 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
25		nn0fz0 10311	. . . . . . . . . . . 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 2316	. . . . . . . . . 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 11179	. . . . . . . . 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 1271	. . . . . . . 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 11070	. . . . . . . . . . . 12 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
32	31	nn0zd 9563	. . . . . . . . . . 11 |- ( B e. Word V -> (♯ ` B ) e. ZZ )
33		elfzmlbp 10324	. . . . . . . . . . 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 11212	. . . . . . . . . 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 6018	. . . . . . 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 10304	. . . . . . . . . . 11 |- ( M e. ( 0 ... L ) <-> ( M e. NN0 /\ L e. NN0 /\ M <_ L ) )
40		nn0cn 9375	. . . . . . . . . . . . . . . 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 9375	. . . . . . . . . . . . . . . 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 9447	. . . . . . . . . . . . . . . 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 1201	. . . . . . . . . . . . . 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 1041	. . . . . . . . . . 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 8380	. . . . . . . . 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 2263	. . . . . 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 6016	. . . . 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 5411	. . . 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 10339	. . . . . 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 9454	. . . . 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 9570	. . . . . 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 10345	. . . . 5 |- ( ( k e. ZZ /\ 0 e. ZZ /\ ( L - M ) e. ZZ ) -> DECID k e. ( 0..^ ( L - M ) ) )
65	60, 61, 63, 64	syl3anc 1271	. . . 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 841	. . . . 5 |- (DECID k e. ( 0..^ ( L - M ) ) -> ( k e. ( 0..^ ( L - M ) ) \/ -. k e. ( 0..^ ( L - M ) ) ) )
67		simprl 529	. . . . . . . . 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 11259	. . . . . . . . 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 6016	. . . . . . . . . . . 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 2309	. . . . . . . . . 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 1004	. . . . . . . . . 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 11127	. . . . . . . . 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 2265	. . . . . . 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 529	. . . . . . . . 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 11258	. . . . . . . . 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 10213	. . . . . . . . . . . . . . . 16 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> N e. ( ZZ>= ` L ) )
92		eluzelz 9727	. . . . . . . . . . . . . . . . 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 1229	. . . . . . . . . . . . . . . . . . . 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 1041	. . . . . . . . . . . . . . . . . 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 11253	. . . . . . . . . . . 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 6018	. . . . . . . . . . . 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 2309	. . . . . . . . . 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 1004	. . . . . . . . . 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 11128	. . . . . . . . 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 2265	. . . . . . 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 721	. . . . 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 5734	. 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 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   <.cop 3669   class class class wbr 4082   Fn wfn 5312   ` cfv 5317  (class class class)co 6000   CCcc 7993   0cc0 7995   + caddc 7998   <_ cle 8178   - cmin 8313   NN0cn0 9365   ZZcz 9442   ZZ>=cuz 9718   ...cfz 10200  ..^cfzo 10334  ♯chash 10992  Word cword 11066   ++ cconcat 11120   substr csubstr 11172   prefix cpfx 11199

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111

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 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-1o 6560  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-fzo 10335  df-ihash 10993  df-word 11067  df-concat 11121  df-substr 11173  df-pfx 11200

This theorem is referenced by:  pfxccat3  11261  pfxccatpfx2  11264  pfxccatin12d  11272