Theorem pfxccatin12 11224

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 11221	. . . 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 11147	. . . 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 10182	. . . . . . 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 10181	. . . . . . 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 11141	. . . . . 6 |- ( ( A e. Word V /\ M e. ZZ /\ L e. ZZ ) -> ( A substr <. M , L >. ) e. Word V )
11	5, 7, 9, 10	syl3anc 1250	. . . . 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 10184	. . . . . . . 8 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> L <_ N )
14		elfzelz 10182	. . . . . . . . 9 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> N e. ZZ )
15		znn0sub 9473	. . . . . . . . 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 11169	. . . . . 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 11095	. . . . 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 11035	. . . . . . . . . . . 12 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
25		nn0fz0 10276	. . . . . . . . . . . 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 2294	. . . . . . . . . 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 11143	. . . . . . . . 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 1250	. . . . . . . 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 11035	. . . . . . . . . . . 12 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
32	31	nn0zd 9528	. . . . . . . . . . 11 |- ( B e. Word V -> (♯ ` B ) e. ZZ )
33		elfzmlbp 10289	. . . . . . . . . . 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 11176	. . . . . . . . . 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 5985	. . . . . . 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 10269	. . . . . . . . . . 11 |- ( M e. ( 0 ... L ) <-> ( M e. NN0 /\ L e. NN0 /\ M <_ L ) )
40		nn0cn 9340	. . . . . . . . . . . . . . . 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 9340	. . . . . . . . . . . . . . . 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 9412	. . . . . . . . . . . . . . . 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 1180	. . . . . . . . . . . . . 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 1020	. . . . . . . . . . 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 8345	. . . . . . . . 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 2241	. . . . . 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 5983	. . . . 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 5384	. . . 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 10304	. . . . . 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 9419	. . . . 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 9535	. . . . . 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 10310	. . . . 5 |- ( ( k e. ZZ /\ 0 e. ZZ /\ ( L - M ) e. ZZ ) -> DECID k e. ( 0..^ ( L - M ) ) )
65	60, 61, 63, 64	syl3anc 1250	. . . 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 838	. . . . 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 11223	. . . . . . . . 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 5983	. . . . . . . . . . . 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 2287	. . . . . . . . . 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 983	. . . . . . . . . 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 11091	. . . . . . . . 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 2243	. . . . . . 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 11222	. . . . . . . . 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 10178	. . . . . . . . . . . . . . . 16 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> N e. ( ZZ>= ` L ) )
92		eluzelz 9692	. . . . . . . . . . . . . . . . 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 1208	. . . . . . . . . . . . . . . . . . . 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 1020	. . . . . . . . . . . . . . . . . 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 11217	. . . . . . . . . . . 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 5985	. . . . . . . . . . . 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 2287	. . . . . . . . . 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 983	. . . . . . . . . 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 11092	. . . . . . . . 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 2243	. . . . . . 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 718	. . . . 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 5703	. 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 710  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2178   <.cop 3646   class class class wbr 4059   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   CCcc 7958   0cc0 7960   + caddc 7963   <_ cle 8143   - cmin 8278   NN0cn0 9330   ZZcz 9407   ZZ>=cuz 9683   ...cfz 10165  ..^cfzo 10299  ♯chash 10957  Word cword 11031   ++ cconcat 11084   substr csubstr 11136   prefix cpfx 11163

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-er 6643  df-en 6851  df-dom 6852  df-fin 6853  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166  df-fzo 10300  df-ihash 10958  df-word 11032  df-concat 11085  df-substr 11137  df-pfx 11164

This theorem is referenced by:  pfxccat3  11225  pfxccatpfx2  11228  pfxccatin12d  11236