Theorem pfxccatin12 11527

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 11524	. . . 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 11450	. . . 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 531	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> A e. Word V )
6		elfzelz 10445	. . . . . . 7 |- ( M e. ( 0 ... L ) -> M e. ZZ )
7	6	ad2antrl 494	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> M e. ZZ )
8		elfzel1 10444	. . . . . . 7 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> L e. ZZ )
9	8	ad2antll 495	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> L e. ZZ )
10		swrdclg 11444	. . . . . 6 |- ( ( A e. Word V /\ M e. ZZ /\ L e. ZZ ) -> ( A substr <. M , L >. ) e. Word V )
11	5, 7, 9, 10	syl3anc 1278	. . . . 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 533	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> B e. Word V )
13		elfzle1 10448	. . . . . . . 8 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> L <_ N )
14		elfzelz 10445	. . . . . . . . 9 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> N e. ZZ )
15		znn0sub 9721	. . . . . . . . 9 |- ( ( L e. ZZ /\ N e. ZZ ) -> ( L <_ N <-> ( N - L ) e. NN0 ) )
16	8, 14, 15	syl2anc 415	. . . . . . . 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 495	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( N - L ) e. NN0 )
19		pfxclg 11472	. . . . . 6 |- ( ( B e. Word V /\ ( N - L ) e. NN0 ) -> ( B prefix ( N - L ) ) e. Word V )
20	12, 18, 19	syl2anc 415	. . . . 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 11391	. . . . 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 415	. . . 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 535	. . . . . . . . 9 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> M e. ( 0 ... L ) )
24		lencl 11330	. . . . . . . . . . . 12 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
25		nn0fz0 10543	. . . . . . . . . . . 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 2325	. . . . . . . . . 10 |- ( A e. Word V -> L e. ( 0 ... (♯ ` A ) ) )
28	27	ad2antrr 492	. . . . . . . . 9 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> L e. ( 0 ... (♯ ` A ) ) )
29		swrdlen 11446	. . . . . . . . 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 1278	. . . . . . . 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 11330	. . . . . . . . . . . 12 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
32	31	nn0zd 9777	. . . . . . . . . . 11 |- ( B e. Word V -> (♯ ` B ) e. ZZ )
33		elfzmlbp 10556	. . . . . . . . . . 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 11479	. . . . . . . . . 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 512	. . . . . . . 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 6106	. . . . . . 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 10536	. . . . . . . . . . 11 |- ( M e. ( 0 ... L ) <-> ( M e. NN0 /\ L e. NN0 /\ M <_ L ) )
40		nn0cn 9584	. . . . . . . . . . . . . . . 16 |- ( L e. NN0 -> L e. CC )
41	40	ad2antll 495	. . . . . . . . . . . . . . 15 |- ( ( N e. ZZ /\ ( M e. NN0 /\ L e. NN0 ) ) -> L e. CC )
42		nn0cn 9584	. . . . . . . . . . . . . . . 16 |- ( M e. NN0 -> M e. CC )
43	42	ad2antrl 494	. . . . . . . . . . . . . . 15 |- ( ( N e. ZZ /\ ( M e. NN0 /\ L e. NN0 ) ) -> M e. CC )
44		zcn 9660	. . . . . . . . . . . . . . . 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 1208	. . . . . . . . . . . . . 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 1048	. . . . . . . . . . 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 8571	. . . . . . . . 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 2272	. . . . . 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 6104	. . . . 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 5475	. . . 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 10571	. . . . . 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 9667	. . . . 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 9784	. . . . . 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 10577	. . . . 5 |- ( ( k e. ZZ /\ 0 e. ZZ /\ ( L - M ) e. ZZ ) -> DECID k e. ( 0..^ ( L - M ) ) )
65	60, 61, 63, 64	syl3anc 1278	. . . 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 848	. . . . 5 |- (DECID k e. ( 0..^ ( L - M ) ) -> ( k e. ( 0..^ ( L - M ) ) \/ -. k e. ( 0..^ ( L - M ) ) ) )
67		simprl 535	. . . . . . . . 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 11526	. . . . . . . . 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 494	. . . . . . . . . 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 6104	. . . . . . . . . . . 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 494	. . . . . . . . . . 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 2318	. . . . . . . . . 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 1011	. . . . . . . . . 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 421	. . . . . . . . 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 11387	. . . . . . . . 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 2274	. . . . . . 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 535	. . . . . . . . 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 11525	. . . . . . . . 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 494	. . . . . . . . . 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 10441	. . . . . . . . . . . . . . . 16 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> N e. ( ZZ>= ` L ) )
92		eluzelz 9947	. . . . . . . . . . . . . . . . 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 1236	. . . . . . . . . . . . . . . . . . . 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 1048	. . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . 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 11520	. . . . . . . . . . . 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 6106	. . . . . . . . . . . 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 494	. . . . . . . . . . 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 2318	. . . . . . . . . 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 1011	. . . . . . . . . 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 421	. . . . . . . . 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 11388	. . . . . . . . 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 2274	. . . . . . 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 728	. . . . 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 5812	. 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 ) ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   <.cop 3712   class class class wbr 4131   Fn wfn 5375   ` cfv 5380  (class class class)co 6088   CCcc 8183   0cc0 8185   + caddc 8188   <_ cle 8367   - cmin 8504   NN0cn0 9574   ZZcz 9655   ZZ>=cuz 9937   ...cfz 10428  ..^cfzo 10566  ♯chash 11236  Word cword 11326   ++ cconcat 11380   substr csubstr 11439   prefix cpfx 11466

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-ihash 11237  df-word 11327  df-concat 11381  df-substr 11440  df-pfx 11467

This theorem is used by:  pfxccat3  11528  pfxccatpfx2  11531  pfxccatin12d  11539