Theorem swrdccatin2 11359

Description: The subword of a concatenation of two words within the second of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Revised by Alexander van der Vekens, 27-May-2018.)

Hypothesis

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

Assertion

Ref	Expression
swrdccatin2	|- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) ) )

Proof of Theorem swrdccatin2

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		swrdccatin2.l	. . . . . . . 8 |- L = (♯ ` A )
2		oveq1 6035	. . . . . . . . . 10 |- ( L = (♯ ` A ) -> ( L ... N ) = ( (♯ ` A ) ... N ) )
3	2	eleq2d 2301	. . . . . . . . 9 |- ( L = (♯ ` A ) -> ( M e. ( L ... N ) <-> M e. ( (♯ ` A ) ... N ) ) )
4		id 19	. . . . . . . . . . 11 |- ( L = (♯ ` A ) -> L = (♯ ` A ) )
5		oveq1 6035	. . . . . . . . . . 11 |- ( L = (♯ ` A ) -> ( L + (♯ ` B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
6	4, 5	oveq12d 6046	. . . . . . . . . 10 |- ( L = (♯ ` A ) -> ( L ... ( L + (♯ ` B ) ) ) = ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) )
7	6	eleq2d 2301	. . . . . . . . 9 |- ( L = (♯ ` A ) -> ( N e. ( L ... ( L + (♯ ` B ) ) ) <-> N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) ) )
8	3, 7	anbi12d 473	. . . . . . . 8 |- ( L = (♯ ` A ) -> ( ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) <-> ( M e. ( (♯ ` A ) ... N ) /\ N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) ) ) )
9	1, 8	ax-mp 5	. . . . . . 7 |- ( ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) <-> ( M e. ( (♯ ` A ) ... N ) /\ N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) ) )
10		lencl 11166	. . . . . . . . 9 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
11		elnn0uz 9838	. . . . . . . . . . . 12 |- ( (♯ ` A ) e. NN0 <-> (♯ ` A ) e. ( ZZ>= ` 0 ) )
12		fzss1 10343	. . . . . . . . . . . 12 |- ( (♯ ` A ) e. ( ZZ>= ` 0 ) -> ( (♯ ` A ) ... N ) C_ ( 0 ... N ) )
13	11, 12	sylbi 121	. . . . . . . . . . 11 |- ( (♯ ` A ) e. NN0 -> ( (♯ ` A ) ... N ) C_ ( 0 ... N ) )
14	13	sseld 3227	. . . . . . . . . 10 |- ( (♯ ` A ) e. NN0 -> ( M e. ( (♯ ` A ) ... N ) -> M e. ( 0 ... N ) ) )
15		fzss1 10343	. . . . . . . . . . . 12 |- ( (♯ ` A ) e. ( ZZ>= ` 0 ) -> ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) C_ ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) )
16	11, 15	sylbi 121	. . . . . . . . . . 11 |- ( (♯ ` A ) e. NN0 -> ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) C_ ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) )
17	16	sseld 3227	. . . . . . . . . 10 |- ( (♯ ` A ) e. NN0 -> ( N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) -> N e. ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) ) )
18	14, 17	anim12d 335	. . . . . . . . 9 |- ( (♯ ` A ) e. NN0 -> ( ( M e. ( (♯ ` A ) ... N ) /\ N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) ) ) )
19	10, 18	syl 14	. . . . . . . 8 |- ( A e. Word V -> ( ( M e. ( (♯ ` A ) ... N ) /\ N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) ) ) )
20	19	adantr 276	. . . . . . 7 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( (♯ ` A ) ... N ) /\ N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) ) ) )
21	9, 20	biimtrid 152	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) ) ) )
22	21	imp 124	. . . . 5 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) ) )
23		swrdccatfn 11354	. . . . 5 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) )
24	22, 23	syldan 282	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) )
25		fzmmmeqm 10338	. . . . . . 7 |- ( M e. ( L ... N ) -> ( ( N - L ) - ( M - L ) ) = ( N - M ) )
26	25	oveq2d 6044	. . . . . 6 |- ( M e. ( L ... N ) -> ( 0..^ ( ( N - L ) - ( M - L ) ) ) = ( 0..^ ( N - M ) ) )
27	26	fneq2d 5428	. . . . 5 |- ( M e. ( L ... N ) -> ( ( ( A ++ B ) substr <. M , N >. ) Fn ( 0..^ ( ( N - L ) - ( M - L ) ) ) <-> ( ( A ++ B ) substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) ) )
28	27	ad2antrl 490	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) Fn ( 0..^ ( ( N - L ) - ( M - L ) ) ) <-> ( ( A ++ B ) substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) ) )
29	24, 28	mpbird 167	. . 3 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) Fn ( 0..^ ( ( N - L ) - ( M - L ) ) ) )
30		simplr 529	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> B e. Word V )
31		elfzmlbm 10411	. . . . 5 |- ( M e. ( L ... N ) -> ( M - L ) e. ( 0 ... ( N - L ) ) )
32	31	ad2antrl 490	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( M - L ) e. ( 0 ... ( N - L ) ) )
33		lencl 11166	. . . . . . . 8 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
34	33	nn0zd 9644	. . . . . . 7 |- ( B e. Word V -> (♯ ` B ) e. ZZ )
35	34	adantl 277	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V ) -> (♯ ` B ) e. ZZ )
36		elfzmlbp 10412	. . . . . 6 |- ( ( (♯ ` B ) e. ZZ /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( N - L ) e. ( 0 ... (♯ ` B ) ) )
37	35, 36	sylan 283	. . . . 5 |- ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( N - L ) e. ( 0 ... (♯ ` B ) ) )
38	37	adantrl 478	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( N - L ) e. ( 0 ... (♯ ` B ) ) )
39		swrdvalfn 11286	. . . 4 |- ( ( B e. Word V /\ ( M - L ) e. ( 0 ... ( N - L ) ) /\ ( N - L ) e. ( 0 ... (♯ ` B ) ) ) -> ( B substr <. ( M - L ) , ( N - L ) >. ) Fn ( 0..^ ( ( N - L ) - ( M - L ) ) ) )
40	30, 32, 38, 39	syl3anc 1274	. . 3 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( B substr <. ( M - L ) , ( N - L ) >. ) Fn ( 0..^ ( ( N - L ) - ( M - L ) ) ) )
41		simpll 527	. . . . . . 7 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) ) -> ( A e. Word V /\ B e. Word V ) )
42		elfzelz 10305	. . . . . . . . . 10 |- ( M e. ( L ... N ) -> M e. ZZ )
43		zaddcl 9563	. . . . . . . . . . 11 |- ( ( k e. ZZ /\ M e. ZZ ) -> ( k + M ) e. ZZ )
44	43	expcom 116	. . . . . . . . . 10 |- ( M e. ZZ -> ( k e. ZZ -> ( k + M ) e. ZZ ) )
45	42, 44	syl 14	. . . . . . . . 9 |- ( M e. ( L ... N ) -> ( k e. ZZ -> ( k + M ) e. ZZ ) )
46	45	ad2antrl 490	. . . . . . . 8 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( k e. ZZ -> ( k + M ) e. ZZ ) )
47		elfzoelz 10427	. . . . . . . 8 |- ( k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) -> k e. ZZ )
48	46, 47	impel 280	. . . . . . 7 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) ) -> ( k + M ) e. ZZ )
49		df-3an 1007	. . . . . . 7 |- ( ( A e. Word V /\ B e. Word V /\ ( k + M ) e. ZZ ) <-> ( ( A e. Word V /\ B e. Word V ) /\ ( k + M ) e. ZZ ) )
50	41, 48, 49	sylanbrc 417	. . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) ) -> ( A e. Word V /\ B e. Word V /\ ( k + M ) e. ZZ ) )
51		ccatsymb 11228	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V /\ ( k + M ) e. ZZ ) -> ( ( A ++ B ) ` ( k + M ) ) = if ( ( k + M ) < (♯ ` A ) , ( A ` ( k + M ) ) , ( B ` ( ( k + M ) - (♯ ` A ) ) ) ) )
52	50, 51	syl 14	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) ) -> ( ( A ++ B ) ` ( k + M ) ) = if ( ( k + M ) < (♯ ` A ) , ( A ` ( k + M ) ) , ( B ` ( ( k + M ) - (♯ ` A ) ) ) ) )
53		elfz2 10295	. . . . . . . . 9 |- ( M e. ( L ... N ) <-> ( ( L e. ZZ /\ N e. ZZ /\ M e. ZZ ) /\ ( L <_ M /\ M <_ N ) ) )
54		zre 9527	. . . . . . . . . . . . . . . . 17 |- ( L e. ZZ -> L e. RR )
55		zre 9527	. . . . . . . . . . . . . . . . 17 |- ( M e. ZZ -> M e. RR )
56	54, 55	anim12i 338	. . . . . . . . . . . . . . . 16 |- ( ( L e. ZZ /\ M e. ZZ ) -> ( L e. RR /\ M e. RR ) )
57		elnn0z 9536	. . . . . . . . . . . . . . . . 17 |- ( k e. NN0 <-> ( k e. ZZ /\ 0 <_ k ) )
58		zre 9527	. . . . . . . . . . . . . . . . . 18 |- ( k e. ZZ -> k e. RR )
59		0re 8222	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 0 e. RR
60	59	jctl 314	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( L e. RR -> ( 0 e. RR /\ L e. RR ) )
61	60	ad2antrl 490	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( k e. RR /\ ( L e. RR /\ M e. RR ) ) -> ( 0 e. RR /\ L e. RR ) )
62		id 19	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( k e. RR /\ M e. RR ) -> ( k e. RR /\ M e. RR ) )
63	62	adantrl 478	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( k e. RR /\ ( L e. RR /\ M e. RR ) ) -> ( k e. RR /\ M e. RR ) )
64		le2add 8666	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( 0 e. RR /\ L e. RR ) /\ ( k e. RR /\ M e. RR ) ) -> ( ( 0 <_ k /\ L <_ M ) -> ( 0 + L ) <_ ( k + M ) ) )
65	61, 63, 64	syl2anc 411	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( k e. RR /\ ( L e. RR /\ M e. RR ) ) -> ( ( 0 <_ k /\ L <_ M ) -> ( 0 + L ) <_ ( k + M ) ) )
66		recn 8208	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( L e. RR -> L e. CC )
67	66	addlidd 8371	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( L e. RR -> ( 0 + L ) = L )
68	67	ad2antrl 490	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( k e. RR /\ ( L e. RR /\ M e. RR ) ) -> ( 0 + L ) = L )
69	68	breq1d 4103	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( k e. RR /\ ( L e. RR /\ M e. RR ) ) -> ( ( 0 + L ) <_ ( k + M ) <-> L <_ ( k + M ) ) )
70	65, 69	sylibd 149	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( k e. RR /\ ( L e. RR /\ M e. RR ) ) -> ( ( 0 <_ k /\ L <_ M ) -> L <_ ( k + M ) ) )
71		simprl 531	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( k e. RR /\ ( L e. RR /\ M e. RR ) ) -> L e. RR )
72		readdcl 8201	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( k e. RR /\ M e. RR ) -> ( k + M ) e. RR )
73	72	adantrl 478	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( k e. RR /\ ( L e. RR /\ M e. RR ) ) -> ( k + M ) e. RR )
74	71, 73	lenltd 8339	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( k e. RR /\ ( L e. RR /\ M e. RR ) ) -> ( L <_ ( k + M ) <-> -. ( k + M ) < L ) )
75	70, 74	sylibd 149	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( k e. RR /\ ( L e. RR /\ M e. RR ) ) -> ( ( 0 <_ k /\ L <_ M ) -> -. ( k + M ) < L ) )
76	75	expd 258	. . . . . . . . . . . . . . . . . . . 20 |- ( ( k e. RR /\ ( L e. RR /\ M e. RR ) ) -> ( 0 <_ k -> ( L <_ M -> -. ( k + M ) < L ) ) )
77	76	com12 30	. . . . . . . . . . . . . . . . . . 19 |- ( 0 <_ k -> ( ( k e. RR /\ ( L e. RR /\ M e. RR ) ) -> ( L <_ M -> -. ( k + M ) < L ) ) )
78	77	expd 258	. . . . . . . . . . . . . . . . . 18 |- ( 0 <_ k -> ( k e. RR -> ( ( L e. RR /\ M e. RR ) -> ( L <_ M -> -. ( k + M ) < L ) ) ) )
79	58, 78	mpan9 281	. . . . . . . . . . . . . . . . 17 |- ( ( k e. ZZ /\ 0 <_ k ) -> ( ( L e. RR /\ M e. RR ) -> ( L <_ M -> -. ( k + M ) < L ) ) )
80	57, 79	sylbi 121	. . . . . . . . . . . . . . . 16 |- ( k e. NN0 -> ( ( L e. RR /\ M e. RR ) -> ( L <_ M -> -. ( k + M ) < L ) ) )
81	56, 80	mpan9 281	. . . . . . . . . . . . . . 15 |- ( ( ( L e. ZZ /\ M e. ZZ ) /\ k e. NN0 ) -> ( L <_ M -> -. ( k + M ) < L ) )
82	1	breq2i 4101	. . . . . . . . . . . . . . . 16 |- ( ( k + M ) < L <-> ( k + M ) < (♯ ` A ) )
83	82	notbii 674	. . . . . . . . . . . . . . 15 |- ( -. ( k + M ) < L <-> -. ( k + M ) < (♯ ` A ) )
84	81, 83	imbitrdi 161	. . . . . . . . . . . . . 14 |- ( ( ( L e. ZZ /\ M e. ZZ ) /\ k e. NN0 ) -> ( L <_ M -> -. ( k + M ) < (♯ ` A ) ) )
85	84	ex 115	. . . . . . . . . . . . 13 |- ( ( L e. ZZ /\ M e. ZZ ) -> ( k e. NN0 -> ( L <_ M -> -. ( k + M ) < (♯ ` A ) ) ) )
86	85	com23 78	. . . . . . . . . . . 12 |- ( ( L e. ZZ /\ M e. ZZ ) -> ( L <_ M -> ( k e. NN0 -> -. ( k + M ) < (♯ ` A ) ) ) )
87	86	3adant2 1043	. . . . . . . . . . 11 |- ( ( L e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( L <_ M -> ( k e. NN0 -> -. ( k + M ) < (♯ ` A ) ) ) )
88	87	imp 124	. . . . . . . . . 10 |- ( ( ( L e. ZZ /\ N e. ZZ /\ M e. ZZ ) /\ L <_ M ) -> ( k e. NN0 -> -. ( k + M ) < (♯ ` A ) ) )
89	88	adantrr 479	. . . . . . . . 9 |- ( ( ( L e. ZZ /\ N e. ZZ /\ M e. ZZ ) /\ ( L <_ M /\ M <_ N ) ) -> ( k e. NN0 -> -. ( k + M ) < (♯ ` A ) ) )
90	53, 89	sylbi 121	. . . . . . . 8 |- ( M e. ( L ... N ) -> ( k e. NN0 -> -. ( k + M ) < (♯ ` A ) ) )
91	90	ad2antrl 490	. . . . . . 7 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( k e. NN0 -> -. ( k + M ) < (♯ ` A ) ) )
92		elfzonn0 10471	. . . . . . 7 |- ( k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) -> k e. NN0 )
93	91, 92	impel 280	. . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) ) -> -. ( k + M ) < (♯ ` A ) )
94	93	iffalsed 3619	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) ) -> if ( ( k + M ) < (♯ ` A ) , ( A ` ( k + M ) ) , ( B ` ( ( k + M ) - (♯ ` A ) ) ) ) = ( B ` ( ( k + M ) - (♯ ` A ) ) ) )
95		zcn 9528	. . . . . . . . . . . . . . . 16 |- ( k e. ZZ -> k e. CC )
96	95	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( L e. ZZ /\ M e. ZZ ) /\ k e. ZZ ) -> k e. CC )
97		zcn 9528	. . . . . . . . . . . . . . . 16 |- ( M e. ZZ -> M e. CC )
98	97	ad2antlr 489	. . . . . . . . . . . . . . 15 |- ( ( ( L e. ZZ /\ M e. ZZ ) /\ k e. ZZ ) -> M e. CC )
99		zcn 9528	. . . . . . . . . . . . . . . 16 |- ( L e. ZZ -> L e. CC )
100	99	ad2antrr 488	. . . . . . . . . . . . . . 15 |- ( ( ( L e. ZZ /\ M e. ZZ ) /\ k e. ZZ ) -> L e. CC )
101	96, 98, 100	addsubassd 8552	. . . . . . . . . . . . . 14 |- ( ( ( L e. ZZ /\ M e. ZZ ) /\ k e. ZZ ) -> ( ( k + M ) - L ) = ( k + ( M - L ) ) )
102		oveq2 6036	. . . . . . . . . . . . . . 15 |- ( L = (♯ ` A ) -> ( ( k + M ) - L ) = ( ( k + M ) - (♯ ` A ) ) )
103	102	eqeq1d 2240	. . . . . . . . . . . . . 14 |- ( L = (♯ ` A ) -> ( ( ( k + M ) - L ) = ( k + ( M - L ) ) <-> ( ( k + M ) - (♯ ` A ) ) = ( k + ( M - L ) ) ) )
104	101, 103	imbitrid 154	. . . . . . . . . . . . 13 |- ( L = (♯ ` A ) -> ( ( ( L e. ZZ /\ M e. ZZ ) /\ k e. ZZ ) -> ( ( k + M ) - (♯ ` A ) ) = ( k + ( M - L ) ) ) )
105	1, 104	ax-mp 5	. . . . . . . . . . . 12 |- ( ( ( L e. ZZ /\ M e. ZZ ) /\ k e. ZZ ) -> ( ( k + M ) - (♯ ` A ) ) = ( k + ( M - L ) ) )
106	105	ex 115	. . . . . . . . . . 11 |- ( ( L e. ZZ /\ M e. ZZ ) -> ( k e. ZZ -> ( ( k + M ) - (♯ ` A ) ) = ( k + ( M - L ) ) ) )
107	106	3adant2 1043	. . . . . . . . . 10 |- ( ( L e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( k e. ZZ -> ( ( k + M ) - (♯ ` A ) ) = ( k + ( M - L ) ) ) )
108	107	adantr 276	. . . . . . . . 9 |- ( ( ( L e. ZZ /\ N e. ZZ /\ M e. ZZ ) /\ ( L <_ M /\ M <_ N ) ) -> ( k e. ZZ -> ( ( k + M ) - (♯ ` A ) ) = ( k + ( M - L ) ) ) )
109	53, 108	sylbi 121	. . . . . . . 8 |- ( M e. ( L ... N ) -> ( k e. ZZ -> ( ( k + M ) - (♯ ` A ) ) = ( k + ( M - L ) ) ) )
110	109	ad2antrl 490	. . . . . . 7 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( k e. ZZ -> ( ( k + M ) - (♯ ` A ) ) = ( k + ( M - L ) ) ) )
111	110, 47	impel 280	. . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) ) -> ( ( k + M ) - (♯ ` A ) ) = ( k + ( M - L ) ) )
112	111	fveq2d 5652	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) ) -> ( B ` ( ( k + M ) - (♯ ` A ) ) ) = ( B ` ( k + ( M - L ) ) ) )
113	52, 94, 112	3eqtrd 2268	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) ) -> ( ( A ++ B ) ` ( k + M ) ) = ( B ` ( k + ( M - L ) ) ) )
114		ccatcl 11219	. . . . . . 7 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) e. Word V )
115	114	adantr 276	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( A ++ B ) e. Word V )
116	11	biimpi 120	. . . . . . . . . 10 |- ( (♯ ` A ) e. NN0 -> (♯ ` A ) e. ( ZZ>= ` 0 ) )
117	1, 116	eqeltrid 2318	. . . . . . . . 9 |- ( (♯ ` A ) e. NN0 -> L e. ( ZZ>= ` 0 ) )
118		fzss1 10343	. . . . . . . . 9 |- ( L e. ( ZZ>= ` 0 ) -> ( L ... N ) C_ ( 0 ... N ) )
119	10, 117, 118	3syl 17	. . . . . . . 8 |- ( A e. Word V -> ( L ... N ) C_ ( 0 ... N ) )
120	119	sselda 3228	. . . . . . 7 |- ( ( A e. Word V /\ M e. ( L ... N ) ) -> M e. ( 0 ... N ) )
121	120	ad2ant2r 509	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> M e. ( 0 ... N ) )
122	1, 7	ax-mp 5	. . . . . . . . 9 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) <-> N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) )
123	10, 116, 15	3syl 17	. . . . . . . . . . . . . 14 |- ( A e. Word V -> ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) C_ ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) )
124	123	adantr 276	. . . . . . . . . . . . 13 |- ( ( A e. Word V /\ B e. Word V ) -> ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) C_ ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) )
125	124	sseld 3227	. . . . . . . . . . . 12 |- ( ( A e. Word V /\ B e. Word V ) -> ( N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) -> N e. ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) ) )
126	125	impcom 125	. . . . . . . . . . 11 |- ( ( N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) /\ ( A e. Word V /\ B e. Word V ) ) -> N e. ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) )
127		ccatlen 11221	. . . . . . . . . . . . . 14 |- ( ( A e. Word V /\ B e. Word V ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
128	127	oveq2d 6044	. . . . . . . . . . . . 13 |- ( ( A e. Word V /\ B e. Word V ) -> ( 0 ... (♯ ` ( A ++ B ) ) ) = ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) )
129	128	eleq2d 2301	. . . . . . . . . . . 12 |- ( ( A e. Word V /\ B e. Word V ) -> ( N e. ( 0 ... (♯ ` ( A ++ B ) ) ) <-> N e. ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) ) )
130	129	adantl 277	. . . . . . . . . . 11 |- ( ( N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) /\ ( A e. Word V /\ B e. Word V ) ) -> ( N e. ( 0 ... (♯ ` ( A ++ B ) ) ) <-> N e. ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) ) )
131	126, 130	mpbird 167	. . . . . . . . . 10 |- ( ( N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) /\ ( A e. Word V /\ B e. Word V ) ) -> N e. ( 0 ... (♯ ` ( A ++ B ) ) ) )
132	131	ex 115	. . . . . . . . 9 |- ( N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) -> ( ( A e. Word V /\ B e. Word V ) -> N e. ( 0 ... (♯ ` ( A ++ B ) ) ) ) )
133	122, 132	sylbi 121	. . . . . . . 8 |- ( N e. ( L ... ( L + (♯ ` B ) ) ) -> ( ( A e. Word V /\ B e. Word V ) -> N e. ( 0 ... (♯ ` ( A ++ B ) ) ) ) )
134	133	impcom 125	. . . . . . 7 |- ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> N e. ( 0 ... (♯ ` ( A ++ B ) ) ) )
135	134	adantrl 478	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> N e. ( 0 ... (♯ ` ( A ++ B ) ) ) )
136	115, 121, 135	3jca 1204	. . . . 5 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( A ++ B ) e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` ( A ++ B ) ) ) ) )
137	26	eleq2d 2301	. . . . . . 7 |- ( M e. ( L ... N ) -> ( k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) <-> k e. ( 0..^ ( N - M ) ) ) )
138	137	ad2antrl 490	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) <-> k e. ( 0..^ ( N - M ) ) ) )
139	138	biimpa 296	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) ) -> k e. ( 0..^ ( N - M ) ) )
140		swrdfv 11283	. . . . 5 |- ( ( ( ( A ++ B ) e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` ( A ++ B ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( A ++ B ) ` ( k + M ) ) )
141	136, 139, 140	syl2an2r 599	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( A ++ B ) ` ( k + M ) ) )
142	34, 36	sylan 283	. . . . . . 7 |- ( ( B e. Word V /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( N - L ) e. ( 0 ... (♯ ` B ) ) )
143	142	ad2ant2l 508	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( N - L ) e. ( 0 ... (♯ ` B ) ) )
144	30, 32, 143	3jca 1204	. . . . 5 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( B e. Word V /\ ( M - L ) e. ( 0 ... ( N - L ) ) /\ ( N - L ) e. ( 0 ... (♯ ` B ) ) ) )
145		swrdfv 11283	. . . . 5 |- ( ( ( B e. Word V /\ ( M - L ) e. ( 0 ... ( N - L ) ) /\ ( N - L ) e. ( 0 ... (♯ ` B ) ) ) /\ k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) ) -> ( ( B substr <. ( M - L ) , ( N - L ) >. ) ` k ) = ( B ` ( k + ( M - L ) ) ) )
146	144, 145	sylan 283	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) ) -> ( ( B substr <. ( M - L ) , ( N - L ) >. ) ` k ) = ( B ` ( k + ( M - L ) ) ) )
147	113, 141, 146	3eqtr4d 2274	. . 3 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) /\ k e. ( 0..^ ( ( N - L ) - ( M - L ) ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( B substr <. ( M - L ) , ( N - L ) >. ) ` k ) )
148	29, 40, 147	eqfnfvd 5756	. 2 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) )
149	148	ex 115	1 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   C_ wss 3201   ifcif 3607   <.cop 3676   class class class wbr 4093   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   CCcc 8073   RRcr 8074   0cc0 8075   + caddc 8078   < clt 8256   <_ cle 8257   - cmin 8392   NN0cn0 9444   ZZcz 9523   ZZ>=cuz 9799   ...cfz 10288  ..^cfzo 10422  ♯chash 11083  Word cword 11162   ++ cconcat 11216   substr csubstr 11275

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-inn 9186  df-n0 9445  df-z 9524  df-uz 9800  df-fz 10289  df-fzo 10423  df-ihash 11084  df-word 11163  df-concat 11217  df-substr 11276

This theorem is referenced by:  pfxccat3  11364  swrdccatin2d  11374