Theorem swrdccatin2 11256

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 6007	. . . . . . . . . 10 |- ( L = (♯ ` A ) -> ( L ... N ) = ( (♯ ` A ) ... N ) )
3	2	eleq2d 2299	. . . . . . . . 9 |- ( L = (♯ ` A ) -> ( M e. ( L ... N ) <-> M e. ( (♯ ` A ) ... N ) ) )
4		id 19	. . . . . . . . . . 11 |- ( L = (♯ ` A ) -> L = (♯ ` A ) )
5		oveq1 6007	. . . . . . . . . . 11 |- ( L = (♯ ` A ) -> ( L + (♯ ` B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
6	4, 5	oveq12d 6018	. . . . . . . . . 10 |- ( L = (♯ ` A ) -> ( L ... ( L + (♯ ` B ) ) ) = ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) )
7	6	eleq2d 2299	. . . . . . . . 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 11070	. . . . . . . . 9 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
11		elnn0uz 9756	. . . . . . . . . . . 12 |- ( (♯ ` A ) e. NN0 <-> (♯ ` A ) e. ( ZZ>= ` 0 ) )
12		fzss1 10255	. . . . . . . . . . . 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 3223	. . . . . . . . . 10 |- ( (♯ ` A ) e. NN0 -> ( M e. ( (♯ ` A ) ... N ) -> M e. ( 0 ... N ) ) )
15		fzss1 10255	. . . . . . . . . . . 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 3223	. . . . . . . . . 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 11251	. . . . 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 10250	. . . . . . 7 |- ( M e. ( L ... N ) -> ( ( N - L ) - ( M - L ) ) = ( N - M ) )
26	25	oveq2d 6016	. . . . . 6 |- ( M e. ( L ... N ) -> ( 0..^ ( ( N - L ) - ( M - L ) ) ) = ( 0..^ ( N - M ) ) )
27	26	fneq2d 5411	. . . . 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 528	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> B e. Word V )
31		elfzmlbm 10323	. . . . 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 11070	. . . . . . . 8 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
34	33	nn0zd 9563	. . . . . . 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 10324	. . . . . 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 11183	. . . 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 1271	. . 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 10217	. . . . . . . . . 10 |- ( M e. ( L ... N ) -> M e. ZZ )
43		zaddcl 9482	. . . . . . . . . . 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 10339	. . . . . . . 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 1004	. . . . . . 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 11132	. . . . . 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 10207	. . . . . . . . 9 |- ( M e. ( L ... N ) <-> ( ( L e. ZZ /\ N e. ZZ /\ M e. ZZ ) /\ ( L <_ M /\ M <_ N ) ) )
54		zre 9446	. . . . . . . . . . . . . . . . 17 |- ( L e. ZZ -> L e. RR )
55		zre 9446	. . . . . . . . . . . . . . . . 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 9455	. . . . . . . . . . . . . . . . 17 |- ( k e. NN0 <-> ( k e. ZZ /\ 0 <_ k ) )
58		zre 9446	. . . . . . . . . . . . . . . . . 18 |- ( k e. ZZ -> k e. RR )
59		0re 8142	. . . . . . . . . . . . . . . . . . . . . . . . . 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 8587	. . . . . . . . . . . . . . . . . . . . . . . 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 8128	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( L e. RR -> L e. CC )
67	66	addlidd 8292	. . . . . . . . . . . . . . . . . . . . . . . . 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 4092	. . . . . . . . . . . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( k e. RR /\ ( L e. RR /\ M e. RR ) ) -> L e. RR )
72		readdcl 8121	. . . . . . . . . . . . . . . . . . . . . . . 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 8260	. . . . . . . . . . . . . . . . . . . . . 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 4090	. . . . . . . . . . . . . . . 16 |- ( ( k + M ) < L <-> ( k + M ) < (♯ ` A ) )
83	82	notbii 672	. . . . . . . . . . . . . . 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 1040	. . . . . . . . . . 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 10382	. . . . . . 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 3612	. . . . 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 9447	. . . . . . . . . . . . . . . 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 9447	. . . . . . . . . . . . . . . 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 9447	. . . . . . . . . . . . . . . 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 8473	. . . . . . . . . . . . . 14 |- ( ( ( L e. ZZ /\ M e. ZZ ) /\ k e. ZZ ) -> ( ( k + M ) - L ) = ( k + ( M - L ) ) )
102		oveq2 6008	. . . . . . . . . . . . . . 15 |- ( L = (♯ ` A ) -> ( ( k + M ) - L ) = ( ( k + M ) - (♯ ` A ) ) )
103	102	eqeq1d 2238	. . . . . . . . . . . . . 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 1040	. . . . . . . . . 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 5630	. . . . 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 2266	. . . 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 11123	. . . . . . 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 2316	. . . . . . . . 9 |- ( (♯ ` A ) e. NN0 -> L e. ( ZZ>= ` 0 ) )
118		fzss1 10255	. . . . . . . . 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 3224	. . . . . . 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 3223	. . . . . . . . . . . 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 11125	. . . . . . . . . . . . . 14 |- ( ( A e. Word V /\ B e. Word V ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
128	127	oveq2d 6016	. . . . . . . . . . . . 13 |- ( ( A e. Word V /\ B e. Word V ) -> ( 0 ... (♯ ` ( A ++ B ) ) ) = ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) )
129	128	eleq2d 2299	. . . . . . . . . . . 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 1201	. . . . 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 2299	. . . . . . 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 11180	. . . . 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 597	. . . 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 1201	. . . . 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 11180	. . . . 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 2272	. . 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 5734	. 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 1002   = wceq 1395   e. wcel 2200   C_ wss 3197   ifcif 3602   <.cop 3669   class class class wbr 4082   Fn wfn 5312   ` cfv 5317  (class class class)co 6000   CCcc 7993   RRcr 7994   0cc0 7995   + caddc 7998   < clt 8177   <_ 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

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

This theorem is referenced by:  pfxccat3  11261  swrdccatin2d  11271