Theorem swrdccatin1 11305

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

Assertion

Ref	Expression
swrdccatin1	|- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) )

Proof of Theorem swrdccatin1

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6025	. . . . . 6 |- ( (♯ ` A ) = 0 -> ( 0 ... (♯ ` A ) ) = ( 0 ... 0 ) )
2	1	eleq2d 2301	. . . . 5 |- ( (♯ ` A ) = 0 -> ( N e. ( 0 ... (♯ ` A ) ) <-> N e. ( 0 ... 0 ) ) )
3	2	adantl 277	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) = 0 ) -> ( N e. ( 0 ... (♯ ` A ) ) <-> N e. ( 0 ... 0 ) ) )
4		elfz1eq 10269	. . . . . . 7 |- ( N e. ( 0 ... 0 ) -> N = 0 )
5		elfz1eq 10269	. . . . . . . . . . 11 |- ( M e. ( 0 ... 0 ) -> M = 0 )
6		ccatcl 11169	. . . . . . . . . . . . . . 15 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) e. Word V )
7		0z 9489	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
8		swrd00g 11229	. . . . . . . . . . . . . . 15 |- ( ( ( A ++ B ) e. Word V /\ 0 e. ZZ ) -> ( ( A ++ B ) substr <. 0 , 0 >. ) = (/) )
9	6, 7, 8	sylancl 413	. . . . . . . . . . . . . 14 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( A ++ B ) substr <. 0 , 0 >. ) = (/) )
10		simpl 109	. . . . . . . . . . . . . . 15 |- ( ( A e. Word V /\ B e. Word V ) -> A e. Word V )
11		swrd00g 11229	. . . . . . . . . . . . . . 15 |- ( ( A e. Word V /\ 0 e. ZZ ) -> ( A substr <. 0 , 0 >. ) = (/) )
12	10, 7, 11	sylancl 413	. . . . . . . . . . . . . 14 |- ( ( A e. Word V /\ B e. Word V ) -> ( A substr <. 0 , 0 >. ) = (/) )
13	9, 12	eqtr4d 2267	. . . . . . . . . . . . 13 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( A ++ B ) substr <. 0 , 0 >. ) = ( A substr <. 0 , 0 >. ) )
14	13	adantr 276	. . . . . . . . . . . 12 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M = 0 ) -> ( ( A ++ B ) substr <. 0 , 0 >. ) = ( A substr <. 0 , 0 >. ) )
15		opeq1 3862	. . . . . . . . . . . . . 14 |- ( M = 0 -> <. M , 0 >. = <. 0 , 0 >. )
16	15	oveq2d 6033	. . . . . . . . . . . . 13 |- ( M = 0 -> ( ( A ++ B ) substr <. M , 0 >. ) = ( ( A ++ B ) substr <. 0 , 0 >. ) )
17	16	adantl 277	. . . . . . . . . . . 12 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M = 0 ) -> ( ( A ++ B ) substr <. M , 0 >. ) = ( ( A ++ B ) substr <. 0 , 0 >. ) )
18	15	oveq2d 6033	. . . . . . . . . . . . 13 |- ( M = 0 -> ( A substr <. M , 0 >. ) = ( A substr <. 0 , 0 >. ) )
19	18	adantl 277	. . . . . . . . . . . 12 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M = 0 ) -> ( A substr <. M , 0 >. ) = ( A substr <. 0 , 0 >. ) )
20	14, 17, 19	3eqtr4d 2274	. . . . . . . . . . 11 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M = 0 ) -> ( ( A ++ B ) substr <. M , 0 >. ) = ( A substr <. M , 0 >. ) )
21	5, 20	sylan2 286	. . . . . . . . . 10 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... 0 ) ) -> ( ( A ++ B ) substr <. M , 0 >. ) = ( A substr <. M , 0 >. ) )
22	21	ex 115	. . . . . . . . 9 |- ( ( A e. Word V /\ B e. Word V ) -> ( M e. ( 0 ... 0 ) -> ( ( A ++ B ) substr <. M , 0 >. ) = ( A substr <. M , 0 >. ) ) )
23	22	adantr 276	. . . . . . . 8 |- ( ( ( A e. Word V /\ B e. Word V ) /\ N = 0 ) -> ( M e. ( 0 ... 0 ) -> ( ( A ++ B ) substr <. M , 0 >. ) = ( A substr <. M , 0 >. ) ) )
24		oveq2 6025	. . . . . . . . . . 11 |- ( N = 0 -> ( 0 ... N ) = ( 0 ... 0 ) )
25	24	eleq2d 2301	. . . . . . . . . 10 |- ( N = 0 -> ( M e. ( 0 ... N ) <-> M e. ( 0 ... 0 ) ) )
26		opeq2 3863	. . . . . . . . . . . 12 |- ( N = 0 -> <. M , N >. = <. M , 0 >. )
27	26	oveq2d 6033	. . . . . . . . . . 11 |- ( N = 0 -> ( ( A ++ B ) substr <. M , N >. ) = ( ( A ++ B ) substr <. M , 0 >. ) )
28	26	oveq2d 6033	. . . . . . . . . . 11 |- ( N = 0 -> ( A substr <. M , N >. ) = ( A substr <. M , 0 >. ) )
29	27, 28	eqeq12d 2246	. . . . . . . . . 10 |- ( N = 0 -> ( ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) <-> ( ( A ++ B ) substr <. M , 0 >. ) = ( A substr <. M , 0 >. ) ) )
30	25, 29	imbi12d 234	. . . . . . . . 9 |- ( N = 0 -> ( ( M e. ( 0 ... N ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) <-> ( M e. ( 0 ... 0 ) -> ( ( A ++ B ) substr <. M , 0 >. ) = ( A substr <. M , 0 >. ) ) ) )
31	30	adantl 277	. . . . . . . 8 |- ( ( ( A e. Word V /\ B e. Word V ) /\ N = 0 ) -> ( ( M e. ( 0 ... N ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) <-> ( M e. ( 0 ... 0 ) -> ( ( A ++ B ) substr <. M , 0 >. ) = ( A substr <. M , 0 >. ) ) ) )
32	23, 31	mpbird 167	. . . . . . 7 |- ( ( ( A e. Word V /\ B e. Word V ) /\ N = 0 ) -> ( M e. ( 0 ... N ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) )
33	4, 32	sylan2 286	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... 0 ) ) -> ( M e. ( 0 ... N ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) )
34	33	ex 115	. . . . 5 |- ( ( A e. Word V /\ B e. Word V ) -> ( N e. ( 0 ... 0 ) -> ( M e. ( 0 ... N ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) ) )
35	34	adantr 276	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) = 0 ) -> ( N e. ( 0 ... 0 ) -> ( M e. ( 0 ... N ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) ) )
36	3, 35	sylbid 150	. . 3 |- ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) = 0 ) -> ( N e. ( 0 ... (♯ ` A ) ) -> ( M e. ( 0 ... N ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) ) )
37	36	impcomd 255	. 2 |- ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) = 0 ) -> ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) )
38	6	ad2antrr 488	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) -> ( A ++ B ) e. Word V )
39		simprl 531	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) -> M e. ( 0 ... N ) )
40		elfzelfzccat 11176	. . . . . . 7 |- ( ( A e. Word V /\ B e. Word V ) -> ( N e. ( 0 ... (♯ ` A ) ) -> N e. ( 0 ... (♯ ` ( A ++ B ) ) ) ) )
41	40	imp 124	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ N e. ( 0 ... (♯ ` A ) ) ) -> N e. ( 0 ... (♯ ` ( A ++ B ) ) ) )
42	41	ad2ant2rl 511	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) -> N e. ( 0 ... (♯ ` ( A ++ B ) ) ) )
43		swrdvalfn 11236	. . . . 5 |- ( ( ( A ++ B ) e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` ( A ++ B ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) )
44	38, 39, 42, 43	syl3anc 1273	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) )
45		3anass 1008	. . . . . . . 8 |- ( ( A e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) <-> ( A e. Word V /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) )
46	45	simplbi2 385	. . . . . . 7 |- ( A e. Word V -> ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) -> ( A e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) )
47	46	ad2antrr 488	. . . . . 6 |- ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) -> ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) -> ( A e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) )
48	47	imp 124	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) -> ( A e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) )
49		swrdvalfn 11236	. . . . 5 |- ( ( A e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) -> ( A substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) )
50	48, 49	syl 14	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) -> ( A substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) )
51		simp-4l 543	. . . . . 6 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> A e. Word V )
52		simp-4r 544	. . . . . 6 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> B e. Word V )
53		elfznn0 10348	. . . . . . . . . 10 |- ( M e. ( 0 ... N ) -> M e. NN0 )
54		nn0addcl 9436	. . . . . . . . . . 11 |- ( ( k e. NN0 /\ M e. NN0 ) -> ( k + M ) e. NN0 )
55	54	expcom 116	. . . . . . . . . 10 |- ( M e. NN0 -> ( k e. NN0 -> ( k + M ) e. NN0 ) )
56	53, 55	syl 14	. . . . . . . . 9 |- ( M e. ( 0 ... N ) -> ( k e. NN0 -> ( k + M ) e. NN0 ) )
57	56	ad2antrl 490	. . . . . . . 8 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) -> ( k e. NN0 -> ( k + M ) e. NN0 ) )
58		elfzonn0 10424	. . . . . . . 8 |- ( k e. ( 0..^ ( N - M ) ) -> k e. NN0 )
59	57, 58	impel 280	. . . . . . 7 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( k + M ) e. NN0 )
60		lencl 11116	. . . . . . . . . . 11 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
61		elnnne0 9415	. . . . . . . . . . . 12 |- ( (♯ ` A ) e. NN <-> ( (♯ ` A ) e. NN0 /\ (♯ ` A ) =/= 0 ) )
62	61	simplbi2 385	. . . . . . . . . . 11 |- ( (♯ ` A ) e. NN0 -> ( (♯ ` A ) =/= 0 -> (♯ ` A ) e. NN ) )
63	60, 62	syl 14	. . . . . . . . . 10 |- ( A e. Word V -> ( (♯ ` A ) =/= 0 -> (♯ ` A ) e. NN ) )
64	63	adantr 276	. . . . . . . . 9 |- ( ( A e. Word V /\ B e. Word V ) -> ( (♯ ` A ) =/= 0 -> (♯ ` A ) e. NN ) )
65	64	imp 124	. . . . . . . 8 |- ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) -> (♯ ` A ) e. NN )
66	65	ad2antrr 488	. . . . . . 7 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> (♯ ` A ) e. NN )
67		elfzo0 10420	. . . . . . . . 9 |- ( k e. ( 0..^ ( N - M ) ) <-> ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) )
68		elfz2nn0 10346	. . . . . . . . . . . 12 |- ( N e. ( 0 ... (♯ ` A ) ) <-> ( N e. NN0 /\ (♯ ` A ) e. NN0 /\ N <_ (♯ ` A ) ) )
69		nn0re 9410	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k e. NN0 -> k e. RR )
70	69	ad2antrl 490	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> k e. RR )
71		nn0re 9410	. . . . . . . . . . . . . . . . . . . . . 22 |- ( M e. NN0 -> M e. RR )
72	71	ad2antll 491	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> M e. RR )
73		nn0re 9410	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> N e. RR )
74	73	ad2antrr 488	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> N e. RR )
75	70, 72, 74	ltaddsubd 8724	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> ( ( k + M ) < N <-> k < ( N - M ) ) )
76		nn0readdcl 9460	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( k e. NN0 /\ M e. NN0 ) -> ( k + M ) e. RR )
77	76	adantl 277	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> ( k + M ) e. RR )
78		nn0re 9410	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( (♯ ` A ) e. NN0 -> (♯ ` A ) e. RR )
79	78	ad2antlr 489	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> (♯ ` A ) e. RR )
80		ltletr 8268	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( k + M ) e. RR /\ N e. RR /\ (♯ ` A ) e. RR ) -> ( ( ( k + M ) < N /\ N <_ (♯ ` A ) ) -> ( k + M ) < (♯ ` A ) ) )
81	77, 74, 79, 80	syl3anc 1273	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> ( ( ( k + M ) < N /\ N <_ (♯ ` A ) ) -> ( k + M ) < (♯ ` A ) ) )
82	81	expd 258	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> ( ( k + M ) < N -> ( N <_ (♯ ` A ) -> ( k + M ) < (♯ ` A ) ) ) )
83	75, 82	sylbird 170	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> ( k < ( N - M ) -> ( N <_ (♯ ` A ) -> ( k + M ) < (♯ ` A ) ) ) )
84	83	ex 115	. . . . . . . . . . . . . . . . . 18 |- ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) -> ( ( k e. NN0 /\ M e. NN0 ) -> ( k < ( N - M ) -> ( N <_ (♯ ` A ) -> ( k + M ) < (♯ ` A ) ) ) ) )
85	84	com24 87	. . . . . . . . . . . . . . . . 17 |- ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) -> ( N <_ (♯ ` A ) -> ( k < ( N - M ) -> ( ( k e. NN0 /\ M e. NN0 ) -> ( k + M ) < (♯ ` A ) ) ) ) )
86	85	3impia 1226	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ (♯ ` A ) e. NN0 /\ N <_ (♯ ` A ) ) -> ( k < ( N - M ) -> ( ( k e. NN0 /\ M e. NN0 ) -> ( k + M ) < (♯ ` A ) ) ) )
87	86	com13 80	. . . . . . . . . . . . . . 15 |- ( ( k e. NN0 /\ M e. NN0 ) -> ( k < ( N - M ) -> ( ( N e. NN0 /\ (♯ ` A ) e. NN0 /\ N <_ (♯ ` A ) ) -> ( k + M ) < (♯ ` A ) ) ) )
88	87	impancom 260	. . . . . . . . . . . . . 14 |- ( ( k e. NN0 /\ k < ( N - M ) ) -> ( M e. NN0 -> ( ( N e. NN0 /\ (♯ ` A ) e. NN0 /\ N <_ (♯ ` A ) ) -> ( k + M ) < (♯ ` A ) ) ) )
89	88	3adant2 1042	. . . . . . . . . . . . 13 |- ( ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) -> ( M e. NN0 -> ( ( N e. NN0 /\ (♯ ` A ) e. NN0 /\ N <_ (♯ ` A ) ) -> ( k + M ) < (♯ ` A ) ) ) )
90	89	com13 80	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ (♯ ` A ) e. NN0 /\ N <_ (♯ ` A ) ) -> ( M e. NN0 -> ( ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) -> ( k + M ) < (♯ ` A ) ) ) )
91	68, 90	sylbi 121	. . . . . . . . . . 11 |- ( N e. ( 0 ... (♯ ` A ) ) -> ( M e. NN0 -> ( ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) -> ( k + M ) < (♯ ` A ) ) ) )
92	53, 91	mpan9 281	. . . . . . . . . 10 |- ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) -> ( ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) -> ( k + M ) < (♯ ` A ) ) )
93	92	adantl 277	. . . . . . . . 9 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) -> ( ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) -> ( k + M ) < (♯ ` A ) ) )
94	67, 93	biimtrid 152	. . . . . . . 8 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) -> ( k e. ( 0..^ ( N - M ) ) -> ( k + M ) < (♯ ` A ) ) )
95	94	imp 124	. . . . . . 7 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( k + M ) < (♯ ` A ) )
96		elfzo0 10420	. . . . . . 7 |- ( ( k + M ) e. ( 0..^ (♯ ` A ) ) <-> ( ( k + M ) e. NN0 /\ (♯ ` A ) e. NN /\ ( k + M ) < (♯ ` A ) ) )
97	59, 66, 95, 96	syl3anbrc 1207	. . . . . 6 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( k + M ) e. ( 0..^ (♯ ` A ) ) )
98		ccatval1 11173	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V /\ ( k + M ) e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ B ) ` ( k + M ) ) = ( A ` ( k + M ) ) )
99	51, 52, 97, 98	syl3anc 1273	. . . . 5 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( A ++ B ) ` ( k + M ) ) = ( A ` ( k + M ) ) )
100	6	ad3antrrr 492	. . . . . 6 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( A ++ B ) e. Word V )
101		simplrl 537	. . . . . 6 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> M e. ( 0 ... N ) )
102	42	adantr 276	. . . . . 6 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> N e. ( 0 ... (♯ ` ( A ++ B ) ) ) )
103		simpr 110	. . . . . 6 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> k e. ( 0..^ ( N - M ) ) )
104		swrdfv 11233	. . . . . 6 |- ( ( ( ( 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 ) ) )
105	100, 101, 102, 103, 104	syl31anc 1276	. . . . 5 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( A ++ B ) ` ( k + M ) ) )
106		swrdfv 11233	. . . . . 6 |- ( ( ( A e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( A substr <. M , N >. ) ` k ) = ( A ` ( k + M ) ) )
107	48, 106	sylan 283	. . . . 5 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( A substr <. M , N >. ) ` k ) = ( A ` ( k + M ) ) )
108	99, 105, 107	3eqtr4d 2274	. . . 4 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( A substr <. M , N >. ) ` k ) )
109	44, 50, 108	eqfnfvd 5747	. . 3 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) )
110	109	ex 115	. 2 |- ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` A ) =/= 0 ) -> ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) )
111	60	nn0zd 9599	. . . . 5 |- ( A e. Word V -> (♯ ` A ) e. ZZ )
112		zdceq 9554	. . . . 5 |- ( ( (♯ ` A ) e. ZZ /\ 0 e. ZZ ) -> DECID (♯ ` A ) = 0 )
113	111, 7, 112	sylancl 413	. . . 4 |- ( A e. Word V -> DECID (♯ ` A ) = 0 )
114		dcne 2413	. . . 4 |- (DECID (♯ ` A ) = 0 <-> ( (♯ ` A ) = 0 \/ (♯ ` A ) =/= 0 ) )
115	113, 114	sylib 122	. . 3 |- ( A e. Word V -> ( (♯ ` A ) = 0 \/ (♯ ` A ) =/= 0 ) )
116	115	adantr 276	. 2 |- ( ( A e. Word V /\ B e. Word V ) -> ( (♯ ` A ) = 0 \/ (♯ ` A ) =/= 0 ) )
117	37, 110, 116	mpjaodan 805	1 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   (/)c0 3494   <.cop 3672   class class class wbr 4088   Fn wfn 5321   ` cfv 5326  (class class class)co 6017   RRcr 8030   0cc0 8031   + caddc 8034   < clt 8213   <_ cle 8214   - cmin 8349   NNcn 9142   NN0cn0 9401   ZZcz 9478   ...cfz 10242  ..^cfzo 10376  ♯chash 11036  Word cword 11112   ++ cconcat 11166   substr csubstr 11225

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-concat 11167  df-substr 11226

This theorem is referenced by:  pfxccat3  11314  pfxccatpfx1  11316  swrdccatin1d  11323