Theorem swrdccatin1 11442

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 6066	. . . . . 6 |- ( (♯ ` A ) = 0 -> ( 0 ... (♯ ` A ) ) = ( 0 ... 0 ) )
2	1	eleq2d 2304	. . . . 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 10389	. . . . . . 7 |- ( N e. ( 0 ... 0 ) -> N = 0 )
5		elfz1eq 10389	. . . . . . . . . . 11 |- ( M e. ( 0 ... 0 ) -> M = 0 )
6		ccatcl 11306	. . . . . . . . . . . . . . 15 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) e. Word V )
7		0z 9605	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
8		swrd00g 11366	. . . . . . . . . . . . . . 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 11366	. . . . . . . . . . . . . . 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 2270	. . . . . . . . . . . . 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 3888	. . . . . . . . . . . . . 14 |- ( M = 0 -> <. M , 0 >. = <. 0 , 0 >. )
16	15	oveq2d 6074	. . . . . . . . . . . . 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 6074	. . . . . . . . . . . . 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 2277	. . . . . . . . . . 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 6066	. . . . . . . . . . 11 |- ( N = 0 -> ( 0 ... N ) = ( 0 ... 0 ) )
25	24	eleq2d 2304	. . . . . . . . . 10 |- ( N = 0 -> ( M e. ( 0 ... N ) <-> M e. ( 0 ... 0 ) ) )
26		opeq2 3889	. . . . . . . . . . . 12 |- ( N = 0 -> <. M , N >. = <. M , 0 >. )
27	26	oveq2d 6074	. . . . . . . . . . 11 |- ( N = 0 -> ( ( A ++ B ) substr <. M , N >. ) = ( ( A ++ B ) substr <. M , 0 >. ) )
28	26	oveq2d 6074	. . . . . . . . . . 11 |- ( N = 0 -> ( A substr <. M , N >. ) = ( A substr <. M , 0 >. ) )
29	27, 28	eqeq12d 2249	. . . . . . . . . 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 11313	. . . . . . 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 11373	. . . . 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 1274	. . . 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 1009	. . . . . . . 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 11373	. . . . 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 10470	. . . . . . . . . 10 |- ( M e. ( 0 ... N ) -> M e. NN0 )
54		nn0addcl 9548	. . . . . . . . . . 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 10547	. . . . . . . 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 11253	. . . . . . . . . . 11 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
61		elnnne0 9527	. . . . . . . . . . . 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 10542	. . . . . . . . 9 |- ( k e. ( 0..^ ( N - M ) ) <-> ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) )
68		elfz2nn0 10468	. . . . . . . . . . . 12 |- ( N e. ( 0 ... (♯ ` A ) ) <-> ( N e. NN0 /\ (♯ ` A ) e. NN0 /\ N <_ (♯ ` A ) ) )
69		nn0re 9522	. . . . . . . . . . . . . . . . . . . . . 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 9522	. . . . . . . . . . . . . . . . . . . . . 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 9522	. . . . . . . . . . . . . . . . . . . . . 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 8836	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> ( ( k + M ) < N <-> k < ( N - M ) ) )
76		nn0readdcl 9576	. . . . . . . . . . . . . . . . . . . . . . 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 9522	. . . . . . . . . . . . . . . . . . . . . . 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 8379	. . . . . . . . . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . . . 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 1043	. . . . . . . . . . . . 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 10542	. . . . . . 7 |- ( ( k + M ) e. ( 0..^ (♯ ` A ) ) <-> ( ( k + M ) e. NN0 /\ (♯ ` A ) e. NN /\ ( k + M ) < (♯ ` A ) ) )
97	59, 66, 95, 96	syl3anbrc 1208	. . . . . 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 11310	. . . . . 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 1274	. . . . 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 11370	. . . . . 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 1277	. . . . 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 11370	. . . . . 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 2277	. . . 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 5783	. . 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 9716	. . . . 5 |- ( A e. Word V -> (♯ ` A ) e. ZZ )
112		zdceq 9670	. . . . 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 2425	. . . 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 806	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 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   (/)c0 3512   <.cop 3697   class class class wbr 4114   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   RRcr 8142   0cc0 8143   + caddc 8146   < clt 8324   <_ cle 8325   - cmin 8460   NNcn 9254   NN0cn0 9513   ZZcz 9594   ...cfz 10361  ..^cfzo 10498  ♯chash 11163  Word cword 11249   ++ cconcat 11303   substr csubstr 11362

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-concat 11304  df-substr 11363

This theorem is referenced by:  pfxccat3  11451  pfxccatpfx1  11453  swrdccatin1d  11460