Theorem swrdccatin1 11252

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 6008	. . . . . 6 |- ( (♯ ` A ) = 0 -> ( 0 ... (♯ ` A ) ) = ( 0 ... 0 ) )
2	1	eleq2d 2299	. . . . 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 10227	. . . . . . 7 |- ( N e. ( 0 ... 0 ) -> N = 0 )
5		elfz1eq 10227	. . . . . . . . . . 11 |- ( M e. ( 0 ... 0 ) -> M = 0 )
6		ccatcl 11123	. . . . . . . . . . . . . . 15 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) e. Word V )
7		0z 9453	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
8		swrd00g 11176	. . . . . . . . . . . . . . 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 11176	. . . . . . . . . . . . . . 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 2265	. . . . . . . . . . . . 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 3856	. . . . . . . . . . . . . 14 |- ( M = 0 -> <. M , 0 >. = <. 0 , 0 >. )
16	15	oveq2d 6016	. . . . . . . . . . . . 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 6016	. . . . . . . . . . . . 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 2272	. . . . . . . . . . 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 6008	. . . . . . . . . . 11 |- ( N = 0 -> ( 0 ... N ) = ( 0 ... 0 ) )
25	24	eleq2d 2299	. . . . . . . . . 10 |- ( N = 0 -> ( M e. ( 0 ... N ) <-> M e. ( 0 ... 0 ) ) )
26		opeq2 3857	. . . . . . . . . . . 12 |- ( N = 0 -> <. M , N >. = <. M , 0 >. )
27	26	oveq2d 6016	. . . . . . . . . . 11 |- ( N = 0 -> ( ( A ++ B ) substr <. M , N >. ) = ( ( A ++ B ) substr <. M , 0 >. ) )
28	26	oveq2d 6016	. . . . . . . . . . 11 |- ( N = 0 -> ( A substr <. M , N >. ) = ( A substr <. M , 0 >. ) )
29	27, 28	eqeq12d 2244	. . . . . . . . . 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 529	. . . . 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 11130	. . . . . . 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 11183	. . . . 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 1271	. . . 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 1006	. . . . . . . 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 11183	. . . . 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 541	. . . . . 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 542	. . . . . 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 10306	. . . . . . . . . 10 |- ( M e. ( 0 ... N ) -> M e. NN0 )
54		nn0addcl 9400	. . . . . . . . . . 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 10382	. . . . . . . 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 11070	. . . . . . . . . . 11 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
61		elnnne0 9379	. . . . . . . . . . . 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 10378	. . . . . . . . 9 |- ( k e. ( 0..^ ( N - M ) ) <-> ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) )
68		elfz2nn0 10304	. . . . . . . . . . . 12 |- ( N e. ( 0 ... (♯ ` A ) ) <-> ( N e. NN0 /\ (♯ ` A ) e. NN0 /\ N <_ (♯ ` A ) ) )
69		nn0re 9374	. . . . . . . . . . . . . . . . . . . . . 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 9374	. . . . . . . . . . . . . . . . . . . . . 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 9374	. . . . . . . . . . . . . . . . . . . . . 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 8688	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> ( ( k + M ) < N <-> k < ( N - M ) ) )
76		nn0readdcl 9424	. . . . . . . . . . . . . . . . . . . . . . 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 9374	. . . . . . . . . . . . . . . . . . . . . . 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 8232	. . . . . . . . . . . . . . . . . . . . . 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 1271	. . . . . . . . . . . . . . . . . . . . 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 1224	. . . . . . . . . . . . . . . 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 1040	. . . . . . . . . . . . 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 10378	. . . . . . 7 |- ( ( k + M ) e. ( 0..^ (♯ ` A ) ) <-> ( ( k + M ) e. NN0 /\ (♯ ` A ) e. NN /\ ( k + M ) < (♯ ` A ) ) )
97	59, 66, 95, 96	syl3anbrc 1205	. . . . . 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 11127	. . . . . 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 1271	. . . . 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 535	. . . . . 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 11180	. . . . . 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 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 ) substr <. M , N >. ) ` k ) = ( ( A ++ B ) ` ( k + M ) ) )
106		swrdfv 11180	. . . . . 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 2272	. . . 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 5734	. . 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 9563	. . . . 5 |- ( A e. Word V -> (♯ ` A ) e. ZZ )
112		zdceq 9518	. . . . 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 2411	. . . 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 803	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 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   (/)c0 3491   <.cop 3669   class class class wbr 4082   Fn wfn 5312   ` cfv 5317  (class class class)co 6000   RRcr 7994   0cc0 7995   + caddc 7998   < clt 8177   <_ cle 8178   - cmin 8313   NNcn 9106   NN0cn0 9365   ZZcz 9442   ...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  pfxccatpfx1  11263  swrdccatin1d  11270