Theorem swrdccatin1 11497

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 6093	. . . . . 6 |- ( (♯ ` A ) = 0 -> ( 0 ... (♯ ` A ) ) = ( 0 ... 0 ) )
2	1	eleq2d 2308	. . . . 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 10439	. . . . . . 7 |- ( N e. ( 0 ... 0 ) -> N = 0 )
5		elfz1eq 10439	. . . . . . . . . . 11 |- ( M e. ( 0 ... 0 ) -> M = 0 )
6		ccatcl 11361	. . . . . . . . . . . . . . 15 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) e. Word V )
7		0z 9655	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
8		swrd00g 11421	. . . . . . . . . . . . . . 15 |- ( ( ( A ++ B ) e. Word V /\ 0 e. ZZ ) -> ( ( A ++ B ) substr <. 0 , 0 >. ) = (/) )
9	6, 7, 8	sylancl 417	. . . . . . . . . . . . . 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 11421	. . . . . . . . . . . . . . 15 |- ( ( A e. Word V /\ 0 e. ZZ ) -> ( A substr <. 0 , 0 >. ) = (/) )
12	10, 7, 11	sylancl 417	. . . . . . . . . . . . . 14 |- ( ( A e. Word V /\ B e. Word V ) -> ( A substr <. 0 , 0 >. ) = (/) )
13	9, 12	eqtr4d 2274	. . . . . . . . . . . . 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 3904	. . . . . . . . . . . . . 14 |- ( M = 0 -> <. M , 0 >. = <. 0 , 0 >. )
16	15	oveq2d 6101	. . . . . . . . . . . . 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 6101	. . . . . . . . . . . . 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 2281	. . . . . . . . . . 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 6093	. . . . . . . . . . 11 |- ( N = 0 -> ( 0 ... N ) = ( 0 ... 0 ) )
25	24	eleq2d 2308	. . . . . . . . . 10 |- ( N = 0 -> ( M e. ( 0 ... N ) <-> M e. ( 0 ... 0 ) ) )
26		opeq2 3905	. . . . . . . . . . . 12 |- ( N = 0 -> <. M , N >. = <. M , 0 >. )
27	26	oveq2d 6101	. . . . . . . . . . 11 |- ( N = 0 -> ( ( A ++ B ) substr <. M , N >. ) = ( ( A ++ B ) substr <. M , 0 >. ) )
28	26	oveq2d 6101	. . . . . . . . . . 11 |- ( N = 0 -> ( A substr <. M , N >. ) = ( A substr <. M , 0 >. ) )
29	27, 28	eqeq12d 2253	. . . . . . . . . 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 492	. . . . 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 535	. . . . 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 11368	. . . . . . 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 515	. . . . 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 11428	. . . . 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 1278	. . . 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 1013	. . . . . . . 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 492	. . . . . 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 11428	. . . . 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 547	. . . . . 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 548	. . . . . 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 10521	. . . . . . . . . 10 |- ( M e. ( 0 ... N ) -> M e. NN0 )
54		nn0addcl 9598	. . . . . . . . . . 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 494	. . . . . . . 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 10598	. . . . . . . 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 11308	. . . . . . . . . . 11 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
61		elnnne0 9577	. . . . . . . . . . . 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 492	. . . . . . 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 10593	. . . . . . . . 9 |- ( k e. ( 0..^ ( N - M ) ) <-> ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) )
68		elfz2nn0 10519	. . . . . . . . . . . 12 |- ( N e. ( 0 ... (♯ ` A ) ) <-> ( N e. NN0 /\ (♯ ` A ) e. NN0 /\ N <_ (♯ ` A ) ) )
69		nn0re 9572	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k e. NN0 -> k e. RR )
70	69	ad2antrl 494	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> k e. RR )
71		nn0re 9572	. . . . . . . . . . . . . . . . . . . . . 22 |- ( M e. NN0 -> M e. RR )
72	71	ad2antll 495	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> M e. RR )
73		nn0re 9572	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> N e. RR )
74	73	ad2antrr 492	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> N e. RR )
75	70, 72, 74	ltaddsubd 8873	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> ( ( k + M ) < N <-> k < ( N - M ) ) )
76		nn0readdcl 9626	. . . . . . . . . . . . . . . . . . . . . . 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 9572	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( (♯ ` A ) e. NN0 -> (♯ ` A ) e. RR )
79	78	ad2antlr 493	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( N e. NN0 /\ (♯ ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> (♯ ` A ) e. RR )
80		ltletr 8415	. . . . . . . . . . . . . . . . . . . . . 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 1278	. . . . . . . . . . . . . . . . . . . . 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 1231	. . . . . . . . . . . . . . . 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 1047	. . . . . . . . . . . . 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 10593	. . . . . . 7 |- ( ( k + M ) e. ( 0..^ (♯ ` A ) ) <-> ( ( k + M ) e. NN0 /\ (♯ ` A ) e. NN /\ ( k + M ) < (♯ ` A ) ) )
97	59, 66, 95, 96	syl3anbrc 1212	. . . . . 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 11365	. . . . . 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 1278	. . . . 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 496	. . . . . 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 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 ) ) ) -> 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 11425	. . . . . 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 1281	. . . . 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 11425	. . . . . 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 2281	. . . 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 5809	. . 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 9766	. . . . 5 |- ( A e. Word V -> (♯ ` A ) e. ZZ )
112		zdceq 9720	. . . . 5 |- ( ( (♯ ` A ) e. ZZ /\ 0 e. ZZ ) -> DECID (♯ ` A ) = 0 )
113	111, 7, 112	sylancl 417	. . . 4 |- ( A e. Word V -> DECID (♯ ` A ) = 0 )
114		dcne 2431	. . . 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 810	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 >. ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   (/)c0 3520   <.cop 3712   class class class wbr 4130   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   RRcr 8178   0cc0 8179   + caddc 8182   < clt 8360   <_ cle 8361   - cmin 8497   NNcn 9304   NN0cn0 9563   ZZcz 9644   ...cfz 10411  ..^cfzo 10549  ♯chash 11214  Word cword 11304   ++ cconcat 11358   substr csubstr 11417

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-concat 11359  df-substr 11418

This theorem is used by:  pfxccat3  11506  pfxccatpfx1  11508  swrdccatin1d  11515