Theorem swrdccatin1 13483

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 6658	. . . . . . 7 |- ( ( # ` A ) = 0 -> ( 0 ... ( # ` A ) ) = ( 0 ... 0 ) )
2	1	eleq2d 2687	. . . . . 6 |- ( ( # ` A ) = 0 -> ( N e. ( 0 ... ( # ` A ) ) <-> N e. ( 0 ... 0 ) ) )
3		elfz1eq 12352	. . . . . . 7 |- ( N e. ( 0 ... 0 ) -> N = 0 )
4		elfz1eq 12352	. . . . . . . . 9 |- ( M e. ( 0 ... 0 ) -> M = 0 )
5		swrd00 13418	. . . . . . . . . . 11 |- ( ( A ++ B ) substr <. 0 , 0 >. ) = (/)
6		swrd00 13418	. . . . . . . . . . 11 |- ( A substr <. 0 , 0 >. ) = (/)
7	5, 6	eqtr4i 2647	. . . . . . . . . 10 |- ( ( A ++ B ) substr <. 0 , 0 >. ) = ( A substr <. 0 , 0 >. )
8		opeq1 4402	. . . . . . . . . . 11 |- ( M = 0 -> <. M , 0 >. = <. 0 , 0 >. )
9	8	oveq2d 6666	. . . . . . . . . 10 |- ( M = 0 -> ( ( A ++ B ) substr <. M , 0 >. ) = ( ( A ++ B ) substr <. 0 , 0 >. ) )
10	8	oveq2d 6666	. . . . . . . . . 10 |- ( M = 0 -> ( A substr <. M , 0 >. ) = ( A substr <. 0 , 0 >. ) )
11	7, 9, 10	3eqtr4a 2682	. . . . . . . . 9 |- ( M = 0 -> ( ( A ++ B ) substr <. M , 0 >. ) = ( A substr <. M , 0 >. ) )
12	4, 11	syl 17	. . . . . . . 8 |- ( M e. ( 0 ... 0 ) -> ( ( A ++ B ) substr <. M , 0 >. ) = ( A substr <. M , 0 >. ) )
13		oveq2 6658	. . . . . . . . . 10 |- ( N = 0 -> ( 0 ... N ) = ( 0 ... 0 ) )
14	13	eleq2d 2687	. . . . . . . . 9 |- ( N = 0 -> ( M e. ( 0 ... N ) <-> M e. ( 0 ... 0 ) ) )
15		opeq2 4403	. . . . . . . . . . 11 |- ( N = 0 -> <. M , N >. = <. M , 0 >. )
16	15	oveq2d 6666	. . . . . . . . . 10 |- ( N = 0 -> ( ( A ++ B ) substr <. M , N >. ) = ( ( A ++ B ) substr <. M , 0 >. ) )
17	15	oveq2d 6666	. . . . . . . . . 10 |- ( N = 0 -> ( A substr <. M , N >. ) = ( A substr <. M , 0 >. ) )
18	16, 17	eqeq12d 2637	. . . . . . . . 9 |- ( N = 0 -> ( ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) <-> ( ( A ++ B ) substr <. M , 0 >. ) = ( A substr <. M , 0 >. ) ) )
19	14, 18	imbi12d 334	. . . . . . . 8 |- ( 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 >. ) ) ) )
20	12, 19	mpbiri 248	. . . . . . 7 |- ( N = 0 -> ( M e. ( 0 ... N ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) )
21	3, 20	syl 17	. . . . . 6 |- ( N e. ( 0 ... 0 ) -> ( M e. ( 0 ... N ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) )
22	2, 21	syl6bi 243	. . . . 5 |- ( ( # ` A ) = 0 -> ( N e. ( 0 ... ( # ` A ) ) -> ( M e. ( 0 ... N ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) ) )
23	22	com23 86	. . . 4 |- ( ( # ` A ) = 0 -> ( M e. ( 0 ... N ) -> ( N e. ( 0 ... ( # ` A ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) ) )
24	23	impd 447	. . 3 |- ( ( # ` A ) = 0 -> ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) )
25	24	a1d 25	. 2 |- ( ( # ` A ) = 0 -> ( ( 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 >. ) ) ) )
26		ccatcl 13359	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) e. Word V )
27	26	adantl 482	. . . . . . 7 |- ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) -> ( A ++ B ) e. Word V )
28	27	adantr 481	. . . . . 6 |- ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) -> ( A ++ B ) e. Word V )
29		simprl 794	. . . . . 6 |- ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) -> M e. ( 0 ... N ) )
30		elfzelfzccat 13364	. . . . . . . . . 10 |- ( ( A e. Word V /\ B e. Word V ) -> ( N e. ( 0 ... ( # ` A ) ) -> N e. ( 0 ... ( # ` ( A ++ B ) ) ) ) )
31	30	adantl 482	. . . . . . . . 9 |- ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) -> ( N e. ( 0 ... ( # ` A ) ) -> N e. ( 0 ... ( # ` ( A ++ B ) ) ) ) )
32	31	com12 32	. . . . . . . 8 |- ( N e. ( 0 ... ( # ` A ) ) -> ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) -> N e. ( 0 ... ( # ` ( A ++ B ) ) ) ) )
33	32	adantl 482	. . . . . . 7 |- ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) -> ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) -> N e. ( 0 ... ( # ` ( A ++ B ) ) ) ) )
34	33	impcom 446	. . . . . 6 |- ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) -> N e. ( 0 ... ( # ` ( A ++ B ) ) ) )
35		swrdvalfn 13426	. . . . . 6 |- ( ( ( A ++ B ) e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` ( A ++ B ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) )
36	28, 29, 34, 35	syl3anc 1326	. . . . 5 |- ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) )
37		3anass 1042	. . . . . . . . 9 |- ( ( A e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) <-> ( A e. Word V /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) )
38	37	simplbi2 655	. . . . . . . 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 ) ) ) ) )
39	38	ad2antrl 764	. . . . . . 7 |- ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) -> ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) -> ( A e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) )
40	39	imp 445	. . . . . 6 |- ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) -> ( A e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) )
41		swrdvalfn 13426	. . . . . 6 |- ( ( A e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) -> ( A substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) )
42	40, 41	syl 17	. . . . 5 |- ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) -> ( A substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) )
43		simprl 794	. . . . . . . 8 |- ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) -> A e. Word V )
44	43	ad2antrr 762	. . . . . . 7 |- ( ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> A e. Word V )
45		simprr 796	. . . . . . . 8 |- ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) -> B e. Word V )
46	45	ad2antrr 762	. . . . . . 7 |- ( ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> B e. Word V )
47		elfzo0 12508	. . . . . . . . . 10 |- ( k e. ( 0..^ ( N - M ) ) <-> ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) )
48		elfz2nn0 12431	. . . . . . . . . . . . . 14 |- ( M e. ( 0 ... N ) <-> ( M e. NN0 /\ N e. NN0 /\ M <_ N ) )
49		nn0addcl 11328	. . . . . . . . . . . . . . . 16 |- ( ( k e. NN0 /\ M e. NN0 ) -> ( k + M ) e. NN0 )
50	49	expcom 451	. . . . . . . . . . . . . . 15 |- ( M e. NN0 -> ( k e. NN0 -> ( k + M ) e. NN0 ) )
51	50	3ad2ant1 1082	. . . . . . . . . . . . . 14 |- ( ( M e. NN0 /\ N e. NN0 /\ M <_ N ) -> ( k e. NN0 -> ( k + M ) e. NN0 ) )
52	48, 51	sylbi 207	. . . . . . . . . . . . 13 |- ( M e. ( 0 ... N ) -> ( k e. NN0 -> ( k + M ) e. NN0 ) )
53	52	ad2antrl 764	. . . . . . . . . . . 12 |- ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) -> ( k e. NN0 -> ( k + M ) e. NN0 ) )
54	53	com12 32	. . . . . . . . . . 11 |- ( k e. NN0 -> ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) -> ( k + M ) e. NN0 ) )
55	54	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) -> ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) -> ( k + M ) e. NN0 ) )
56	47, 55	sylbi 207	. . . . . . . . 9 |- ( k e. ( 0..^ ( N - M ) ) -> ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) -> ( k + M ) e. NN0 ) )
57	56	impcom 446	. . . . . . . 8 |- ( ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( k + M ) e. NN0 )
58		lencl 13324	. . . . . . . . . . . 12 |- ( A e. Word V -> ( # ` A ) e. NN0 )
59		df-ne 2795	. . . . . . . . . . . . 13 |- ( ( # ` A ) =/= 0 <-> -. ( # ` A ) = 0 )
60		elnnne0 11306	. . . . . . . . . . . . . 14 |- ( ( # ` A ) e. NN <-> ( ( # ` A ) e. NN0 /\ ( # ` A ) =/= 0 ) )
61	60	simplbi2 655	. . . . . . . . . . . . 13 |- ( ( # ` A ) e. NN0 -> ( ( # ` A ) =/= 0 -> ( # ` A ) e. NN ) )
62	59, 61	syl5bir 233	. . . . . . . . . . . 12 |- ( ( # ` A ) e. NN0 -> ( -. ( # ` A ) = 0 -> ( # ` A ) e. NN ) )
63	58, 62	syl 17	. . . . . . . . . . 11 |- ( A e. Word V -> ( -. ( # ` A ) = 0 -> ( # ` A ) e. NN ) )
64	63	adantr 481	. . . . . . . . . 10 |- ( ( A e. Word V /\ B e. Word V ) -> ( -. ( # ` A ) = 0 -> ( # ` A ) e. NN ) )
65	64	impcom 446	. . . . . . . . 9 |- ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) -> ( # ` A ) e. NN )
66	65	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( # ` A ) e. NN )
67		elfz2nn0 12431	. . . . . . . . . . . . . . . 16 |- ( N e. ( 0 ... ( # ` A ) ) <-> ( N e. NN0 /\ ( # ` A ) e. NN0 /\ N <_ ( # ` A ) ) )
68		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( k e. NN0 -> k e. RR )
69	68	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( N e. NN0 /\ ( # ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> k e. RR )
70		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( M e. NN0 -> M e. RR )
71	70	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( k e. NN0 /\ M e. NN0 ) -> M e. RR )
72	71	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( N e. NN0 /\ ( # ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> M e. RR )
73		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( N e. NN0 -> N e. RR )
74	73	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( N e. NN0 /\ ( # ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> N e. RR )
75	69, 72, 74	ltaddsubd 10627	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( N e. NN0 /\ ( # ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> ( ( k + M ) < N <-> k < ( N - M ) ) )
76		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( k + M ) e. NN0 -> ( k + M ) e. RR )
77	49, 76	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( k e. NN0 /\ M e. NN0 ) -> ( k + M ) e. RR )
78	77	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( N e. NN0 /\ ( # ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> ( k + M ) e. RR )
79		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( # ` A ) e. NN0 -> ( # ` A ) e. RR )
80	79	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( N e. NN0 /\ ( # ` A ) e. NN0 ) -> ( # ` A ) e. RR )
81	80	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( N e. NN0 /\ ( # ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> ( # ` A ) e. RR )
82		ltletr 10129	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( k + M ) e. RR /\ N e. RR /\ ( # ` A ) e. RR ) -> ( ( ( k + M ) < N /\ N <_ ( # ` A ) ) -> ( k + M ) < ( # ` A ) ) )
83	78, 74, 81, 82	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( N e. NN0 /\ ( # ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> ( ( ( k + M ) < N /\ N <_ ( # ` A ) ) -> ( k + M ) < ( # ` A ) ) )
84	83	expd 452	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( N e. NN0 /\ ( # ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> ( ( k + M ) < N -> ( N <_ ( # ` A ) -> ( k + M ) < ( # ` A ) ) ) )
85	75, 84	sylbird 250	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( N e. NN0 /\ ( # ` A ) e. NN0 ) /\ ( k e. NN0 /\ M e. NN0 ) ) -> ( k < ( N - M ) -> ( N <_ ( # ` A ) -> ( k + M ) < ( # ` A ) ) ) )
86	85	ex 450	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( N e. NN0 /\ ( # ` A ) e. NN0 ) -> ( ( k e. NN0 /\ M e. NN0 ) -> ( k < ( N - M ) -> ( N <_ ( # ` A ) -> ( k + M ) < ( # ` A ) ) ) ) )
87	86	com24 95	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( N e. NN0 /\ ( # ` A ) e. NN0 ) -> ( N <_ ( # ` A ) -> ( k < ( N - M ) -> ( ( k e. NN0 /\ M e. NN0 ) -> ( k + M ) < ( # ` A ) ) ) ) )
88	87	3impia 1261	. . . . . . . . . . . . . . . . . . . 20 |- ( ( N e. NN0 /\ ( # ` A ) e. NN0 /\ N <_ ( # ` A ) ) -> ( k < ( N - M ) -> ( ( k e. NN0 /\ M e. NN0 ) -> ( k + M ) < ( # ` A ) ) ) )
89	88	com13 88	. . . . . . . . . . . . . . . . . . 19 |- ( ( k e. NN0 /\ M e. NN0 ) -> ( k < ( N - M ) -> ( ( N e. NN0 /\ ( # ` A ) e. NN0 /\ N <_ ( # ` A ) ) -> ( k + M ) < ( # ` A ) ) ) )
90	89	impancom 456	. . . . . . . . . . . . . . . . . 18 |- ( ( k e. NN0 /\ k < ( N - M ) ) -> ( M e. NN0 -> ( ( N e. NN0 /\ ( # ` A ) e. NN0 /\ N <_ ( # ` A ) ) -> ( k + M ) < ( # ` A ) ) ) )
91	90	3adant2 1080	. . . . . . . . . . . . . . . . 17 |- ( ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) -> ( M e. NN0 -> ( ( N e. NN0 /\ ( # ` A ) e. NN0 /\ N <_ ( # ` A ) ) -> ( k + M ) < ( # ` A ) ) ) )
92	91	com13 88	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ ( # ` A ) e. NN0 /\ N <_ ( # ` A ) ) -> ( M e. NN0 -> ( ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) -> ( k + M ) < ( # ` A ) ) ) )
93	67, 92	sylbi 207	. . . . . . . . . . . . . . 15 |- ( N e. ( 0 ... ( # ` A ) ) -> ( M e. NN0 -> ( ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) -> ( k + M ) < ( # ` A ) ) ) )
94	93	com12 32	. . . . . . . . . . . . . 14 |- ( M e. NN0 -> ( N e. ( 0 ... ( # ` A ) ) -> ( ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) -> ( k + M ) < ( # ` A ) ) ) )
95	94	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( M e. NN0 /\ N e. NN0 /\ M <_ N ) -> ( N e. ( 0 ... ( # ` A ) ) -> ( ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) -> ( k + M ) < ( # ` A ) ) ) )
96	48, 95	sylbi 207	. . . . . . . . . . . 12 |- ( M e. ( 0 ... N ) -> ( N e. ( 0 ... ( # ` A ) ) -> ( ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) -> ( k + M ) < ( # ` A ) ) ) )
97	96	a1i 11	. . . . . . . . . . 11 |- ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) -> ( M e. ( 0 ... N ) -> ( N e. ( 0 ... ( # ` A ) ) -> ( ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) -> ( k + M ) < ( # ` A ) ) ) ) )
98	97	imp32 449	. . . . . . . . . 10 |- ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) -> ( ( k e. NN0 /\ ( N - M ) e. NN /\ k < ( N - M ) ) -> ( k + M ) < ( # ` A ) ) )
99	47, 98	syl5bi 232	. . . . . . . . 9 |- ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) -> ( k e. ( 0..^ ( N - M ) ) -> ( k + M ) < ( # ` A ) ) )
100	99	imp 445	. . . . . . . 8 |- ( ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( k + M ) < ( # ` A ) )
101		elfzo0 12508	. . . . . . . 8 |- ( ( k + M ) e. ( 0..^ ( # ` A ) ) <-> ( ( k + M ) e. NN0 /\ ( # ` A ) e. NN /\ ( k + M ) < ( # ` A ) ) )
102	57, 66, 100, 101	syl3anbrc 1246	. . . . . . 7 |- ( ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( k + M ) e. ( 0..^ ( # ` A ) ) )
103		ccatval1 13361	. . . . . . 7 |- ( ( A e. Word V /\ B e. Word V /\ ( k + M ) e. ( 0..^ ( # ` A ) ) ) -> ( ( A ++ B ) ` ( k + M ) ) = ( A ` ( k + M ) ) )
104	44, 46, 102, 103	syl3anc 1326	. . . . . 6 |- ( ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( A ++ B ) ` ( k + M ) ) = ( A ` ( k + M ) ) )
105	27	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( A ++ B ) e. Word V )
106	29	adantr 481	. . . . . . . 8 |- ( ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> M e. ( 0 ... N ) )
107	34	adantr 481	. . . . . . . 8 |- ( ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> N e. ( 0 ... ( # ` ( A ++ B ) ) ) )
108	105, 106, 107	3jca 1242	. . . . . . 7 |- ( ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( A ++ B ) e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` ( A ++ B ) ) ) ) )
109		swrdfv 13424	. . . . . . 7 |- ( ( ( ( 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 ) ) )
110	108, 109	sylancom 701	. . . . . 6 |- ( ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( A ++ B ) ` ( k + M ) ) )
111		swrdfv 13424	. . . . . . 7 |- ( ( ( 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 ) ) )
112	40, 111	sylan 488	. . . . . 6 |- ( ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( A substr <. M , N >. ) ` k ) = ( A ` ( k + M ) ) )
113	104, 110, 112	3eqtr4d 2666	. . . . 5 |- ( ( ( ( -. ( # ` A ) = 0 /\ ( A e. Word V /\ B e. Word V ) ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` A ) ) ) ) /\ k e. ( 0..^ ( N - M ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` k ) = ( ( A substr <. M , N >. ) ` k ) )
114	36, 42, 113	eqfnfvd 6314	. . . 4 |- ( ( ( -. ( # ` A ) = 0 /\ ( 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 >. ) )
115	114	ex 450	. . 3 |- ( ( -. ( # ` A ) = 0 /\ ( 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 >. ) ) )
116	115	ex 450	. 2 |- ( -. ( # ` A ) = 0 -> ( ( 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 >. ) ) ) )
117	25, 116	pm2.61i 176	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:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   <.cop 4183   class class class wbr 4653   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   substr csubstr 13295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303

This theorem is referenced by:  swrdccat3  13492  swrdccatin1d  13499  pfxccat3  41426  pfxccatpfx1  41427