Theorem ccatswrd 11123

Description: Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)

Assertion

Ref	Expression
ccatswrd	|- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) = ( S substr <. X , Z >. ) )

Proof of Theorem ccatswrd

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . 6 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> S e. Word A )
2		elfzelz 10147	. . . . . . . 8 |- ( X e. ( 0 ... Y ) -> X e. ZZ )
3	2	3ad2ant1 1021	. . . . . . 7 |- ( ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> X e. ZZ )
4	3	adantl 277	. . . . . 6 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> X e. ZZ )
5		elfzel2 10145	. . . . . . . 8 |- ( X e. ( 0 ... Y ) -> Y e. ZZ )
6	5	3ad2ant1 1021	. . . . . . 7 |- ( ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> Y e. ZZ )
7	6	adantl 277	. . . . . 6 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ZZ )
8		swrdclg 11103	. . . . . 6 |- ( ( S e. Word A /\ X e. ZZ /\ Y e. ZZ ) -> ( S substr <. X , Y >. ) e. Word A )
9	1, 4, 7, 8	syl3anc 1250	. . . . 5 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S substr <. X , Y >. ) e. Word A )
10		elfzel2 10145	. . . . . . . 8 |- ( Y e. ( 0 ... Z ) -> Z e. ZZ )
11	10	3ad2ant2 1022	. . . . . . 7 |- ( ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> Z e. ZZ )
12	11	adantl 277	. . . . . 6 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Z e. ZZ )
13		swrdclg 11103	. . . . . 6 |- ( ( S e. Word A /\ Y e. ZZ /\ Z e. ZZ ) -> ( S substr <. Y , Z >. ) e. Word A )
14	1, 7, 12, 13	syl3anc 1250	. . . . 5 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S substr <. Y , Z >. ) e. Word A )
15		ccatcl 11049	. . . . 5 |- ( ( ( S substr <. X , Y >. ) e. Word A /\ ( S substr <. Y , Z >. ) e. Word A ) -> ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) e. Word A )
16	9, 14, 15	syl2anc 411	. . . 4 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) e. Word A )
17		wrdfn 11009	. . . 4 |- ( ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) e. Word A -> ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ (♯ ` ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ) ) )
18	16, 17	syl 14	. . 3 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ (♯ ` ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ) ) )
19		ccatlen 11051	. . . . . . 7 |- ( ( ( S substr <. X , Y >. ) e. Word A /\ ( S substr <. Y , Z >. ) e. Word A ) -> (♯ ` ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ) = ( (♯ ` ( S substr <. X , Y >. ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) )
20	9, 14, 19	syl2anc 411	. . . . . 6 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ) = ( (♯ ` ( S substr <. X , Y >. ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) )
21		simpr1 1006	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> X e. ( 0 ... Y ) )
22		simpr2 1007	. . . . . . . . 9 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ( 0 ... Z ) )
23		simpr3 1008	. . . . . . . . 9 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Z e. ( 0 ... (♯ ` S ) ) )
24		fzass4 10184	. . . . . . . . . . 11 |- ( ( Y e. ( 0 ... (♯ ` S ) ) /\ Z e. ( Y ... (♯ ` S ) ) ) <-> ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) )
25	24	biimpri 133	. . . . . . . . . 10 |- ( ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> ( Y e. ( 0 ... (♯ ` S ) ) /\ Z e. ( Y ... (♯ ` S ) ) ) )
26	25	simpld 112	. . . . . . . . 9 |- ( ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> Y e. ( 0 ... (♯ ` S ) ) )
27	22, 23, 26	syl2anc 411	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ( 0 ... (♯ ` S ) ) )
28		swrdlen 11105	. . . . . . . 8 |- ( ( S e. Word A /\ X e. ( 0 ... Y ) /\ Y e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S substr <. X , Y >. ) ) = ( Y - X ) )
29	1, 21, 27, 28	syl3anc 1250	. . . . . . 7 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( S substr <. X , Y >. ) ) = ( Y - X ) )
30		swrdlen 11105	. . . . . . . 8 |- ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S substr <. Y , Z >. ) ) = ( Z - Y ) )
31	30	3adant3r1 1215	. . . . . . 7 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( S substr <. Y , Z >. ) ) = ( Z - Y ) )
32	29, 31	oveq12d 5962	. . . . . 6 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( (♯ ` ( S substr <. X , Y >. ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) = ( ( Y - X ) + ( Z - Y ) ) )
33	22	elfzelzd 10148	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ZZ )
34	33	zcnd 9496	. . . . . . 7 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. CC )
35	21	elfzelzd 10148	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> X e. ZZ )
36	35	zcnd 9496	. . . . . . 7 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> X e. CC )
37	23	elfzelzd 10148	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Z e. ZZ )
38	37	zcnd 9496	. . . . . . 7 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Z e. CC )
39	34, 36, 38	npncan3d 8419	. . . . . 6 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( Y - X ) + ( Z - Y ) ) = ( Z - X ) )
40	20, 32, 39	3eqtrd 2242	. . . . 5 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ) = ( Z - X ) )
41	40	oveq2d 5960	. . . 4 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( 0..^ (♯ ` ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ) ) = ( 0..^ ( Z - X ) ) )
42	41	fneq2d 5365	. . 3 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ (♯ ` ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ) ) <-> ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ ( Z - X ) ) ) )
43	18, 42	mpbid 147	. 2 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ ( Z - X ) ) )
44		swrdclg 11103	. . . . 5 |- ( ( S e. Word A /\ X e. ZZ /\ Z e. ZZ ) -> ( S substr <. X , Z >. ) e. Word A )
45	1, 4, 12, 44	syl3anc 1250	. . . 4 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S substr <. X , Z >. ) e. Word A )
46		wrdfn 11009	. . . 4 |- ( ( S substr <. X , Z >. ) e. Word A -> ( S substr <. X , Z >. ) Fn ( 0..^ (♯ ` ( S substr <. X , Z >. ) ) ) )
47	45, 46	syl 14	. . 3 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S substr <. X , Z >. ) Fn ( 0..^ (♯ ` ( S substr <. X , Z >. ) ) ) )
48		fzass4 10184	. . . . . . . . 9 |- ( ( X e. ( 0 ... Z ) /\ Y e. ( X ... Z ) ) <-> ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) ) )
49	48	biimpri 133	. . . . . . . 8 |- ( ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) ) -> ( X e. ( 0 ... Z ) /\ Y e. ( X ... Z ) ) )
50	49	simpld 112	. . . . . . 7 |- ( ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) ) -> X e. ( 0 ... Z ) )
51	21, 22, 50	syl2anc 411	. . . . . 6 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> X e. ( 0 ... Z ) )
52		swrdlen 11105	. . . . . 6 |- ( ( S e. Word A /\ X e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S substr <. X , Z >. ) ) = ( Z - X ) )
53	1, 51, 23, 52	syl3anc 1250	. . . . 5 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( S substr <. X , Z >. ) ) = ( Z - X ) )
54	53	oveq2d 5960	. . . 4 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( 0..^ (♯ ` ( S substr <. X , Z >. ) ) ) = ( 0..^ ( Z - X ) ) )
55	54	fneq2d 5365	. . 3 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S substr <. X , Z >. ) Fn ( 0..^ (♯ ` ( S substr <. X , Z >. ) ) ) <-> ( S substr <. X , Z >. ) Fn ( 0..^ ( Z - X ) ) ) )
56	47, 55	mpbid 147	. 2 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S substr <. X , Z >. ) Fn ( 0..^ ( Z - X ) ) )
57	33, 35	zsubcld 9500	. . . . . 6 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( Y - X ) e. ZZ )
58	57	anim1ci 341	. . . . 5 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Z - X ) ) ) -> ( x e. ( 0..^ ( Z - X ) ) /\ ( Y - X ) e. ZZ ) )
59		fzospliti 10300	. . . . 5 |- ( ( x e. ( 0..^ ( Z - X ) ) /\ ( Y - X ) e. ZZ ) -> ( x e. ( 0..^ ( Y - X ) ) \/ x e. ( ( Y - X )..^ ( Z - X ) ) ) )
60	58, 59	syl 14	. . . 4 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Z - X ) ) ) -> ( x e. ( 0..^ ( Y - X ) ) \/ x e. ( ( Y - X )..^ ( Z - X ) ) ) )
61	9	adantr 276	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Y - X ) ) ) -> ( S substr <. X , Y >. ) e. Word A )
62	14	adantr 276	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Y - X ) ) ) -> ( S substr <. Y , Z >. ) e. Word A )
63	29	oveq2d 5960	. . . . . . . . 9 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( 0..^ (♯ ` ( S substr <. X , Y >. ) ) ) = ( 0..^ ( Y - X ) ) )
64	63	eleq2d 2275	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( x e. ( 0..^ (♯ ` ( S substr <. X , Y >. ) ) ) <-> x e. ( 0..^ ( Y - X ) ) ) )
65	64	biimpar 297	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Y - X ) ) ) -> x e. ( 0..^ (♯ ` ( S substr <. X , Y >. ) ) ) )
66		ccatval1 11053	. . . . . . 7 |- ( ( ( S substr <. X , Y >. ) e. Word A /\ ( S substr <. Y , Z >. ) e. Word A /\ x e. ( 0..^ (♯ ` ( S substr <. X , Y >. ) ) ) ) -> ( ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S substr <. X , Y >. ) ` x ) )
67	61, 62, 65, 66	syl3anc 1250	. . . . . 6 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Y - X ) ) ) -> ( ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S substr <. X , Y >. ) ` x ) )
68		simpll 527	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Y - X ) ) ) -> S e. Word A )
69		simplr1 1042	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Y - X ) ) ) -> X e. ( 0 ... Y ) )
70	27	adantr 276	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Y - X ) ) ) -> Y e. ( 0 ... (♯ ` S ) ) )
71		simpr 110	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Y - X ) ) ) -> x e. ( 0..^ ( Y - X ) ) )
72		swrdfv 11106	. . . . . . 7 |- ( ( ( S e. Word A /\ X e. ( 0 ... Y ) /\ Y e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( Y - X ) ) ) -> ( ( S substr <. X , Y >. ) ` x ) = ( S ` ( x + X ) ) )
73	68, 69, 70, 71, 72	syl31anc 1253	. . . . . 6 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Y - X ) ) ) -> ( ( S substr <. X , Y >. ) ` x ) = ( S ` ( x + X ) ) )
74	67, 73	eqtrd 2238	. . . . 5 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Y - X ) ) ) -> ( ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( S ` ( x + X ) ) )
75	9	adantr 276	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( S substr <. X , Y >. ) e. Word A )
76	14	adantr 276	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( S substr <. Y , Z >. ) e. Word A )
77	32, 39	eqtrd 2238	. . . . . . . . . 10 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( (♯ ` ( S substr <. X , Y >. ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) = ( Z - X ) )
78	29, 77	oveq12d 5962	. . . . . . . . 9 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( (♯ ` ( S substr <. X , Y >. ) )..^ ( (♯ ` ( S substr <. X , Y >. ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) ) = ( ( Y - X )..^ ( Z - X ) ) )
79	78	eleq2d 2275	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( x e. ( (♯ ` ( S substr <. X , Y >. ) )..^ ( (♯ ` ( S substr <. X , Y >. ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) ) <-> x e. ( ( Y - X )..^ ( Z - X ) ) ) )
80	79	biimpar 297	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> x e. ( (♯ ` ( S substr <. X , Y >. ) )..^ ( (♯ ` ( S substr <. X , Y >. ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) ) )
81		ccatval2 11054	. . . . . . 7 |- ( ( ( S substr <. X , Y >. ) e. Word A /\ ( S substr <. Y , Z >. ) e. Word A /\ x e. ( (♯ ` ( S substr <. X , Y >. ) )..^ ( (♯ ` ( S substr <. X , Y >. ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) ) ) -> ( ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S substr <. Y , Z >. ) ` ( x - (♯ ` ( S substr <. X , Y >. ) ) ) ) )
82	75, 76, 80, 81	syl3anc 1250	. . . . . 6 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S substr <. Y , Z >. ) ` ( x - (♯ ` ( S substr <. X , Y >. ) ) ) ) )
83		simpll 527	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> S e. Word A )
84		simplr2 1043	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> Y e. ( 0 ... Z ) )
85		simplr3 1044	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> Z e. ( 0 ... (♯ ` S ) ) )
86	29	oveq2d 5960	. . . . . . . . 9 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( x - (♯ ` ( S substr <. X , Y >. ) ) ) = ( x - ( Y - X ) ) )
87	86	adantr 276	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( x - (♯ ` ( S substr <. X , Y >. ) ) ) = ( x - ( Y - X ) ) )
88	39	oveq2d 5960	. . . . . . . . . . 11 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( Y - X )..^ ( ( Y - X ) + ( Z - Y ) ) ) = ( ( Y - X )..^ ( Z - X ) ) )
89	88	eleq2d 2275	. . . . . . . . . 10 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( x e. ( ( Y - X )..^ ( ( Y - X ) + ( Z - Y ) ) ) <-> x e. ( ( Y - X )..^ ( Z - X ) ) ) )
90	89	biimpar 297	. . . . . . . . 9 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> x e. ( ( Y - X )..^ ( ( Y - X ) + ( Z - Y ) ) ) )
91	37, 33	zsubcld 9500	. . . . . . . . . 10 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( Z - Y ) e. ZZ )
92	91	adantr 276	. . . . . . . . 9 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( Z - Y ) e. ZZ )
93		fzosubel3 10325	. . . . . . . . 9 |- ( ( x e. ( ( Y - X )..^ ( ( Y - X ) + ( Z - Y ) ) ) /\ ( Z - Y ) e. ZZ ) -> ( x - ( Y - X ) ) e. ( 0..^ ( Z - Y ) ) )
94	90, 92, 93	syl2anc 411	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( x - ( Y - X ) ) e. ( 0..^ ( Z - Y ) ) )
95	87, 94	eqeltrd 2282	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( x - (♯ ` ( S substr <. X , Y >. ) ) ) e. ( 0..^ ( Z - Y ) ) )
96		swrdfv 11106	. . . . . . 7 |- ( ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) /\ ( x - (♯ ` ( S substr <. X , Y >. ) ) ) e. ( 0..^ ( Z - Y ) ) ) -> ( ( S substr <. Y , Z >. ) ` ( x - (♯ ` ( S substr <. X , Y >. ) ) ) ) = ( S ` ( ( x - (♯ ` ( S substr <. X , Y >. ) ) ) + Y ) ) )
97	83, 84, 85, 95, 96	syl31anc 1253	. . . . . 6 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( ( S substr <. Y , Z >. ) ` ( x - (♯ ` ( S substr <. X , Y >. ) ) ) ) = ( S ` ( ( x - (♯ ` ( S substr <. X , Y >. ) ) ) + Y ) ) )
98	86	oveq1d 5959	. . . . . . . . 9 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( x - (♯ ` ( S substr <. X , Y >. ) ) ) + Y ) = ( ( x - ( Y - X ) ) + Y ) )
99	98	adantr 276	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( ( x - (♯ ` ( S substr <. X , Y >. ) ) ) + Y ) = ( ( x - ( Y - X ) ) + Y ) )
100		elfzoelz 10269	. . . . . . . . . . 11 |- ( x e. ( ( Y - X )..^ ( Z - X ) ) -> x e. ZZ )
101	100	zcnd 9496	. . . . . . . . . 10 |- ( x e. ( ( Y - X )..^ ( Z - X ) ) -> x e. CC )
102	101	adantl 277	. . . . . . . . 9 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> x e. CC )
103	34, 36	subcld 8383	. . . . . . . . . 10 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( Y - X ) e. CC )
104	103	adantr 276	. . . . . . . . 9 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( Y - X ) e. CC )
105	34	adantr 276	. . . . . . . . 9 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> Y e. CC )
106	102, 104, 105	subadd23d 8405	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( ( x - ( Y - X ) ) + Y ) = ( x + ( Y - ( Y - X ) ) ) )
107	34, 36	nncand 8388	. . . . . . . . . 10 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( Y - ( Y - X ) ) = X )
108	107	oveq2d 5960	. . . . . . . . 9 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( x + ( Y - ( Y - X ) ) ) = ( x + X ) )
109	108	adantr 276	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( x + ( Y - ( Y - X ) ) ) = ( x + X ) )
110	99, 106, 109	3eqtrd 2242	. . . . . . 7 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( ( x - (♯ ` ( S substr <. X , Y >. ) ) ) + Y ) = ( x + X ) )
111	110	fveq2d 5580	. . . . . 6 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( S ` ( ( x - (♯ ` ( S substr <. X , Y >. ) ) ) + Y ) ) = ( S ` ( x + X ) ) )
112	82, 97, 111	3eqtrd 2242	. . . . 5 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( ( Y - X )..^ ( Z - X ) ) ) -> ( ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( S ` ( x + X ) ) )
113	74, 112	jaodan 799	. . . 4 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ ( x e. ( 0..^ ( Y - X ) ) \/ x e. ( ( Y - X )..^ ( Z - X ) ) ) ) -> ( ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( S ` ( x + X ) ) )
114	60, 113	syldan 282	. . 3 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Z - X ) ) ) -> ( ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( S ` ( x + X ) ) )
115		simpll 527	. . . 4 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Z - X ) ) ) -> S e. Word A )
116	51	adantr 276	. . . 4 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Z - X ) ) ) -> X e. ( 0 ... Z ) )
117		simplr3 1044	. . . 4 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Z - X ) ) ) -> Z e. ( 0 ... (♯ ` S ) ) )
118		simpr 110	. . . 4 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Z - X ) ) ) -> x e. ( 0..^ ( Z - X ) ) )
119		swrdfv 11106	. . . 4 |- ( ( ( S e. Word A /\ X e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( Z - X ) ) ) -> ( ( S substr <. X , Z >. ) ` x ) = ( S ` ( x + X ) ) )
120	115, 116, 117, 118, 119	syl31anc 1253	. . 3 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Z - X ) ) ) -> ( ( S substr <. X , Z >. ) ` x ) = ( S ` ( x + X ) ) )
121	114, 120	eqtr4d 2241	. 2 |- ( ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( Z - X ) ) ) -> ( ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S substr <. X , Z >. ) ` x ) )
122	43, 56, 121	eqfnfvd 5680	1 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) = ( S substr <. X , Z >. ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 710   /\ w3a 981   = wceq 1373   e. wcel 2176   <.cop 3636   Fn wfn 5266   ` cfv 5271  (class class class)co 5944   CCcc 7923   0cc0 7925   + caddc 7928   - cmin 8243   ZZcz 9372   ...cfz 10130  ..^cfzo 10264  ♯chash 10920  Word cword 10994   ++ cconcat 11046   substr csubstr 11098

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-1o 6502  df-er 6620  df-en 6828  df-dom 6829  df-fin 6830  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649  df-fz 10131  df-fzo 10265  df-ihash 10921  df-word 10995  df-concat 11047  df-substr 11099

This theorem is referenced by: (None)