Theorem ccatswrd 11420

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 10407	. . . . . . . 8 |- ( X e. ( 0 ... Y ) -> X e. ZZ )
3	2	3ad2ant1 1049	. . . . . . 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 10405	. . . . . . . 8 |- ( X e. ( 0 ... Y ) -> Y e. ZZ )
6	5	3ad2ant1 1049	. . . . . . 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 11400	. . . . . 6 |- ( ( S e. Word A /\ X e. ZZ /\ Y e. ZZ ) -> ( S substr <. X , Y >. ) e. Word A )
9	1, 4, 7, 8	syl3anc 1278	. . . . 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 10405	. . . . . . . 8 |- ( Y e. ( 0 ... Z ) -> Z e. ZZ )
11	10	3ad2ant2 1050	. . . . . . 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 11400	. . . . . 6 |- ( ( S e. Word A /\ Y e. ZZ /\ Z e. ZZ ) -> ( S substr <. Y , Z >. ) e. Word A )
14	1, 7, 12, 13	syl3anc 1278	. . . . 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 11339	. . . . 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 415	. . . 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 11297	. . . 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 11341	. . . . . . 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 415	. . . . . 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 1034	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> X e. ( 0 ... Y ) )
22		simpr2 1035	. . . . . . . . 9 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ( 0 ... Z ) )
23		simpr3 1036	. . . . . . . . 9 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Z e. ( 0 ... (♯ ` S ) ) )
24		fzass4 10446	. . . . . . . . . . 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 415	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ( 0 ... (♯ ` S ) ) )
28		swrdlen 11402	. . . . . . . 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 1278	. . . . . . 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 11402	. . . . . . . 8 |- ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S substr <. Y , Z >. ) ) = ( Z - Y ) )
31	30	3adant3r1 1243	. . . . . . 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 6093	. . . . . 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 10408	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ZZ )
34	33	zcnd 9748	. . . . . . 7 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. CC )
35	21	elfzelzd 10408	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> X e. ZZ )
36	35	zcnd 9748	. . . . . . 7 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> X e. CC )
37	23	elfzelzd 10408	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Z e. ZZ )
38	37	zcnd 9748	. . . . . . 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 8663	. . . . . 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 2275	. . . . 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 6091	. . . 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 5467	. . 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 11400	. . . . 5 |- ( ( S e. Word A /\ X e. ZZ /\ Z e. ZZ ) -> ( S substr <. X , Z >. ) e. Word A )
45	1, 4, 12, 44	syl3anc 1278	. . . 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 11297	. . . 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 10446	. . . . . . . . 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 415	. . . . . 6 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> X e. ( 0 ... Z ) )
52		swrdlen 11402	. . . . . 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 1278	. . . . 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 6091	. . . 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 5467	. . 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 9752	. . . . . 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 10563	. . . . 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 6091	. . . . . . . . 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 2308	. . . . . . . 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 11343	. . . . . . 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 1278	. . . . . 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 531	. . . . . . 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 1070	. . . . . . 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 11403	. . . . . . 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 1281	. . . . . 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 2271	. . . . 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 2271	. . . . . . . . . 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 6093	. . . . . . . . 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 2308	. . . . . . . 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 11344	. . . . . . 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 1278	. . . . . 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 531	. . . . . . 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 1071	. . . . . . 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 1072	. . . . . . 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 6091	. . . . . . . . 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 6091	. . . . . . . . . . 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 2308	. . . . . . . . . 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 9752	. . . . . . . . . 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 10592	. . . . . . . . 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 415	. . . . . . . 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 2315	. . . . . . 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 11403	. . . . . . 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 1281	. . . . . 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 6090	. . . . . . . . 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 10532	. . . . . . . . . . 11 |- ( x e. ( ( Y - X )..^ ( Z - X ) ) -> x e. ZZ )
101	100	zcnd 9748	. . . . . . . . . 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 8627	. . . . . . . . . 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 8649	. . . . . . . 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 8632	. . . . . . . . . 10 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( Y - ( Y - X ) ) = X )
108	107	oveq2d 6091	. . . . . . . . 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 2275	. . . . . . 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 5694	. . . . . 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 2275	. . . . 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 809	. . . 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 531	. . . 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 1072	. . . 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 11403	. . . 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 1281	. . 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 2274	. 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 5800	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 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   <.cop 3708   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   CCcc 8167   0cc0 8169   + caddc 8172   - cmin 8487   ZZcz 9623   ...cfz 10390  ..^cfzo 10527  ♯chash 11192  Word cword 11282   ++ cconcat 11336   substr csubstr 11395

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-concat 11337  df-substr 11396

This theorem is referenced by: (None)