Theorem ccatswrd 11156

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 10177	. . . . . . . 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 10175	. . . . . . . 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 11136	. . . . . 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 10175	. . . . . . . 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 11136	. . . . . 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 11082	. . . . 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 11041	. . . 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 11084	. . . . . . 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 10214	. . . . . . . . . . 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 11138	. . . . . . . 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 11138	. . . . . . . 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 5980	. . . . . 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 10178	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ZZ )
34	33	zcnd 9526	. . . . . . 7 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. CC )
35	21	elfzelzd 10178	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> X e. ZZ )
36	35	zcnd 9526	. . . . . . 7 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> X e. CC )
37	23	elfzelzd 10178	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Z e. ZZ )
38	37	zcnd 9526	. . . . . . 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 8449	. . . . . 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 2243	. . . . 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 5978	. . . 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 5379	. . 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 11136	. . . . 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 11041	. . . 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 10214	. . . . . . . . 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 11138	. . . . . 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 5978	. . . 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 5379	. . 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 9530	. . . . . 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 10330	. . . . 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 5978	. . . . . . . . 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 2276	. . . . . . . 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 11086	. . . . . . 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 11139	. . . . . . 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 2239	. . . . 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 2239	. . . . . . . . . 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 5980	. . . . . . . . 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 2276	. . . . . . . 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 11087	. . . . . . 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 5978	. . . . . . . . 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 5978	. . . . . . . . . . 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 2276	. . . . . . . . . 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 9530	. . . . . . . . . 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 10357	. . . . . . . . 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 2283	. . . . . . 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 11139	. . . . . . 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 5977	. . . . . . . . 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 10299	. . . . . . . . . . 11 |- ( x e. ( ( Y - X )..^ ( Z - X ) ) -> x e. ZZ )
101	100	zcnd 9526	. . . . . . . . . 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 8413	. . . . . . . . . 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 8435	. . . . . . . 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 8418	. . . . . . . . . 10 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( Y - ( Y - X ) ) = X )
108	107	oveq2d 5978	. . . . . . . . 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 2243	. . . . . . 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 5598	. . . . . 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 2243	. . . . 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 11139	. . . 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 2242	. 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 5698	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 2177   <.cop 3641   Fn wfn 5280   ` cfv 5285  (class class class)co 5962   CCcc 7953   0cc0 7955   + caddc 7958   - cmin 8273   ZZcz 9402   ...cfz 10160  ..^cfzo 10294  ♯chash 10952  Word cword 11026   ++ cconcat 11079   substr csubstr 11131

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-0id 8063  ax-rnegex 8064  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071

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 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-frec 6495  df-1o 6520  df-er 6638  df-en 6846  df-dom 6847  df-fin 6848  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-inn 9067  df-n0 9326  df-z 9403  df-uz 9679  df-fz 10161  df-fzo 10295  df-ihash 10953  df-word 11027  df-concat 11080  df-substr 11132

This theorem is referenced by: (None)