Theorem ccatswrd 11362

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 10359	. . . . . . . 8 |- ( X e. ( 0 ... Y ) -> X e. ZZ )
3	2	3ad2ant1 1045	. . . . . . 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 10357	. . . . . . . 8 |- ( X e. ( 0 ... Y ) -> Y e. ZZ )
6	5	3ad2ant1 1045	. . . . . . 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 11342	. . . . . 6 |- ( ( S e. Word A /\ X e. ZZ /\ Y e. ZZ ) -> ( S substr <. X , Y >. ) e. Word A )
9	1, 4, 7, 8	syl3anc 1274	. . . . 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 10357	. . . . . . . 8 |- ( Y e. ( 0 ... Z ) -> Z e. ZZ )
11	10	3ad2ant2 1046	. . . . . . 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 11342	. . . . . 6 |- ( ( S e. Word A /\ Y e. ZZ /\ Z e. ZZ ) -> ( S substr <. Y , Z >. ) e. Word A )
14	1, 7, 12, 13	syl3anc 1274	. . . . 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 11281	. . . . 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 11239	. . . 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 11283	. . . . . . 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 1030	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> X e. ( 0 ... Y ) )
22		simpr2 1031	. . . . . . . . 9 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ( 0 ... Z ) )
23		simpr3 1032	. . . . . . . . 9 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Z e. ( 0 ... (♯ ` S ) ) )
24		fzass4 10396	. . . . . . . . . . 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 11344	. . . . . . . 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 1274	. . . . . . 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 11344	. . . . . . . 8 |- ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S substr <. Y , Z >. ) ) = ( Z - Y ) )
31	30	3adant3r1 1239	. . . . . . 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 6068	. . . . . 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 10360	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ZZ )
34	33	zcnd 9701	. . . . . . 7 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. CC )
35	21	elfzelzd 10360	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> X e. ZZ )
36	35	zcnd 9701	. . . . . . 7 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> X e. CC )
37	23	elfzelzd 10360	. . . . . . . 8 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Z e. ZZ )
38	37	zcnd 9701	. . . . . . 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 8620	. . . . . 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 2269	. . . . 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 6066	. . . 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 5447	. . 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 11342	. . . . 5 |- ( ( S e. Word A /\ X e. ZZ /\ Z e. ZZ ) -> ( S substr <. X , Z >. ) e. Word A )
45	1, 4, 12, 44	syl3anc 1274	. . . 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 11239	. . . 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 10396	. . . . . . . . 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 11344	. . . . . 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 1274	. . . . 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 6066	. . . 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 5447	. . 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 9705	. . . . . 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 10512	. . . . 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 6066	. . . . . . . . 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 2302	. . . . . . . 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 11285	. . . . . . 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 1274	. . . . . 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 1066	. . . . . . 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 11345	. . . . . . 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 1277	. . . . . 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 2265	. . . . 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 2265	. . . . . . . . . 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 6068	. . . . . . . . 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 2302	. . . . . . . 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 11286	. . . . . . 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 1274	. . . . . 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 1067	. . . . . . 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 1068	. . . . . . 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 6066	. . . . . . . . 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 6066	. . . . . . . . . . 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 2302	. . . . . . . . . 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 9705	. . . . . . . . . 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 10541	. . . . . . . . 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 2309	. . . . . . 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 11345	. . . . . . 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 1277	. . . . . 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 6065	. . . . . . . . 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 10481	. . . . . . . . . . 11 |- ( x e. ( ( Y - X )..^ ( Z - X ) ) -> x e. ZZ )
101	100	zcnd 9701	. . . . . . . . . 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 8584	. . . . . . . . . 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 8606	. . . . . . . 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 8589	. . . . . . . . . 10 |- ( ( S e. Word A /\ ( X e. ( 0 ... Y ) /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( Y - ( Y - X ) ) = X )
108	107	oveq2d 6066	. . . . . . . . 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 2269	. . . . . . 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 5674	. . . . . 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 2269	. . . . 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 805	. . . 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 1068	. . . 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 11345	. . . 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 1277	. . 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 2268	. 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 5778	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 716   /\ w3a 1005   = wceq 1398   e. wcel 2203   <.cop 3692   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   CCcc 8125   0cc0 8127   + caddc 8130   - cmin 8444   ZZcz 9577   ...cfz 10342  ..^cfzo 10476  ♯chash 11138  Word cword 11224   ++ cconcat 11278   substr csubstr 11337

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-concat 11279  df-substr 11338

This theorem is referenced by: (None)