Theorem ccatpfx 11348

Description: Concatenating a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.)

Assertion

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

Proof of Theorem ccatpfx

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfznn0 10411	. . . . . . . 8 |- ( Y e. ( 0 ... Z ) -> Y e. NN0 )
2	1	ad2antrl 490	. . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. NN0 )
3		pfxclg 11325	. . . . . . 7 |- ( ( S e. Word A /\ Y e. NN0 ) -> ( S prefix Y ) e. Word A )
4	2, 3	syldan 282	. . . . . 6 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S prefix Y ) e. Word A )
5		simpl 109	. . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> S e. Word A )
6	2	nn0zd 9661	. . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ZZ )
7		elfzelz 10322	. . . . . . . . 9 |- ( Z e. ( 0 ... (♯ ` S ) ) -> Z e. ZZ )
8	7	adantl 277	. . . . . . . 8 |- ( ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> Z e. ZZ )
9	8	adantl 277	. . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Z e. ZZ )
10		swrdclg 11297	. . . . . . 7 |- ( ( S e. Word A /\ Y e. ZZ /\ Z e. ZZ ) -> ( S substr <. Y , Z >. ) e. Word A )
11	5, 6, 9, 10	syl3anc 1274	. . . . . 6 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S substr <. Y , Z >. ) e. Word A )
12		ccatcl 11236	. . . . . 6 |- ( ( ( S prefix Y ) e. Word A /\ ( S substr <. Y , Z >. ) e. Word A ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) e. Word A )
13	4, 11, 12	syl2anc 411	. . . . 5 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) e. Word A )
14		wrdfn 11194	. . . . 5 |- ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) e. Word A -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) ) )
15	13, 14	syl 14	. . . 4 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) ) )
16		ccatlen 11238	. . . . . . . 8 |- ( ( ( S prefix Y ) e. Word A /\ ( S substr <. Y , Z >. ) e. Word A ) -> (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) = ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) )
17	4, 11, 16	syl2anc 411	. . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) = ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) )
18		fzass4 10359	. . . . . . . . . . 11 |- ( ( Y e. ( 0 ... (♯ ` S ) ) /\ Z e. ( Y ... (♯ ` S ) ) ) <-> ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) )
19	18	biimpri 133	. . . . . . . . . 10 |- ( ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> ( Y e. ( 0 ... (♯ ` S ) ) /\ Z e. ( Y ... (♯ ` S ) ) ) )
20	19	simpld 112	. . . . . . . . 9 |- ( ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> Y e. ( 0 ... (♯ ` S ) ) )
21		pfxlen 11332	. . . . . . . . 9 |- ( ( S e. Word A /\ Y e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S prefix Y ) ) = Y )
22	20, 21	sylan2 286	. . . . . . . 8 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( S prefix Y ) ) = Y )
23		swrdlen 11299	. . . . . . . . 9 |- ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S substr <. Y , Z >. ) ) = ( Z - Y ) )
24	23	3expb 1231	. . . . . . . 8 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( S substr <. Y , Z >. ) ) = ( Z - Y ) )
25	22, 24	oveq12d 6046	. . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) = ( Y + ( Z - Y ) ) )
26		elfzelz 10322	. . . . . . . . . 10 |- ( Y e. ( 0 ... Z ) -> Y e. ZZ )
27	26	zcnd 9664	. . . . . . . . 9 |- ( Y e. ( 0 ... Z ) -> Y e. CC )
28	7	zcnd 9664	. . . . . . . . 9 |- ( Z e. ( 0 ... (♯ ` S ) ) -> Z e. CC )
29		pncan3 8446	. . . . . . . . 9 |- ( ( Y e. CC /\ Z e. CC ) -> ( Y + ( Z - Y ) ) = Z )
30	27, 28, 29	syl2an 289	. . . . . . . 8 |- ( ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> ( Y + ( Z - Y ) ) = Z )
31	30	adantl 277	. . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( Y + ( Z - Y ) ) = Z )
32	17, 25, 31	3eqtrd 2268	. . . . . 6 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) = Z )
33	32	oveq2d 6044	. . . . 5 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( 0..^ (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) ) = ( 0..^ Z ) )
34	33	fneq2d 5428	. . . 4 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) ) <-> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ Z ) ) )
35	15, 34	mpbid 147	. . 3 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ Z ) )
36		pfxfn 11330	. . . 4 |- ( ( S e. Word A /\ Z e. ( 0 ... (♯ ` S ) ) ) -> ( S prefix Z ) Fn ( 0..^ Z ) )
37	36	adantrl 478	. . 3 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S prefix Z ) Fn ( 0..^ Z ) )
38		id 19	. . . . . 6 |- ( x e. ( 0..^ Z ) -> x e. ( 0..^ Z ) )
39	26	ad2antrl 490	. . . . . 6 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ZZ )
40		fzospliti 10475	. . . . . 6 |- ( ( x e. ( 0..^ Z ) /\ Y e. ZZ ) -> ( x e. ( 0..^ Y ) \/ x e. ( Y..^ Z ) ) )
41	38, 39, 40	syl2anr 290	. . . . 5 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Z ) ) -> ( x e. ( 0..^ Y ) \/ x e. ( Y..^ Z ) ) )
42	4	adantr 276	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Y ) ) -> ( S prefix Y ) e. Word A )
43	11	adantr 276	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Y ) ) -> ( S substr <. Y , Z >. ) e. Word A )
44	22	oveq2d 6044	. . . . . . . . . 10 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( 0..^ (♯ ` ( S prefix Y ) ) ) = ( 0..^ Y ) )
45	44	eleq2d 2301	. . . . . . . . 9 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( x e. ( 0..^ (♯ ` ( S prefix Y ) ) ) <-> x e. ( 0..^ Y ) ) )
46	45	biimpar 297	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Y ) ) -> x e. ( 0..^ (♯ ` ( S prefix Y ) ) ) )
47		ccatval1 11240	. . . . . . . 8 |- ( ( ( S prefix Y ) e. Word A /\ ( S substr <. Y , Z >. ) e. Word A /\ x e. ( 0..^ (♯ ` ( S prefix Y ) ) ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S prefix Y ) ` x ) )
48	42, 43, 46, 47	syl3anc 1274	. . . . . . 7 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Y ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S prefix Y ) ` x ) )
49	20	adantl 277	. . . . . . . 8 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ( 0 ... (♯ ` S ) ) )
50		id 19	. . . . . . . 8 |- ( x e. ( 0..^ Y ) -> x e. ( 0..^ Y ) )
51		pfxfv 11331	. . . . . . . 8 |- ( ( S e. Word A /\ Y e. ( 0 ... (♯ ` S ) ) /\ x e. ( 0..^ Y ) ) -> ( ( S prefix Y ) ` x ) = ( S ` x ) )
52	5, 49, 50, 51	syl2an3an 1335	. . . . . . 7 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Y ) ) -> ( ( S prefix Y ) ` x ) = ( S ` x ) )
53	48, 52	eqtrd 2264	. . . . . 6 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Y ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( S ` x ) )
54	4	adantr 276	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( S prefix Y ) e. Word A )
55	11	adantr 276	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( S substr <. Y , Z >. ) e. Word A )
56	25, 31	eqtrd 2264	. . . . . . . . . . 11 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) = Z )
57	22, 56	oveq12d 6046	. . . . . . . . . 10 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( (♯ ` ( S prefix Y ) )..^ ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) ) = ( Y..^ Z ) )
58	57	eleq2d 2301	. . . . . . . . 9 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( x e. ( (♯ ` ( S prefix Y ) )..^ ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) ) <-> x e. ( Y..^ Z ) ) )
59	58	biimpar 297	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> x e. ( (♯ ` ( S prefix Y ) )..^ ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) ) )
60		ccatval2 11241	. . . . . . . 8 |- ( ( ( S prefix Y ) e. Word A /\ ( S substr <. Y , Z >. ) e. Word A /\ x e. ( (♯ ` ( S prefix Y ) )..^ ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S substr <. Y , Z >. ) ` ( x - (♯ ` ( S prefix Y ) ) ) ) )
61	54, 55, 59, 60	syl3anc 1274	. . . . . . 7 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S substr <. Y , Z >. ) ` ( x - (♯ ` ( S prefix Y ) ) ) ) )
62		id 19	. . . . . . . . 9 |- ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) )
63	62	3expb 1231	. . . . . . . 8 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) )
64	22	oveq2d 6044	. . . . . . . . . 10 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( x - (♯ ` ( S prefix Y ) ) ) = ( x - Y ) )
65	64	adantr 276	. . . . . . . . 9 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( x - (♯ ` ( S prefix Y ) ) ) = ( x - Y ) )
66		id 19	. . . . . . . . . . 11 |- ( x e. ( Y..^ Z ) -> x e. ( Y..^ Z ) )
67		fzosubel 10502	. . . . . . . . . . 11 |- ( ( x e. ( Y..^ Z ) /\ Y e. ZZ ) -> ( x - Y ) e. ( ( Y - Y )..^ ( Z - Y ) ) )
68	66, 39, 67	syl2anr 290	. . . . . . . . . 10 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( x - Y ) e. ( ( Y - Y )..^ ( Z - Y ) ) )
69	27	subidd 8537	. . . . . . . . . . . . . 14 |- ( Y e. ( 0 ... Z ) -> ( Y - Y ) = 0 )
70	69	oveq1d 6043	. . . . . . . . . . . . 13 |- ( Y e. ( 0 ... Z ) -> ( ( Y - Y )..^ ( Z - Y ) ) = ( 0..^ ( Z - Y ) ) )
71	70	eleq2d 2301	. . . . . . . . . . . 12 |- ( Y e. ( 0 ... Z ) -> ( ( x - Y ) e. ( ( Y - Y )..^ ( Z - Y ) ) <-> ( x - Y ) e. ( 0..^ ( Z - Y ) ) ) )
72	71	ad2antrl 490	. . . . . . . . . . 11 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( x - Y ) e. ( ( Y - Y )..^ ( Z - Y ) ) <-> ( x - Y ) e. ( 0..^ ( Z - Y ) ) ) )
73	72	adantr 276	. . . . . . . . . 10 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( x - Y ) e. ( ( Y - Y )..^ ( Z - Y ) ) <-> ( x - Y ) e. ( 0..^ ( Z - Y ) ) ) )
74	68, 73	mpbid 147	. . . . . . . . 9 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( x - Y ) e. ( 0..^ ( Z - Y ) ) )
75	65, 74	eqeltrd 2308	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( x - (♯ ` ( S prefix Y ) ) ) e. ( 0..^ ( Z - Y ) ) )
76		swrdfv 11300	. . . . . . . 8 |- ( ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) /\ ( x - (♯ ` ( S prefix Y ) ) ) e. ( 0..^ ( Z - Y ) ) ) -> ( ( S substr <. Y , Z >. ) ` ( x - (♯ ` ( S prefix Y ) ) ) ) = ( S ` ( ( x - (♯ ` ( S prefix Y ) ) ) + Y ) ) )
77	63, 75, 76	syl2an2r 599	. . . . . . 7 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( S substr <. Y , Z >. ) ` ( x - (♯ ` ( S prefix Y ) ) ) ) = ( S ` ( ( x - (♯ ` ( S prefix Y ) ) ) + Y ) ) )
78	64	oveq1d 6043	. . . . . . . . . 10 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( x - (♯ ` ( S prefix Y ) ) ) + Y ) = ( ( x - Y ) + Y ) )
79	78	adantr 276	. . . . . . . . 9 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( x - (♯ ` ( S prefix Y ) ) ) + Y ) = ( ( x - Y ) + Y ) )
80		elfzoelz 10444	. . . . . . . . . . 11 |- ( x e. ( Y..^ Z ) -> x e. ZZ )
81	80	zcnd 9664	. . . . . . . . . 10 |- ( x e. ( Y..^ Z ) -> x e. CC )
82	27	ad2antrl 490	. . . . . . . . . 10 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. CC )
83		npcan 8447	. . . . . . . . . 10 |- ( ( x e. CC /\ Y e. CC ) -> ( ( x - Y ) + Y ) = x )
84	81, 82, 83	syl2anr 290	. . . . . . . . 9 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( x - Y ) + Y ) = x )
85	79, 84	eqtrd 2264	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( x - (♯ ` ( S prefix Y ) ) ) + Y ) = x )
86	85	fveq2d 5652	. . . . . . 7 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( S ` ( ( x - (♯ ` ( S prefix Y ) ) ) + Y ) ) = ( S ` x ) )
87	61, 77, 86	3eqtrd 2268	. . . . . 6 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( S ` x ) )
88	53, 87	jaodan 805	. . . . 5 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ ( x e. ( 0..^ Y ) \/ x e. ( Y..^ Z ) ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( S ` x ) )
89	41, 88	syldan 282	. . . 4 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Z ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( S ` x ) )
90		pfxfv 11331	. . . . . 6 |- ( ( S e. Word A /\ Z e. ( 0 ... (♯ ` S ) ) /\ x e. ( 0..^ Z ) ) -> ( ( S prefix Z ) ` x ) = ( S ` x ) )
91	90	3expa 1230	. . . . 5 |- ( ( ( S e. Word A /\ Z e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ Z ) ) -> ( ( S prefix Z ) ` x ) = ( S ` x ) )
92	91	adantlrl 482	. . . 4 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Z ) ) -> ( ( S prefix Z ) ` x ) = ( S ` x ) )
93	89, 92	eqtr4d 2267	. . 3 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Z ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S prefix Z ) ` x ) )
94	35, 37, 93	eqfnfvd 5756	. 2 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) = ( S prefix Z ) )
95	94	3impb 1226	1 |- ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) = ( S prefix Z ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   <.cop 3676   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   CCcc 8090   0cc0 8092   + caddc 8095   - cmin 8409   NN0cn0 9461   ZZcz 9540   ...cfz 10305  ..^cfzo 10439  ♯chash 11100  Word cword 11179   ++ cconcat 11233   substr csubstr 11292   prefix cpfx 11319

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-fz 10306  df-fzo 10440  df-ihash 11101  df-word 11180  df-concat 11234  df-substr 11293  df-pfx 11320

This theorem is referenced by:  pfxcctswrd  11357  wrdeqs1cat  11367