Theorem efgsp1 15324

Description: If F is an extension sequence and A is an extension of the last element of F, then F + <" A "> is an extension sequence. (Contributed by Mario Carneiro, 1-Oct-2015.)

Hypotheses

Ref	Expression
efgval.w	|- W = ( _I ` Word ( I X. 2o ) )
efgval.r	|- .~ = ( ~FG ` I )
efgval2.m	|- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
efgval2.t	|- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
efgred.d	|- D = ( W \ U_ x e. W ran ( T ` x ) )
efgred.s	|- S = ( m e. { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) )

Assertion

Ref	Expression
efgsp1	|- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F concat <" A "> ) e. dom S )

Distinct variable groups:   y, z   t, n, v, w, y, z, m, x   m, M   x, n, M, t, v, w   k, m, t, x, T   k, n, v, w, y, z, W, m, t, x   .~ , m, t, x, y, z   m, I, n, t, v, w, x, y, z   D, m, t

Allowed substitution hints:   A( x, y, z, w, v, t, k, m, n)   D( x, y, z, w, v, k, n)   .~ ( w, v, k, n)   S( x, y, z, w, v, t, k, m, n)   T( y, z, w, v, n)   F( x, y, z, w, v, t, k, m, n)   I( k)   M( y, z, k)

Proof of Theorem efgsp1

Dummy variables a i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . . . . 8 |- W = ( _I ` Word ( I X. 2o ) )
2		efgval.r	. . . . . . . 8 |- .~ = ( ~FG ` I )
3		efgval2.m	. . . . . . . 8 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
4		efgval2.t	. . . . . . . 8 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
5		efgred.d	. . . . . . . 8 |- D = ( W \ U_ x e. W ran ( T ` x ) )
6		efgred.s	. . . . . . . 8 |- S = ( m e. { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) )
7	1, 2, 3, 4, 5, 6	efgsdm 15317	. . . . . . 7 |- ( F e. dom S <-> ( F e. (Word W \ { (/) } ) /\ ( F ` 0 ) e. D /\ A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) ) )
8	7	simp1bi 972	. . . . . 6 |- ( F e. dom S -> F e. (Word W \ { (/) } ) )
9	8	adantr 452	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> F e. (Word W \ { (/) } ) )
10	9	eldifad 3292	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> F e. Word W )
11	1, 2, 3, 4, 5, 6	efgsf 15316	. . . . . . . . . . . 12 |- S : { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } --> W
12	11	fdmi 5555	. . . . . . . . . . . . 13 |- dom S = { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) }
13	12	feq2i 5545	. . . . . . . . . . . 12 |- ( S : dom S --> W <-> S : { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } --> W )
14	11, 13	mpbir 201	. . . . . . . . . . 11 |- S : dom S --> W
15	14	ffvelrni 5828	. . . . . . . . . 10 |- ( F e. dom S -> ( S ` F ) e. W )
16	15	adantr 452	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( S ` F ) e. W )
17	1, 2, 3, 4	efgtf 15309	. . . . . . . . 9 |- ( ( S ` F ) e. W -> ( ( T ` ( S ` F ) ) = ( a e. ( 0 ... ( # ` ( S ` F ) ) ) , i e. ( I X. 2o ) |-> ( ( S ` F ) splice <. a , a , <" i ( M ` i ) "> >. ) ) /\ ( T ` ( S ` F ) ) : ( ( 0 ... ( # ` ( S ` F ) ) ) X. ( I X. 2o ) ) --> W ) )
18	16, 17	syl 16	. . . . . . . 8 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( T ` ( S ` F ) ) = ( a e. ( 0 ... ( # ` ( S ` F ) ) ) , i e. ( I X. 2o ) |-> ( ( S ` F ) splice <. a , a , <" i ( M ` i ) "> >. ) ) /\ ( T ` ( S ` F ) ) : ( ( 0 ... ( # ` ( S ` F ) ) ) X. ( I X. 2o ) ) --> W ) )
19	18	simprd 450	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( T ` ( S ` F ) ) : ( ( 0 ... ( # ` ( S ` F ) ) ) X. ( I X. 2o ) ) --> W )
20		frn 5556	. . . . . . 7 |- ( ( T ` ( S ` F ) ) : ( ( 0 ... ( # ` ( S ` F ) ) ) X. ( I X. 2o ) ) --> W -> ran ( T ` ( S ` F ) ) C_ W )
21	19, 20	syl 16	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ran ( T ` ( S ` F ) ) C_ W )
22		simpr 448	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A e. ran ( T ` ( S ` F ) ) )
23	21, 22	sseldd 3309	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A e. W )
24	23	s1cld 11711	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> <" A "> e. Word W )
25		ccatcl 11698	. . . 4 |- ( ( F e. Word W /\ <" A "> e. Word W ) -> ( F concat <" A "> ) e. Word W )
26	10, 24, 25	syl2anc 643	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F concat <" A "> ) e. Word W )
27		ccatlen 11699	. . . . . . 7 |- ( ( F e. Word W /\ <" A "> e. Word W ) -> ( # ` ( F concat <" A "> ) ) = ( ( # ` F ) + ( # ` <" A "> ) ) )
28	10, 24, 27	syl2anc 643	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` ( F concat <" A "> ) ) = ( ( # ` F ) + ( # ` <" A "> ) ) )
29		s1len 11713	. . . . . . 7 |- ( # ` <" A "> ) = 1
30	29	oveq2i 6051	. . . . . 6 |- ( ( # ` F ) + ( # ` <" A "> ) ) = ( ( # ` F ) + 1 )
31	28, 30	syl6eq 2452	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` ( F concat <" A "> ) ) = ( ( # ` F ) + 1 ) )
32		lencl 11690	. . . . . 6 |- ( F e. Word W -> ( # ` F ) e. NN0 )
33		nn0p1nn 10215	. . . . . 6 |- ( ( # ` F ) e. NN0 -> ( ( # ` F ) + 1 ) e. NN )
34	10, 32, 33	3syl 19	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( # ` F ) + 1 ) e. NN )
35	31, 34	eqeltrd 2478	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` ( F concat <" A "> ) ) e. NN )
36		wrdfin 11689	. . . . 5 |- ( ( F concat <" A "> ) e. Word W -> ( F concat <" A "> ) e. Fin )
37		hashnncl 11600	. . . . 5 |- ( ( F concat <" A "> ) e. Fin -> ( ( # ` ( F concat <" A "> ) ) e. NN <-> ( F concat <" A "> ) =/= (/) ) )
38	26, 36, 37	3syl 19	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( # ` ( F concat <" A "> ) ) e. NN <-> ( F concat <" A "> ) =/= (/) ) )
39	35, 38	mpbid 202	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F concat <" A "> ) =/= (/) )
40		eldifsn 3887	. . 3 |- ( ( F concat <" A "> ) e. (Word W \ { (/) } ) <-> ( ( F concat <" A "> ) e. Word W /\ ( F concat <" A "> ) =/= (/) ) )
41	26, 39, 40	sylanbrc 646	. 2 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F concat <" A "> ) e. (Word W \ { (/) } ) )
42		eldifsni 3888	. . . . . . 7 |- ( F e. (Word W \ { (/) } ) -> F =/= (/) )
43	9, 42	syl 16	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> F =/= (/) )
44		wrdfin 11689	. . . . . . 7 |- ( F e. Word W -> F e. Fin )
45		hashnncl 11600	. . . . . . 7 |- ( F e. Fin -> ( ( # ` F ) e. NN <-> F =/= (/) ) )
46	10, 44, 45	3syl 19	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( # ` F ) e. NN <-> F =/= (/) ) )
47	43, 46	mpbird 224	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` F ) e. NN )
48		lbfzo0 11125	. . . . 5 |- ( 0 e. ( 0..^ ( # ` F ) ) <-> ( # ` F ) e. NN )
49	47, 48	sylibr 204	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> 0 e. ( 0..^ ( # ` F ) ) )
50		ccatval1 11700	. . . 4 |- ( ( F e. Word W /\ <" A "> e. Word W /\ 0 e. ( 0..^ ( # ` F ) ) ) -> ( ( F concat <" A "> ) ` 0 ) = ( F ` 0 ) )
51	10, 24, 49, 50	syl3anc 1184	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` 0 ) = ( F ` 0 ) )
52	7	simp2bi 973	. . . 4 |- ( F e. dom S -> ( F ` 0 ) e. D )
53	52	adantr 452	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F ` 0 ) e. D )
54	51, 53	eqeltrd 2478	. 2 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` 0 ) e. D )
55	7	simp3bi 974	. . . . . 6 |- ( F e. dom S -> A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) )
56	55	adantr 452	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) )
57		1nn0 10193	. . . . . . . . . . . . 13 |- 1 e. NN0
58		nn0uz 10476	. . . . . . . . . . . . 13 |- NN0 = ( ZZ>= ` 0 )
59	57, 58	eleqtri 2476	. . . . . . . . . . . 12 |- 1 e. ( ZZ>= ` 0 )
60		fzoss1 11117	. . . . . . . . . . . 12 |- ( 1 e. ( ZZ>= ` 0 ) -> ( 1..^ ( # ` F ) ) C_ ( 0..^ ( # ` F ) ) )
61	59, 60	ax-mp 8	. . . . . . . . . . 11 |- ( 1..^ ( # ` F ) ) C_ ( 0..^ ( # ` F ) )
62	61	sseli 3304	. . . . . . . . . 10 |- ( i e. ( 1..^ ( # ` F ) ) -> i e. ( 0..^ ( # ` F ) ) )
63		ccatval1 11700	. . . . . . . . . 10 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 0..^ ( # ` F ) ) ) -> ( ( F concat <" A "> ) ` i ) = ( F ` i ) )
64	62, 63	syl3an3 1219	. . . . . . . . 9 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ( ( F concat <" A "> ) ` i ) = ( F ` i ) )
65		elfzoel2 11094	. . . . . . . . . . . . . . . 16 |- ( i e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. ZZ )
66		peano2zm 10276	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) e. ZZ -> ( ( # ` F ) - 1 ) e. ZZ )
67	65, 66	syl 16	. . . . . . . . . . . . . . 15 |- ( i e. ( 1..^ ( # ` F ) ) -> ( ( # ` F ) - 1 ) e. ZZ )
68	65	zred 10331	. . . . . . . . . . . . . . . 16 |- ( i e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. RR )
69	68	lem1d 9900	. . . . . . . . . . . . . . 15 |- ( i e. ( 1..^ ( # ` F ) ) -> ( ( # ` F ) - 1 ) <_ ( # ` F ) )
70		eluz2 10450	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) <-> ( ( ( # ` F ) - 1 ) e. ZZ /\ ( # ` F ) e. ZZ /\ ( ( # ` F ) - 1 ) <_ ( # ` F ) ) )
71	67, 65, 69, 70	syl3anbrc 1138	. . . . . . . . . . . . . 14 |- ( i e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) )
72		fzoss2 11118	. . . . . . . . . . . . . 14 |- ( ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) -> ( 0..^ ( ( # ` F ) - 1 ) ) C_ ( 0..^ ( # ` F ) ) )
73	71, 72	syl 16	. . . . . . . . . . . . 13 |- ( i e. ( 1..^ ( # ` F ) ) -> ( 0..^ ( ( # ` F ) - 1 ) ) C_ ( 0..^ ( # ` F ) ) )
74		elfzoelz 11095	. . . . . . . . . . . . . . 15 |- ( i e. ( 1..^ ( # ` F ) ) -> i e. ZZ )
75		elfzom1b 11146	. . . . . . . . . . . . . . 15 |- ( ( i e. ZZ /\ ( # ` F ) e. ZZ ) -> ( i e. ( 1..^ ( # ` F ) ) <-> ( i - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) ) )
76	74, 65, 75	syl2anc 643	. . . . . . . . . . . . . 14 |- ( i e. ( 1..^ ( # ` F ) ) -> ( i e. ( 1..^ ( # ` F ) ) <-> ( i - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) ) )
77	76	ibi 233	. . . . . . . . . . . . 13 |- ( i e. ( 1..^ ( # ` F ) ) -> ( i - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) )
78	73, 77	sseldd 3309	. . . . . . . . . . . 12 |- ( i e. ( 1..^ ( # ` F ) ) -> ( i - 1 ) e. ( 0..^ ( # ` F ) ) )
79		ccatval1 11700	. . . . . . . . . . . 12 |- ( ( F e. Word W /\ <" A "> e. Word W /\ ( i - 1 ) e. ( 0..^ ( # ` F ) ) ) -> ( ( F concat <" A "> ) ` ( i - 1 ) ) = ( F ` ( i - 1 ) ) )
80	78, 79	syl3an3 1219	. . . . . . . . . . 11 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ( ( F concat <" A "> ) ` ( i - 1 ) ) = ( F ` ( i - 1 ) ) )
81	80	fveq2d 5691	. . . . . . . . . 10 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) = ( T ` ( F ` ( i - 1 ) ) ) )
82	81	rneqd 5056	. . . . . . . . 9 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) = ran ( T ` ( F ` ( i - 1 ) ) ) )
83	64, 82	eleq12d 2472	. . . . . . . 8 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ( ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) <-> ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) ) )
84	83	3expa 1153	. . . . . . 7 |- ( ( ( F e. Word W /\ <" A "> e. Word W ) /\ i e. ( 1..^ ( # ` F ) ) ) -> ( ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) <-> ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) ) )
85	84	ralbidva 2682	. . . . . 6 |- ( ( F e. Word W /\ <" A "> e. Word W ) -> ( A. i e. ( 1..^ ( # ` F ) ) ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) <-> A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) ) )
86	10, 24, 85	syl2anc 643	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( A. i e. ( 1..^ ( # ` F ) ) ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) <-> A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) ) )
87	56, 86	mpbird 224	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A. i e. ( 1..^ ( # ` F ) ) ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) )
88	10, 32	syl 16	. . . . . . . . . 10 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` F ) e. NN0 )
89	88	nn0cnd 10232	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` F ) e. CC )
90	89	addid2d 9223	. . . . . . . 8 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 0 + ( # ` F ) ) = ( # ` F ) )
91	90	fveq2d 5691	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` ( 0 + ( # ` F ) ) ) = ( ( F concat <" A "> ) ` ( # ` F ) ) )
92		1nn 9967	. . . . . . . . . . 11 |- 1 e. NN
93	29, 92	eqeltri 2474	. . . . . . . . . 10 |- ( # ` <" A "> ) e. NN
94	93	a1i 11	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` <" A "> ) e. NN )
95		lbfzo0 11125	. . . . . . . . 9 |- ( 0 e. ( 0..^ ( # ` <" A "> ) ) <-> ( # ` <" A "> ) e. NN )
96	94, 95	sylibr 204	. . . . . . . 8 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> 0 e. ( 0..^ ( # ` <" A "> ) ) )
97		ccatval3 11702	. . . . . . . 8 |- ( ( F e. Word W /\ <" A "> e. Word W /\ 0 e. ( 0..^ ( # ` <" A "> ) ) ) -> ( ( F concat <" A "> ) ` ( 0 + ( # ` F ) ) ) = ( <" A "> ` 0 ) )
98	10, 24, 96, 97	syl3anc 1184	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` ( 0 + ( # ` F ) ) ) = ( <" A "> ` 0 ) )
99	91, 98	eqtr3d 2438	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` ( # ` F ) ) = ( <" A "> ` 0 ) )
100		s1fv 11715	. . . . . . . 8 |- ( A e. ran ( T ` ( S ` F ) ) -> ( <" A "> ` 0 ) = A )
101	100	adantl 453	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( <" A "> ` 0 ) = A )
102		fzo0end 11143	. . . . . . . . . . . 12 |- ( ( # ` F ) e. NN -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
103	47, 102	syl 16	. . . . . . . . . . 11 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
104		ccatval1 11700	. . . . . . . . . . 11 |- ( ( F e. Word W /\ <" A "> e. Word W /\ ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) ) -> ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) = ( F ` ( ( # ` F ) - 1 ) ) )
105	10, 24, 103, 104	syl3anc 1184	. . . . . . . . . 10 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) = ( F ` ( ( # ` F ) - 1 ) ) )
106	1, 2, 3, 4, 5, 6	efgsval 15318	. . . . . . . . . . 11 |- ( F e. dom S -> ( S ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
107	106	adantr 452	. . . . . . . . . 10 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( S ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
108	105, 107	eqtr4d 2439	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) = ( S ` F ) )
109	108	fveq2d 5691	. . . . . . . 8 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( T ` ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) ) = ( T ` ( S ` F ) ) )
110	109	rneqd 5056	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ran ( T ` ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) ) = ran ( T ` ( S ` F ) ) )
111	22, 101, 110	3eltr4d 2485	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( <" A "> ` 0 ) e. ran ( T ` ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
112	99, 111	eqeltrd 2478	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` ( # ` F ) ) e. ran ( T ` ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
113		fvex 5701	. . . . . 6 |- ( # ` F ) e. _V
114		fveq2 5687	. . . . . . 7 |- ( i = ( # ` F ) -> ( ( F concat <" A "> ) ` i ) = ( ( F concat <" A "> ) ` ( # ` F ) ) )
115		oveq1 6047	. . . . . . . . . 10 |- ( i = ( # ` F ) -> ( i - 1 ) = ( ( # ` F ) - 1 ) )
116	115	fveq2d 5691	. . . . . . . . 9 |- ( i = ( # ` F ) -> ( ( F concat <" A "> ) ` ( i - 1 ) ) = ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) )
117	116	fveq2d 5691	. . . . . . . 8 |- ( i = ( # ` F ) -> ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) = ( T ` ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
118	117	rneqd 5056	. . . . . . 7 |- ( i = ( # ` F ) -> ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) = ran ( T ` ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
119	114, 118	eleq12d 2472	. . . . . 6 |- ( i = ( # ` F ) -> ( ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) <-> ( ( F concat <" A "> ) ` ( # ` F ) ) e. ran ( T ` ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) ) ) )
120	113, 119	ralsn 3809	. . . . 5 |- ( A. i e. { ( # ` F ) } ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) <-> ( ( F concat <" A "> ) ` ( # ` F ) ) e. ran ( T ` ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
121	112, 120	sylibr 204	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A. i e. { ( # ` F ) } ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) )
122		ralunb 3488	. . . 4 |- ( A. i e. ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) <-> ( A. i e. ( 1..^ ( # ` F ) ) ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) /\ A. i e. { ( # ` F ) } ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) ) )
123	87, 121, 122	sylanbrc 646	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A. i e. ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) )
124	31	oveq2d 6056	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 1..^ ( # ` ( F concat <" A "> ) ) ) = ( 1..^ ( ( # ` F ) + 1 ) ) )
125		nnuz 10477	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
126	47, 125	syl6eleq 2494	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` F ) e. ( ZZ>= ` 1 ) )
127		fzosplitsn 11150	. . . . . 6 |- ( ( # ` F ) e. ( ZZ>= ` 1 ) -> ( 1..^ ( ( # ` F ) + 1 ) ) = ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) )
128	126, 127	syl 16	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 1..^ ( ( # ` F ) + 1 ) ) = ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) )
129	124, 128	eqtrd 2436	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 1..^ ( # ` ( F concat <" A "> ) ) ) = ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) )
130	129	raleqdv 2870	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( A. i e. ( 1..^ ( # ` ( F concat <" A "> ) ) ) ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) <-> A. i e. ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) ) )
131	123, 130	mpbird 224	. 2 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A. i e. ( 1..^ ( # ` ( F concat <" A "> ) ) ) ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) )
132	1, 2, 3, 4, 5, 6	efgsdm 15317	. 2 |- ( ( F concat <" A "> ) e. dom S <-> ( ( F concat <" A "> ) e. (Word W \ { (/) } ) /\ ( ( F concat <" A "> ) ` 0 ) e. D /\ A. i e. ( 1..^ ( # ` ( F concat <" A "> ) ) ) ( ( F concat <" A "> ) ` i ) e. ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) ) )
133	41, 54, 131, 132	syl3anbrc 1138	1 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F concat <" A "> ) e. dom S )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   {crab 2670   \ cdif 3277   u. cun 3278   C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777   <.cotp 3778   U_ciun 4053   class class class wbr 4172   e. cmpt 4226   _I cid 4453   X. cxp 4835   dom cdm 4837   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040   e. cmpt2 6042   1oc1o 6676   2oc2o 6677   Fincfn 7068   0cc0 8946   1c1 8947   + caddc 8949   <_ cle 9077   - cmin 9247   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999  ..^cfzo 11090   #chash 11573  Word cword 11672   concat cconcat 11673   <"cs1 11674   splice csplice 11676   <"cs2 11760   ~_FG cefg 15293

This theorem is referenced by:  efgsfo  15326  efgredlemd  15331  efgrelexlemb  15337

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-substr 11681  df-splice 11682  df-s2 11767