Theorem efgsp1 16734

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 16727	. . . . . . 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 1012	. . . . . 6 |- ( F e. dom S -> F e. (Word W \ { (/) } ) )
9	8	adantr 465	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> F e. (Word W \ { (/) } ) )
10	9	eldifad 3473	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> F e. Word W )
11	1, 2, 3, 4, 5, 6	efgsf 16726	. . . . . . . . . . . 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 5726	. . . . . . . . . . . . 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 5714	. . . . . . . . . . . 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 209	. . . . . . . . . . 11 |- S : dom S --> W
15	14	ffvelrni 6015	. . . . . . . . . 10 |- ( F e. dom S -> ( S ` F ) e. W )
16	15	adantr 465	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( S ` F ) e. W )
17	1, 2, 3, 4	efgtf 16719	. . . . . . . . 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 463	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( T ` ( S ` F ) ) : ( ( 0 ... ( # ` ( S ` F ) ) ) X. ( I X. 2o ) ) --> W )
20		frn 5727	. . . . . . 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 461	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A e. ran ( T ` ( S ` F ) ) )
23	21, 22	sseldd 3490	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A e. W )
24	23	s1cld 12597	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> <" A "> e. Word W )
25		ccatcl 12575	. . . 4 |- ( ( F e. Word W /\ <" A "> e. Word W ) -> ( F concat <" A "> ) e. Word W )
26	10, 24, 25	syl2anc 661	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F concat <" A "> ) e. Word W )
27		ccatlen 12576	. . . . . . 7 |- ( ( F e. Word W /\ <" A "> e. Word W ) -> ( # ` ( F concat <" A "> ) ) = ( ( # ` F ) + ( # ` <" A "> ) ) )
28	10, 24, 27	syl2anc 661	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` ( F concat <" A "> ) ) = ( ( # ` F ) + ( # ` <" A "> ) ) )
29		s1len 12599	. . . . . . 7 |- ( # ` <" A "> ) = 1
30	29	oveq2i 6292	. . . . . 6 |- ( ( # ` F ) + ( # ` <" A "> ) ) = ( ( # ` F ) + 1 )
31	28, 30	syl6eq 2500	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` ( F concat <" A "> ) ) = ( ( # ` F ) + 1 ) )
32		lencl 12544	. . . . . 6 |- ( F e. Word W -> ( # ` F ) e. NN0 )
33		nn0p1nn 10842	. . . . . 6 |- ( ( # ` F ) e. NN0 -> ( ( # ` F ) + 1 ) e. NN )
34	10, 32, 33	3syl 20	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( # ` F ) + 1 ) e. NN )
35	31, 34	eqeltrd 2531	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` ( F concat <" A "> ) ) e. NN )
36		wrdfin 12543	. . . . 5 |- ( ( F concat <" A "> ) e. Word W -> ( F concat <" A "> ) e. Fin )
37		hashnncl 12418	. . . . 5 |- ( ( F concat <" A "> ) e. Fin -> ( ( # ` ( F concat <" A "> ) ) e. NN <-> ( F concat <" A "> ) =/= (/) ) )
38	26, 36, 37	3syl 20	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( # ` ( F concat <" A "> ) ) e. NN <-> ( F concat <" A "> ) =/= (/) ) )
39	35, 38	mpbid 210	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F concat <" A "> ) =/= (/) )
40		eldifsn 4140	. . 3 |- ( ( F concat <" A "> ) e. (Word W \ { (/) } ) <-> ( ( F concat <" A "> ) e. Word W /\ ( F concat <" A "> ) =/= (/) ) )
41	26, 39, 40	sylanbrc 664	. 2 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F concat <" A "> ) e. (Word W \ { (/) } ) )
42		eldifsni 4141	. . . . . . 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 12543	. . . . . . 7 |- ( F e. Word W -> F e. Fin )
45		hashnncl 12418	. . . . . . 7 |- ( F e. Fin -> ( ( # ` F ) e. NN <-> F =/= (/) ) )
46	10, 44, 45	3syl 20	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( # ` F ) e. NN <-> F =/= (/) ) )
47	43, 46	mpbird 232	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` F ) e. NN )
48		lbfzo0 11844	. . . . 5 |- ( 0 e. ( 0..^ ( # ` F ) ) <-> ( # ` F ) e. NN )
49	47, 48	sylibr 212	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> 0 e. ( 0..^ ( # ` F ) ) )
50		ccatval1 12577	. . . 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 1229	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` 0 ) = ( F ` 0 ) )
52	7	simp2bi 1013	. . . 4 |- ( F e. dom S -> ( F ` 0 ) e. D )
53	52	adantr 465	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F ` 0 ) e. D )
54	51, 53	eqeltrd 2531	. 2 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` 0 ) e. D )
55	7	simp3bi 1014	. . . . . 6 |- ( F e. dom S -> A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) )
56	55	adantr 465	. . . . 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		fzo0ss1 11837	. . . . . . . . . . 11 |- ( 1..^ ( # ` F ) ) C_ ( 0..^ ( # ` F ) )
58	57	sseli 3485	. . . . . . . . . 10 |- ( i e. ( 1..^ ( # ` F ) ) -> i e. ( 0..^ ( # ` F ) ) )
59		ccatval1 12577	. . . . . . . . . 10 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 0..^ ( # ` F ) ) ) -> ( ( F concat <" A "> ) ` i ) = ( F ` i ) )
60	58, 59	syl3an3 1264	. . . . . . . . 9 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ( ( F concat <" A "> ) ` i ) = ( F ` i ) )
61		elfzoel2 11810	. . . . . . . . . . . . . . . 16 |- ( i e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. ZZ )
62		peano2zm 10914	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) e. ZZ -> ( ( # ` F ) - 1 ) e. ZZ )
63	61, 62	syl 16	. . . . . . . . . . . . . . 15 |- ( i e. ( 1..^ ( # ` F ) ) -> ( ( # ` F ) - 1 ) e. ZZ )
64	61	zred 10976	. . . . . . . . . . . . . . . 16 |- ( i e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. RR )
65	64	lem1d 10486	. . . . . . . . . . . . . . 15 |- ( i e. ( 1..^ ( # ` F ) ) -> ( ( # ` F ) - 1 ) <_ ( # ` F ) )
66		eluz2 11098	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) <-> ( ( ( # ` F ) - 1 ) e. ZZ /\ ( # ` F ) e. ZZ /\ ( ( # ` F ) - 1 ) <_ ( # ` F ) ) )
67	63, 61, 65, 66	syl3anbrc 1181	. . . . . . . . . . . . . 14 |- ( i e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) )
68		fzoss2 11835	. . . . . . . . . . . . . 14 |- ( ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) -> ( 0..^ ( ( # ` F ) - 1 ) ) C_ ( 0..^ ( # ` F ) ) )
69	67, 68	syl 16	. . . . . . . . . . . . 13 |- ( i e. ( 1..^ ( # ` F ) ) -> ( 0..^ ( ( # ` F ) - 1 ) ) C_ ( 0..^ ( # ` F ) ) )
70		elfzoelz 11811	. . . . . . . . . . . . . . 15 |- ( i e. ( 1..^ ( # ` F ) ) -> i e. ZZ )
71		elfzom1b 11893	. . . . . . . . . . . . . . 15 |- ( ( i e. ZZ /\ ( # ` F ) e. ZZ ) -> ( i e. ( 1..^ ( # ` F ) ) <-> ( i - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) ) )
72	70, 61, 71	syl2anc 661	. . . . . . . . . . . . . 14 |- ( i e. ( 1..^ ( # ` F ) ) -> ( i e. ( 1..^ ( # ` F ) ) <-> ( i - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) ) )
73	72	ibi 241	. . . . . . . . . . . . 13 |- ( i e. ( 1..^ ( # ` F ) ) -> ( i - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) )
74	69, 73	sseldd 3490	. . . . . . . . . . . 12 |- ( i e. ( 1..^ ( # ` F ) ) -> ( i - 1 ) e. ( 0..^ ( # ` F ) ) )
75		ccatval1 12577	. . . . . . . . . . . 12 |- ( ( F e. Word W /\ <" A "> e. Word W /\ ( i - 1 ) e. ( 0..^ ( # ` F ) ) ) -> ( ( F concat <" A "> ) ` ( i - 1 ) ) = ( F ` ( i - 1 ) ) )
76	74, 75	syl3an3 1264	. . . . . . . . . . 11 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ( ( F concat <" A "> ) ` ( i - 1 ) ) = ( F ` ( i - 1 ) ) )
77	76	fveq2d 5860	. . . . . . . . . 10 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) = ( T ` ( F ` ( i - 1 ) ) ) )
78	77	rneqd 5220	. . . . . . . . 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 ) ) ) )
79	60, 78	eleq12d 2525	. . . . . . . 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 ) ) ) ) )
80	79	3expa 1197	. . . . . . 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 ) ) ) ) )
81	80	ralbidva 2879	. . . . . 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 ) ) ) ) )
82	10, 24, 81	syl2anc 661	. . . . 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 ) ) ) ) )
83	56, 82	mpbird 232	. . . 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 ) ) ) )
84	10, 32	syl 16	. . . . . . . . . 10 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` F ) e. NN0 )
85	84	nn0cnd 10861	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` F ) e. CC )
86	85	addid2d 9784	. . . . . . . 8 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 0 + ( # ` F ) ) = ( # ` F ) )
87	86	fveq2d 5860	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` ( 0 + ( # ` F ) ) ) = ( ( F concat <" A "> ) ` ( # ` F ) ) )
88		1nn 10554	. . . . . . . . . . 11 |- 1 e. NN
89	29, 88	eqeltri 2527	. . . . . . . . . 10 |- ( # ` <" A "> ) e. NN
90	89	a1i 11	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` <" A "> ) e. NN )
91		lbfzo0 11844	. . . . . . . . 9 |- ( 0 e. ( 0..^ ( # ` <" A "> ) ) <-> ( # ` <" A "> ) e. NN )
92	90, 91	sylibr 212	. . . . . . . 8 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> 0 e. ( 0..^ ( # ` <" A "> ) ) )
93		ccatval3 12579	. . . . . . . 8 |- ( ( F e. Word W /\ <" A "> e. Word W /\ 0 e. ( 0..^ ( # ` <" A "> ) ) ) -> ( ( F concat <" A "> ) ` ( 0 + ( # ` F ) ) ) = ( <" A "> ` 0 ) )
94	10, 24, 92, 93	syl3anc 1229	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` ( 0 + ( # ` F ) ) ) = ( <" A "> ` 0 ) )
95	87, 94	eqtr3d 2486	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` ( # ` F ) ) = ( <" A "> ` 0 ) )
96		s1fv 12601	. . . . . . . 8 |- ( A e. ran ( T ` ( S ` F ) ) -> ( <" A "> ` 0 ) = A )
97	96	adantl 466	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( <" A "> ` 0 ) = A )
98		fzo0end 11886	. . . . . . . . . . . 12 |- ( ( # ` F ) e. NN -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
99	47, 98	syl 16	. . . . . . . . . . 11 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
100		ccatval1 12577	. . . . . . . . . . 11 |- ( ( F e. Word W /\ <" A "> e. Word W /\ ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) ) -> ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) = ( F ` ( ( # ` F ) - 1 ) ) )
101	10, 24, 99, 100	syl3anc 1229	. . . . . . . . . 10 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) = ( F ` ( ( # ` F ) - 1 ) ) )
102	1, 2, 3, 4, 5, 6	efgsval 16728	. . . . . . . . . . 11 |- ( F e. dom S -> ( S ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
103	102	adantr 465	. . . . . . . . . 10 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( S ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
104	101, 103	eqtr4d 2487	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) = ( S ` F ) )
105	104	fveq2d 5860	. . . . . . . 8 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( T ` ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) ) = ( T ` ( S ` F ) ) )
106	105	rneqd 5220	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ran ( T ` ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) ) = ran ( T ` ( S ` F ) ) )
107	22, 97, 106	3eltr4d 2546	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( <" A "> ` 0 ) e. ran ( T ` ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
108	95, 107	eqeltrd 2531	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F concat <" A "> ) ` ( # ` F ) ) e. ran ( T ` ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
109		fvex 5866	. . . . . 6 |- ( # ` F ) e. _V
110		fveq2 5856	. . . . . . 7 |- ( i = ( # ` F ) -> ( ( F concat <" A "> ) ` i ) = ( ( F concat <" A "> ) ` ( # ` F ) ) )
111		oveq1 6288	. . . . . . . . . 10 |- ( i = ( # ` F ) -> ( i - 1 ) = ( ( # ` F ) - 1 ) )
112	111	fveq2d 5860	. . . . . . . . 9 |- ( i = ( # ` F ) -> ( ( F concat <" A "> ) ` ( i - 1 ) ) = ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) )
113	112	fveq2d 5860	. . . . . . . 8 |- ( i = ( # ` F ) -> ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) = ( T ` ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
114	113	rneqd 5220	. . . . . . 7 |- ( i = ( # ` F ) -> ran ( T ` ( ( F concat <" A "> ) ` ( i - 1 ) ) ) = ran ( T ` ( ( F concat <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
115	110, 114	eleq12d 2525	. . . . . 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 ) ) ) ) )
116	109, 115	ralsn 4053	. . . . 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 ) ) ) )
117	108, 116	sylibr 212	. . . 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 ) ) ) )
118		ralunb 3670	. . . 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 ) ) ) ) )
119	83, 117, 118	sylanbrc 664	. . 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 ) ) ) )
120	31	oveq2d 6297	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 1..^ ( # ` ( F concat <" A "> ) ) ) = ( 1..^ ( ( # ` F ) + 1 ) ) )
121		nnuz 11127	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
122	47, 121	syl6eleq 2541	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` F ) e. ( ZZ>= ` 1 ) )
123		fzosplitsn 11900	. . . . . 6 |- ( ( # ` F ) e. ( ZZ>= ` 1 ) -> ( 1..^ ( ( # ` F ) + 1 ) ) = ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) )
124	122, 123	syl 16	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 1..^ ( ( # ` F ) + 1 ) ) = ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) )
125	120, 124	eqtrd 2484	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 1..^ ( # ` ( F concat <" A "> ) ) ) = ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) )
126	125	raleqdv 3046	. . 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 ) ) ) ) )
127	119, 126	mpbird 232	. 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 ) ) ) )
128	1, 2, 3, 4, 5, 6	efgsdm 16727	. 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 ) ) ) ) )
129	41, 54, 127, 128	syl3anbrc 1181	1 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F concat <" A "> ) e. dom S )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 974   = wceq 1383   e. wcel 1804   =/= wne 2638   A.wral 2793   {crab 2797   \ cdif 3458   u. cun 3459   C_ wss 3461   (/)c0 3770   {csn 4014   <.cop 4020   <.cotp 4022   U_ciun 4315   class class class wbr 4437   |-> cmpt 4495   _I cid 4780   X. cxp 4987   dom cdm 4989   ran crn 4990   -->wf 5574   ` cfv 5578  (class class class)co 6281   |-> cmpt2 6283   1oc1o 7125   2oc2o 7126   Fincfn 7518   0cc0 9495   1c1 9496   + caddc 9498   <_ cle 9632   - cmin 9810   NNcn 10543   NN0cn0 10802   ZZcz 10871   ZZ>=cuz 11092   ...cfz 11683  ..^cfzo 11806   #chash 12387  Word cword 12516   concat cconcat 12518   <"cs1 12519   splice csplice 12521   <"cs2 12788   ~_FG cefg 16703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-fzo 11807  df-hash 12388  df-word 12524  df-concat 12526  df-s1 12527  df-substr 12528  df-splice 12529  df-s2 12795

This theorem is referenced by:  efgsfo  16736  efgredlemd  16741  efgrelexlemb  16747