Theorem efgsp1 17077

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 ++ <" 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 17070	. . . . . . 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 463	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> F e. (Word W \ { (/) } ) )
10	9	eldifad 3425	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> F e. Word W )
11	1, 2, 3, 4, 5, 6	efgsf 17069	. . . . . . . . . . . 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 5718	. . . . . . . . . . . . 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 5706	. . . . . . . . . . . 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 6007	. . . . . . . . . 10 |- ( F e. dom S -> ( S ` F ) e. W )
16	15	adantr 463	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( S ` F ) e. W )
17	1, 2, 3, 4	efgtf 17062	. . . . . . . . 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 17	. . . . . . . 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 461	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( T ` ( S ` F ) ) : ( ( 0 ... ( # ` ( S ` F ) ) ) X. ( I X. 2o ) ) --> W )
20		frn 5719	. . . . . . 7 |- ( ( T ` ( S ` F ) ) : ( ( 0 ... ( # ` ( S ` F ) ) ) X. ( I X. 2o ) ) --> W -> ran ( T ` ( S ` F ) ) C_ W )
21	19, 20	syl 17	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ran ( T ` ( S ` F ) ) C_ W )
22		simpr 459	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A e. ran ( T ` ( S ` F ) ) )
23	21, 22	sseldd 3442	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A e. W )
24	23	s1cld 12667	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> <" A "> e. Word W )
25		ccatcl 12645	. . . 4 |- ( ( F e. Word W /\ <" A "> e. Word W ) -> ( F ++ <" A "> ) e. Word W )
26	10, 24, 25	syl2anc 659	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F ++ <" A "> ) e. Word W )
27		ccatlen 12646	. . . . . . 7 |- ( ( F e. Word W /\ <" A "> e. Word W ) -> ( # ` ( F ++ <" A "> ) ) = ( ( # ` F ) + ( # ` <" A "> ) ) )
28	10, 24, 27	syl2anc 659	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` ( F ++ <" A "> ) ) = ( ( # ` F ) + ( # ` <" A "> ) ) )
29		s1len 12669	. . . . . . 7 |- ( # ` <" A "> ) = 1
30	29	oveq2i 6288	. . . . . 6 |- ( ( # ` F ) + ( # ` <" A "> ) ) = ( ( # ` F ) + 1 )
31	28, 30	syl6eq 2459	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` ( F ++ <" A "> ) ) = ( ( # ` F ) + 1 ) )
32		lencl 12612	. . . . . 6 |- ( F e. Word W -> ( # ` F ) e. NN0 )
33		nn0p1nn 10875	. . . . . 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 2490	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` ( F ++ <" A "> ) ) e. NN )
36		wrdfin 12611	. . . . 5 |- ( ( F ++ <" A "> ) e. Word W -> ( F ++ <" A "> ) e. Fin )
37		hashnncl 12482	. . . . 5 |- ( ( F ++ <" A "> ) e. Fin -> ( ( # ` ( F ++ <" A "> ) ) e. NN <-> ( F ++ <" A "> ) =/= (/) ) )
38	26, 36, 37	3syl 20	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( # ` ( F ++ <" A "> ) ) e. NN <-> ( F ++ <" A "> ) =/= (/) ) )
39	35, 38	mpbid 210	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F ++ <" A "> ) =/= (/) )
40		eldifsn 4096	. . 3 |- ( ( F ++ <" A "> ) e. (Word W \ { (/) } ) <-> ( ( F ++ <" A "> ) e. Word W /\ ( F ++ <" A "> ) =/= (/) ) )
41	26, 39, 40	sylanbrc 662	. 2 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F ++ <" A "> ) e. (Word W \ { (/) } ) )
42		eldifsni 4097	. . . . . . 7 |- ( F e. (Word W \ { (/) } ) -> F =/= (/) )
43	9, 42	syl 17	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> F =/= (/) )
44		wrdfin 12611	. . . . . . 7 |- ( F e. Word W -> F e. Fin )
45		hashnncl 12482	. . . . . . 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 11892	. . . . 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 12647	. . . 4 |- ( ( F e. Word W /\ <" A "> e. Word W /\ 0 e. ( 0..^ ( # ` F ) ) ) -> ( ( F ++ <" A "> ) ` 0 ) = ( F ` 0 ) )
51	10, 24, 49, 50	syl3anc 1230	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" A "> ) ` 0 ) = ( F ` 0 ) )
52	7	simp2bi 1013	. . . 4 |- ( F e. dom S -> ( F ` 0 ) e. D )
53	52	adantr 463	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F ` 0 ) e. D )
54	51, 53	eqeltrd 2490	. 2 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" 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 463	. . . . 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 11885	. . . . . . . . . . 11 |- ( 1..^ ( # ` F ) ) C_ ( 0..^ ( # ` F ) )
58	57	sseli 3437	. . . . . . . . . 10 |- ( i e. ( 1..^ ( # ` F ) ) -> i e. ( 0..^ ( # ` F ) ) )
59		ccatval1 12647	. . . . . . . . . 10 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 0..^ ( # ` F ) ) ) -> ( ( F ++ <" A "> ) ` i ) = ( F ` i ) )
60	58, 59	syl3an3 1265	. . . . . . . . 9 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ( ( F ++ <" A "> ) ` i ) = ( F ` i ) )
61		elfzoel2 11856	. . . . . . . . . . . . . . . 16 |- ( i e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. ZZ )
62		peano2zm 10947	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) e. ZZ -> ( ( # ` F ) - 1 ) e. ZZ )
63	61, 62	syl 17	. . . . . . . . . . . . . . 15 |- ( i e. ( 1..^ ( # ` F ) ) -> ( ( # ` F ) - 1 ) e. ZZ )
64	61	zred 11007	. . . . . . . . . . . . . . . 16 |- ( i e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. RR )
65	64	lem1d 10518	. . . . . . . . . . . . . . 15 |- ( i e. ( 1..^ ( # ` F ) ) -> ( ( # ` F ) - 1 ) <_ ( # ` F ) )
66		eluz2 11132	. . . . . . . . . . . . . . 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 11883	. . . . . . . . . . . . . 14 |- ( ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) -> ( 0..^ ( ( # ` F ) - 1 ) ) C_ ( 0..^ ( # ` F ) ) )
69	67, 68	syl 17	. . . . . . . . . . . . 13 |- ( i e. ( 1..^ ( # ` F ) ) -> ( 0..^ ( ( # ` F ) - 1 ) ) C_ ( 0..^ ( # ` F ) ) )
70		elfzoelz 11857	. . . . . . . . . . . . . . 15 |- ( i e. ( 1..^ ( # ` F ) ) -> i e. ZZ )
71		elfzom1b 11946	. . . . . . . . . . . . . . 15 |- ( ( i e. ZZ /\ ( # ` F ) e. ZZ ) -> ( i e. ( 1..^ ( # ` F ) ) <-> ( i - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) ) )
72	70, 61, 71	syl2anc 659	. . . . . . . . . . . . . 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 3442	. . . . . . . . . . . 12 |- ( i e. ( 1..^ ( # ` F ) ) -> ( i - 1 ) e. ( 0..^ ( # ` F ) ) )
75		ccatval1 12647	. . . . . . . . . . . 12 |- ( ( F e. Word W /\ <" A "> e. Word W /\ ( i - 1 ) e. ( 0..^ ( # ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( i - 1 ) ) = ( F ` ( i - 1 ) ) )
76	74, 75	syl3an3 1265	. . . . . . . . . . 11 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( i - 1 ) ) = ( F ` ( i - 1 ) ) )
77	76	fveq2d 5852	. . . . . . . . . 10 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) = ( T ` ( F ` ( i - 1 ) ) ) )
78	77	rneqd 5050	. . . . . . . . 9 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) = ran ( T ` ( F ` ( i - 1 ) ) ) )
79	60, 78	eleq12d 2484	. . . . . . . 8 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ( ( ( F ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" 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 ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) <-> ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) ) )
81	80	ralbidva 2839	. . . . . 6 |- ( ( F e. Word W /\ <" A "> e. Word W ) -> ( A. i e. ( 1..^ ( # ` F ) ) ( ( F ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) <-> A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) ) )
82	10, 24, 81	syl2anc 659	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( A. i e. ( 1..^ ( # ` F ) ) ( ( F ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" 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 ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) )
84	10, 32	syl 17	. . . . . . . . . 10 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` F ) e. NN0 )
85	84	nn0cnd 10894	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` F ) e. CC )
86	85	addid2d 9814	. . . . . . . 8 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 0 + ( # ` F ) ) = ( # ` F ) )
87	86	fveq2d 5852	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( 0 + ( # ` F ) ) ) = ( ( F ++ <" A "> ) ` ( # ` F ) ) )
88		1nn 10586	. . . . . . . . . . 11 |- 1 e. NN
89	29, 88	eqeltri 2486	. . . . . . . . . 10 |- ( # ` <" A "> ) e. NN
90	89	a1i 11	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` <" A "> ) e. NN )
91		lbfzo0 11892	. . . . . . . . 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 12649	. . . . . . . 8 |- ( ( F e. Word W /\ <" A "> e. Word W /\ 0 e. ( 0..^ ( # ` <" A "> ) ) ) -> ( ( F ++ <" A "> ) ` ( 0 + ( # ` F ) ) ) = ( <" A "> ` 0 ) )
94	10, 24, 92, 93	syl3anc 1230	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( 0 + ( # ` F ) ) ) = ( <" A "> ` 0 ) )
95	87, 94	eqtr3d 2445	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( # ` F ) ) = ( <" A "> ` 0 ) )
96		s1fv 12671	. . . . . . . 8 |- ( A e. ran ( T ` ( S ` F ) ) -> ( <" A "> ` 0 ) = A )
97	96	adantl 464	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( <" A "> ` 0 ) = A )
98		fzo0end 11939	. . . . . . . . . . . 12 |- ( ( # ` F ) e. NN -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
99	47, 98	syl 17	. . . . . . . . . . 11 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
100		ccatval1 12647	. . . . . . . . . . 11 |- ( ( F e. Word W /\ <" A "> e. Word W /\ ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) = ( F ` ( ( # ` F ) - 1 ) ) )
101	10, 24, 99, 100	syl3anc 1230	. . . . . . . . . 10 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) = ( F ` ( ( # ` F ) - 1 ) ) )
102	1, 2, 3, 4, 5, 6	efgsval 17071	. . . . . . . . . . 11 |- ( F e. dom S -> ( S ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
103	102	adantr 463	. . . . . . . . . 10 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( S ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
104	101, 103	eqtr4d 2446	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) = ( S ` F ) )
105	104	fveq2d 5852	. . . . . . . 8 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( T ` ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) ) = ( T ` ( S ` F ) ) )
106	105	rneqd 5050	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ran ( T ` ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) ) = ran ( T ` ( S ` F ) ) )
107	22, 97, 106	3eltr4d 2505	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( <" A "> ` 0 ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
108	95, 107	eqeltrd 2490	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( # ` F ) ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
109		fvex 5858	. . . . . 6 |- ( # ` F ) e. _V
110		fveq2 5848	. . . . . . 7 |- ( i = ( # ` F ) -> ( ( F ++ <" A "> ) ` i ) = ( ( F ++ <" A "> ) ` ( # ` F ) ) )
111		oveq1 6284	. . . . . . . . . 10 |- ( i = ( # ` F ) -> ( i - 1 ) = ( ( # ` F ) - 1 ) )
112	111	fveq2d 5852	. . . . . . . . 9 |- ( i = ( # ` F ) -> ( ( F ++ <" A "> ) ` ( i - 1 ) ) = ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) )
113	112	fveq2d 5852	. . . . . . . 8 |- ( i = ( # ` F ) -> ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) = ( T ` ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
114	113	rneqd 5050	. . . . . . 7 |- ( i = ( # ` F ) -> ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) = ran ( T ` ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
115	110, 114	eleq12d 2484	. . . . . 6 |- ( i = ( # ` F ) -> ( ( ( F ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) <-> ( ( F ++ <" A "> ) ` ( # ` F ) ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) ) ) )
116	109, 115	ralsn 4010	. . . . 5 |- ( A. i e. { ( # ` F ) } ( ( F ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) <-> ( ( F ++ <" A "> ) ` ( # ` F ) ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
117	108, 116	sylibr 212	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A. i e. { ( # ` F ) } ( ( F ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) )
118		ralunb 3623	. . . 4 |- ( A. i e. ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) ( ( F ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) <-> ( A. i e. ( 1..^ ( # ` F ) ) ( ( F ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) /\ A. i e. { ( # ` F ) } ( ( F ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) ) )
119	83, 117, 118	sylanbrc 662	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A. i e. ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) ( ( F ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) )
120	31	oveq2d 6293	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 1..^ ( # ` ( F ++ <" A "> ) ) ) = ( 1..^ ( ( # ` F ) + 1 ) ) )
121		nnuz 11161	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
122	47, 121	syl6eleq 2500	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` F ) e. ( ZZ>= ` 1 ) )
123		fzosplitsn 11953	. . . . . 6 |- ( ( # ` F ) e. ( ZZ>= ` 1 ) -> ( 1..^ ( ( # ` F ) + 1 ) ) = ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) )
124	122, 123	syl 17	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 1..^ ( ( # ` F ) + 1 ) ) = ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) )
125	120, 124	eqtrd 2443	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 1..^ ( # ` ( F ++ <" A "> ) ) ) = ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) )
126	125	raleqdv 3009	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( A. i e. ( 1..^ ( # ` ( F ++ <" A "> ) ) ) ( ( F ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) <-> A. i e. ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) ( ( F ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) ) )
127	119, 126	mpbird 232	. 2 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A. i e. ( 1..^ ( # ` ( F ++ <" A "> ) ) ) ( ( F ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) )
128	1, 2, 3, 4, 5, 6	efgsdm 17070	. 2 |- ( ( F ++ <" A "> ) e. dom S <-> ( ( F ++ <" A "> ) e. (Word W \ { (/) } ) /\ ( ( F ++ <" A "> ) ` 0 ) e. D /\ A. i e. ( 1..^ ( # ` ( F ++ <" A "> ) ) ) ( ( F ++ <" A "> ) ` i ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) ) )
129	41, 54, 127, 128	syl3anbrc 1181	1 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F ++ <" A "> ) e. dom S )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   =/= wne 2598   A.wral 2753   {crab 2757   \ cdif 3410   u. cun 3411   C_ wss 3413   (/)c0 3737   {csn 3971   <.cop 3977   <.cotp 3979   U_ciun 4270   class class class wbr 4394   |-> cmpt 4452   _I cid 4732   X. cxp 4820   dom cdm 4822   ran crn 4823   -->wf 5564   ` cfv 5568  (class class class)co 6277   |-> cmpt2 6279   1oc1o 7159   2oc2o 7160   Fincfn 7553   0cc0 9521   1c1 9522   + caddc 9524   <_ cle 9658   - cmin 9840   NNcn 10575   NN0cn0 10835   ZZcz 10904   ZZ>=cuz 11126   ...cfz 11724  ..^cfzo 11852   #chash 12450  Word cword 12581   ++ cconcat 12583   <"cs1 12584   splice csplice 12586   <"cs2 12860   ~_FG cefg 17046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-ot 3980  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-concat 12591  df-s1 12592  df-substr 12593  df-splice 12594  df-s2 12867

This theorem is referenced by:  efgsfo  17079  efgredlemd  17084  efgrelexlemb  17090