Theorem efgsp1 18150

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 18143	. . . . . . 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 1076	. . . . . 6 |- ( F e. dom S -> F e. (Word W \ { (/) } ) )
9	8	adantr 481	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> F e. (Word W \ { (/) } ) )
10	9	eldifad 3586	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> F e. Word W )
11	1, 2, 3, 4, 5, 6	efgsf 18142	. . . . . . . . . . . 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 6052	. . . . . . . . . . . . 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 6037	. . . . . . . . . . . 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 221	. . . . . . . . . . 11 |- S : dom S --> W
15	14	ffvelrni 6358	. . . . . . . . . 10 |- ( F e. dom S -> ( S ` F ) e. W )
16	15	adantr 481	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( S ` F ) e. W )
17	1, 2, 3, 4	efgtf 18135	. . . . . . . . 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 479	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( T ` ( S ` F ) ) : ( ( 0 ... ( # ` ( S ` F ) ) ) X. ( I X. 2o ) ) --> W )
20		frn 6053	. . . . . . 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 477	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A e. ran ( T ` ( S ` F ) ) )
23	21, 22	sseldd 3604	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> A e. W )
24	23	s1cld 13383	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> <" A "> e. Word W )
25		ccatcl 13359	. . . 4 |- ( ( F e. Word W /\ <" A "> e. Word W ) -> ( F ++ <" A "> ) e. Word W )
26	10, 24, 25	syl2anc 693	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F ++ <" A "> ) e. Word W )
27		ccatlen 13360	. . . . . . 7 |- ( ( F e. Word W /\ <" A "> e. Word W ) -> ( # ` ( F ++ <" A "> ) ) = ( ( # ` F ) + ( # ` <" A "> ) ) )
28	10, 24, 27	syl2anc 693	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` ( F ++ <" A "> ) ) = ( ( # ` F ) + ( # ` <" A "> ) ) )
29		s1len 13385	. . . . . . 7 |- ( # ` <" A "> ) = 1
30	29	oveq2i 6661	. . . . . 6 |- ( ( # ` F ) + ( # ` <" A "> ) ) = ( ( # ` F ) + 1 )
31	28, 30	syl6eq 2672	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` ( F ++ <" A "> ) ) = ( ( # ` F ) + 1 ) )
32		lencl 13324	. . . . . 6 |- ( F e. Word W -> ( # ` F ) e. NN0 )
33		nn0p1nn 11332	. . . . . 6 |- ( ( # ` F ) e. NN0 -> ( ( # ` F ) + 1 ) e. NN )
34	10, 32, 33	3syl 18	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( # ` F ) + 1 ) e. NN )
35	31, 34	eqeltrd 2701	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` ( F ++ <" A "> ) ) e. NN )
36		wrdfin 13323	. . . . 5 |- ( ( F ++ <" A "> ) e. Word W -> ( F ++ <" A "> ) e. Fin )
37		hashnncl 13157	. . . . 5 |- ( ( F ++ <" A "> ) e. Fin -> ( ( # ` ( F ++ <" A "> ) ) e. NN <-> ( F ++ <" A "> ) =/= (/) ) )
38	26, 36, 37	3syl 18	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( # ` ( F ++ <" A "> ) ) e. NN <-> ( F ++ <" A "> ) =/= (/) ) )
39	35, 38	mpbid 222	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F ++ <" A "> ) =/= (/) )
40		eldifsn 4317	. . 3 |- ( ( F ++ <" A "> ) e. (Word W \ { (/) } ) <-> ( ( F ++ <" A "> ) e. Word W /\ ( F ++ <" A "> ) =/= (/) ) )
41	26, 39, 40	sylanbrc 698	. 2 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F ++ <" A "> ) e. (Word W \ { (/) } ) )
42		eldifsni 4320	. . . . . . 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 13323	. . . . . . 7 |- ( F e. Word W -> F e. Fin )
45		hashnncl 13157	. . . . . . 7 |- ( F e. Fin -> ( ( # ` F ) e. NN <-> F =/= (/) ) )
46	10, 44, 45	3syl 18	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( # ` F ) e. NN <-> F =/= (/) ) )
47	43, 46	mpbird 247	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` F ) e. NN )
48		lbfzo0 12507	. . . . 5 |- ( 0 e. ( 0..^ ( # ` F ) ) <-> ( # ` F ) e. NN )
49	47, 48	sylibr 224	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> 0 e. ( 0..^ ( # ` F ) ) )
50		ccatval1 13361	. . . 4 |- ( ( F e. Word W /\ <" A "> e. Word W /\ 0 e. ( 0..^ ( # ` F ) ) ) -> ( ( F ++ <" A "> ) ` 0 ) = ( F ` 0 ) )
51	10, 24, 49, 50	syl3anc 1326	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" A "> ) ` 0 ) = ( F ` 0 ) )
52	7	simp2bi 1077	. . . 4 |- ( F e. dom S -> ( F ` 0 ) e. D )
53	52	adantr 481	. . 3 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F ` 0 ) e. D )
54	51, 53	eqeltrd 2701	. 2 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" A "> ) ` 0 ) e. D )
55	7	simp3bi 1078	. . . . . 6 |- ( F e. dom S -> A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) )
56	55	adantr 481	. . . . 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 12498	. . . . . . . . . . 11 |- ( 1..^ ( # ` F ) ) C_ ( 0..^ ( # ` F ) )
58	57	sseli 3599	. . . . . . . . . 10 |- ( i e. ( 1..^ ( # ` F ) ) -> i e. ( 0..^ ( # ` F ) ) )
59		ccatval1 13361	. . . . . . . . . 10 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 0..^ ( # ` F ) ) ) -> ( ( F ++ <" A "> ) ` i ) = ( F ` i ) )
60	58, 59	syl3an3 1361	. . . . . . . . 9 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ( ( F ++ <" A "> ) ` i ) = ( F ` i ) )
61		elfzoel2 12469	. . . . . . . . . . . . . . . 16 |- ( i e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. ZZ )
62		peano2zm 11420	. . . . . . . . . . . . . . . 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 11482	. . . . . . . . . . . . . . . 16 |- ( i e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. RR )
65	64	lem1d 10957	. . . . . . . . . . . . . . 15 |- ( i e. ( 1..^ ( # ` F ) ) -> ( ( # ` F ) - 1 ) <_ ( # ` F ) )
66		eluz2 11693	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) <-> ( ( ( # ` F ) - 1 ) e. ZZ /\ ( # ` F ) e. ZZ /\ ( ( # ` F ) - 1 ) <_ ( # ` F ) ) )
67	63, 61, 65, 66	syl3anbrc 1246	. . . . . . . . . . . . . 14 |- ( i e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) )
68		fzoss2 12496	. . . . . . . . . . . . . 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 12470	. . . . . . . . . . . . . . 15 |- ( i e. ( 1..^ ( # ` F ) ) -> i e. ZZ )
71		elfzom1b 12567	. . . . . . . . . . . . . . 15 |- ( ( i e. ZZ /\ ( # ` F ) e. ZZ ) -> ( i e. ( 1..^ ( # ` F ) ) <-> ( i - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) ) )
72	70, 61, 71	syl2anc 693	. . . . . . . . . . . . . 14 |- ( i e. ( 1..^ ( # ` F ) ) -> ( i e. ( 1..^ ( # ` F ) ) <-> ( i - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) ) )
73	72	ibi 256	. . . . . . . . . . . . 13 |- ( i e. ( 1..^ ( # ` F ) ) -> ( i - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) )
74	69, 73	sseldd 3604	. . . . . . . . . . . 12 |- ( i e. ( 1..^ ( # ` F ) ) -> ( i - 1 ) e. ( 0..^ ( # ` F ) ) )
75		ccatval1 13361	. . . . . . . . . . . 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 1361	. . . . . . . . . . 11 |- ( ( F e. Word W /\ <" A "> e. Word W /\ i e. ( 1..^ ( # ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( i - 1 ) ) = ( F ` ( i - 1 ) ) )
77	76	fveq2d 6195	. . . . . . . . . 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 5353	. . . . . . . . 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 2695	. . . . . . . 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 1265	. . . . . . 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 2985	. . . . . 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 693	. . . . 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 247	. . . 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 11353	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` F ) e. CC )
86	85	addid2d 10237	. . . . . . . 8 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 0 + ( # ` F ) ) = ( # ` F ) )
87	86	fveq2d 6195	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( 0 + ( # ` F ) ) ) = ( ( F ++ <" A "> ) ` ( # ` F ) ) )
88		1nn 11031	. . . . . . . . . . 11 |- 1 e. NN
89	29, 88	eqeltri 2697	. . . . . . . . . 10 |- ( # ` <" A "> ) e. NN
90	89	a1i 11	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` <" A "> ) e. NN )
91		lbfzo0 12507	. . . . . . . . 9 |- ( 0 e. ( 0..^ ( # ` <" A "> ) ) <-> ( # ` <" A "> ) e. NN )
92	90, 91	sylibr 224	. . . . . . . 8 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> 0 e. ( 0..^ ( # ` <" A "> ) ) )
93		ccatval3 13363	. . . . . . . 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 1326	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( 0 + ( # ` F ) ) ) = ( <" A "> ` 0 ) )
95	87, 94	eqtr3d 2658	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( # ` F ) ) = ( <" A "> ` 0 ) )
96		s1fv 13390	. . . . . . . 8 |- ( A e. ran ( T ` ( S ` F ) ) -> ( <" A "> ` 0 ) = A )
97	96	adantl 482	. . . . . . 7 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( <" A "> ` 0 ) = A )
98		fzo0end 12560	. . . . . . . . . . . 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 13361	. . . . . . . . . . 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 1326	. . . . . . . . . 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 18144	. . . . . . . . . . 11 |- ( F e. dom S -> ( S ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
103	102	adantr 481	. . . . . . . . . 10 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( S ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
104	101, 103	eqtr4d 2659	. . . . . . . . 9 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) = ( S ` F ) )
105	104	fveq2d 6195	. . . . . . . 8 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( T ` ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) ) = ( T ` ( S ` F ) ) )
106	105	rneqd 5353	. . . . . . 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 2716	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( <" A "> ` 0 ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
108	95, 107	eqeltrd 2701	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( ( F ++ <" A "> ) ` ( # ` F ) ) e. ran ( T ` ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
109		fvex 6201	. . . . . 6 |- ( # ` F ) e. _V
110		fveq2 6191	. . . . . . 7 |- ( i = ( # ` F ) -> ( ( F ++ <" A "> ) ` i ) = ( ( F ++ <" A "> ) ` ( # ` F ) ) )
111		oveq1 6657	. . . . . . . . . 10 |- ( i = ( # ` F ) -> ( i - 1 ) = ( ( # ` F ) - 1 ) )
112	111	fveq2d 6195	. . . . . . . . 9 |- ( i = ( # ` F ) -> ( ( F ++ <" A "> ) ` ( i - 1 ) ) = ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) )
113	112	fveq2d 6195	. . . . . . . 8 |- ( i = ( # ` F ) -> ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) = ( T ` ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
114	113	rneqd 5353	. . . . . . 7 |- ( i = ( # ` F ) -> ran ( T ` ( ( F ++ <" A "> ) ` ( i - 1 ) ) ) = ran ( T ` ( ( F ++ <" A "> ) ` ( ( # ` F ) - 1 ) ) ) )
115	110, 114	eleq12d 2695	. . . . . 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 4222	. . . . 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 224	. . . 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 3794	. . . 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 698	. . 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 6666	. . . . 5 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 1..^ ( # ` ( F ++ <" A "> ) ) ) = ( 1..^ ( ( # ` F ) + 1 ) ) )
121		nnuz 11723	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
122	47, 121	syl6eleq 2711	. . . . . 6 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( # ` F ) e. ( ZZ>= ` 1 ) )
123		fzosplitsn 12576	. . . . . 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 2656	. . . 4 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( 1..^ ( # ` ( F ++ <" A "> ) ) ) = ( ( 1..^ ( # ` F ) ) u. { ( # ` F ) } ) )
126	125	raleqdv 3144	. . 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 247	. 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 18143	. 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 1246	1 |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F ++ <" A "> ) e. dom S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   <.cotp 4185   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294   splice csplice 13296   <"cs2 13586   ~_FG cefg 18119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-s2 13593

This theorem is referenced by:  efgsfo  18152  efgredlemd  18157  efgrelexlemb  18163