Theorem efgsdm 18143

Description: Elementhood in the domain of S, the set of sequences of extensions starting at an irreducible word. (Contributed by Mario Carneiro, 27-Sep-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
efgsdm	|- ( 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 ) ) ) ) )

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

Allowed substitution hints:   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 efgsdm

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6190	. . . . 5 |- ( f = F -> ( f ` 0 ) = ( F ` 0 ) )
2	1	eleq1d 2686	. . . 4 |- ( f = F -> ( ( f ` 0 ) e. D <-> ( F ` 0 ) e. D ) )
3		fveq2 6191	. . . . . 6 |- ( f = F -> ( # ` f ) = ( # ` F ) )
4	3	oveq2d 6666	. . . . 5 |- ( f = F -> ( 1..^ ( # ` f ) ) = ( 1..^ ( # ` F ) ) )
5		fveq1 6190	. . . . . 6 |- ( f = F -> ( f ` i ) = ( F ` i ) )
6		fveq1 6190	. . . . . . . 8 |- ( f = F -> ( f ` ( i - 1 ) ) = ( F ` ( i - 1 ) ) )
7	6	fveq2d 6195	. . . . . . 7 |- ( f = F -> ( T ` ( f ` ( i - 1 ) ) ) = ( T ` ( F ` ( i - 1 ) ) ) )
8	7	rneqd 5353	. . . . . 6 |- ( f = F -> ran ( T ` ( f ` ( i - 1 ) ) ) = ran ( T ` ( F ` ( i - 1 ) ) ) )
9	5, 8	eleq12d 2695	. . . . 5 |- ( f = F -> ( ( f ` i ) e. ran ( T ` ( f ` ( i - 1 ) ) ) <-> ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) ) )
10	4, 9	raleqbidv 3152	. . . 4 |- ( f = F -> ( A. i e. ( 1..^ ( # ` f ) ) ( f ` i ) e. ran ( T ` ( f ` ( i - 1 ) ) ) <-> A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) ) )
11	2, 10	anbi12d 747	. . 3 |- ( f = F -> ( ( ( f ` 0 ) e. D /\ A. i e. ( 1..^ ( # ` f ) ) ( f ` i ) e. ran ( T ` ( f ` ( i - 1 ) ) ) ) <-> ( ( F ` 0 ) e. D /\ A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) ) ) )
12		efgval.w	. . . . . 6 |- W = ( _I ` Word ( I X. 2o ) )
13		efgval.r	. . . . . 6 |- .~ = ( ~FG ` I )
14		efgval2.m	. . . . . 6 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
15		efgval2.t	. . . . . 6 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
16		efgred.d	. . . . . 6 |- D = ( W \ U_ x e. W ran ( T ` x ) )
17		efgred.s	. . . . . 6 |- 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 ) ) )
18	12, 13, 14, 15, 16, 17	efgsf 18142	. . . . 5 |- S : { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } --> W
19	18	fdmi 6052	. . . 4 |- dom S = { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) }
20		fveq1 6190	. . . . . . 7 |- ( t = f -> ( t ` 0 ) = ( f ` 0 ) )
21	20	eleq1d 2686	. . . . . 6 |- ( t = f -> ( ( t ` 0 ) e. D <-> ( f ` 0 ) e. D ) )
22		fveq2 6191	. . . . . . . . 9 |- ( k = i -> ( t ` k ) = ( t ` i ) )
23		oveq1 6657	. . . . . . . . . . . 12 |- ( k = i -> ( k - 1 ) = ( i - 1 ) )
24	23	fveq2d 6195	. . . . . . . . . . 11 |- ( k = i -> ( t ` ( k - 1 ) ) = ( t ` ( i - 1 ) ) )
25	24	fveq2d 6195	. . . . . . . . . 10 |- ( k = i -> ( T ` ( t ` ( k - 1 ) ) ) = ( T ` ( t ` ( i - 1 ) ) ) )
26	25	rneqd 5353	. . . . . . . . 9 |- ( k = i -> ran ( T ` ( t ` ( k - 1 ) ) ) = ran ( T ` ( t ` ( i - 1 ) ) ) )
27	22, 26	eleq12d 2695	. . . . . . . 8 |- ( k = i -> ( ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) <-> ( t ` i ) e. ran ( T ` ( t ` ( i - 1 ) ) ) ) )
28	27	cbvralv 3171	. . . . . . 7 |- ( A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) <-> A. i e. ( 1..^ ( # ` t ) ) ( t ` i ) e. ran ( T ` ( t ` ( i - 1 ) ) ) )
29		fveq2 6191	. . . . . . . . 9 |- ( t = f -> ( # ` t ) = ( # ` f ) )
30	29	oveq2d 6666	. . . . . . . 8 |- ( t = f -> ( 1..^ ( # ` t ) ) = ( 1..^ ( # ` f ) ) )
31		fveq1 6190	. . . . . . . . 9 |- ( t = f -> ( t ` i ) = ( f ` i ) )
32		fveq1 6190	. . . . . . . . . . 11 |- ( t = f -> ( t ` ( i - 1 ) ) = ( f ` ( i - 1 ) ) )
33	32	fveq2d 6195	. . . . . . . . . 10 |- ( t = f -> ( T ` ( t ` ( i - 1 ) ) ) = ( T ` ( f ` ( i - 1 ) ) ) )
34	33	rneqd 5353	. . . . . . . . 9 |- ( t = f -> ran ( T ` ( t ` ( i - 1 ) ) ) = ran ( T ` ( f ` ( i - 1 ) ) ) )
35	31, 34	eleq12d 2695	. . . . . . . 8 |- ( t = f -> ( ( t ` i ) e. ran ( T ` ( t ` ( i - 1 ) ) ) <-> ( f ` i ) e. ran ( T ` ( f ` ( i - 1 ) ) ) ) )
36	30, 35	raleqbidv 3152	. . . . . . 7 |- ( t = f -> ( A. i e. ( 1..^ ( # ` t ) ) ( t ` i ) e. ran ( T ` ( t ` ( i - 1 ) ) ) <-> A. i e. ( 1..^ ( # ` f ) ) ( f ` i ) e. ran ( T ` ( f ` ( i - 1 ) ) ) ) )
37	28, 36	syl5bb 272	. . . . . 6 |- ( t = f -> ( A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) <-> A. i e. ( 1..^ ( # ` f ) ) ( f ` i ) e. ran ( T ` ( f ` ( i - 1 ) ) ) ) )
38	21, 37	anbi12d 747	. . . . 5 |- ( t = f -> ( ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) <-> ( ( f ` 0 ) e. D /\ A. i e. ( 1..^ ( # ` f ) ) ( f ` i ) e. ran ( T ` ( f ` ( i - 1 ) ) ) ) ) )
39	38	cbvrabv 3199	. . . 4 |- { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } = { f e. (Word W \ { (/) } ) | ( ( f ` 0 ) e. D /\ A. i e. ( 1..^ ( # ` f ) ) ( f ` i ) e. ran ( T ` ( f ` ( i - 1 ) ) ) ) }
40	19, 39	eqtri 2644	. . 3 |- dom S = { f e. (Word W \ { (/) } ) | ( ( f ` 0 ) e. D /\ A. i e. ( 1..^ ( # ` f ) ) ( f ` i ) e. ran ( T ` ( f ` ( i - 1 ) ) ) ) }
41	11, 40	elrab2 3366	. 2 |- ( 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 ) ) ) ) ) )
42		3anass 1042	. 2 |- ( ( F e. (Word W \ { (/) } ) /\ ( F ` 0 ) e. D /\ A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) ) <-> ( F e. (Word W \ { (/) } ) /\ ( ( F ` 0 ) e. D /\ A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) ) ) )
43	41, 42	bitr4i 267	1 |- ( 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 ) ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   (/)c0 3915   {csn 4177   <.cop 4183   <.cotp 4185   U_ciun 4520   |-> cmpt 4729   _I cid 5023   X. cxp 5112   dom cdm 5114   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   0cc0 9936   1c1 9937   - cmin 10266   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   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-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-oadd 7564  df-er 7742  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

This theorem is referenced by:  efgsdmi  18145  efgsrel  18147  efgs1  18148  efgs1b  18149  efgsp1  18150  efgsres  18151  efgsfo  18152  efgredlema  18153  efgredlemf  18154  efgredlemd  18157  efgredlemc  18158  efgredlem  18160  efgrelexlemb  18163  efgredeu  18165  efgred2  18166