Theorem efgsdmi 18145

Description: Property of the last link in the chain of extensions. (Contributed by Mario Carneiro, 29-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
efgsdmi	|- ( ( F e. dom S /\ ( ( # ` F ) - 1 ) e. NN ) -> ( S ` F ) e. ran ( T ` ( F ` ( ( ( # ` F ) - 1 ) - 1 ) ) ) )

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:   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 efgsdmi

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . 4 |- W = ( _I ` Word ( I X. 2o ) )
2		efgval.r	. . . 4 |- .~ = ( ~FG ` I )
3		efgval2.m	. . . 4 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
4		efgval2.t	. . . 4 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
5		efgred.d	. . . 4 |- D = ( W \ U_ x e. W ran ( T ` x ) )
6		efgred.s	. . . 4 |- 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	efgsval 18144	. . 3 |- ( F e. dom S -> ( S ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
8	7	adantr 481	. 2 |- ( ( F e. dom S /\ ( ( # ` F ) - 1 ) e. NN ) -> ( S ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
9		simpr 477	. . . . . . 7 |- ( ( F e. dom S /\ ( ( # ` F ) - 1 ) e. NN ) -> ( ( # ` F ) - 1 ) e. NN )
10		nnuz 11723	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
11	9, 10	syl6eleq 2711	. . . . . 6 |- ( ( F e. dom S /\ ( ( # ` F ) - 1 ) e. NN ) -> ( ( # ` F ) - 1 ) e. ( ZZ>= ` 1 ) )
12		eluzfz1 12348	. . . . . 6 |- ( ( ( # ` F ) - 1 ) e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... ( ( # ` F ) - 1 ) ) )
13	11, 12	syl 17	. . . . 5 |- ( ( F e. dom S /\ ( ( # ` F ) - 1 ) e. NN ) -> 1 e. ( 1 ... ( ( # ` F ) - 1 ) ) )
14	1, 2, 3, 4, 5, 6	efgsdm 18143	. . . . . . . . 9 |- ( 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 ) ) ) ) )
15	14	simp1bi 1076	. . . . . . . 8 |- ( F e. dom S -> F e. (Word W \ { (/) } ) )
16	15	adantr 481	. . . . . . 7 |- ( ( F e. dom S /\ ( ( # ` F ) - 1 ) e. NN ) -> F e. (Word W \ { (/) } ) )
17	16	eldifad 3586	. . . . . 6 |- ( ( F e. dom S /\ ( ( # ` F ) - 1 ) e. NN ) -> F e. Word W )
18		lencl 13324	. . . . . 6 |- ( F e. Word W -> ( # ` F ) e. NN0 )
19		nn0z 11400	. . . . . 6 |- ( ( # ` F ) e. NN0 -> ( # ` F ) e. ZZ )
20		fzoval 12471	. . . . . 6 |- ( ( # ` F ) e. ZZ -> ( 1..^ ( # ` F ) ) = ( 1 ... ( ( # ` F ) - 1 ) ) )
21	17, 18, 19, 20	4syl 19	. . . . 5 |- ( ( F e. dom S /\ ( ( # ` F ) - 1 ) e. NN ) -> ( 1..^ ( # ` F ) ) = ( 1 ... ( ( # ` F ) - 1 ) ) )
22	13, 21	eleqtrrd 2704	. . . 4 |- ( ( F e. dom S /\ ( ( # ` F ) - 1 ) e. NN ) -> 1 e. ( 1..^ ( # ` F ) ) )
23		fzoend 12559	. . . 4 |- ( 1 e. ( 1..^ ( # ` F ) ) -> ( ( # ` F ) - 1 ) e. ( 1..^ ( # ` F ) ) )
24	22, 23	syl 17	. . 3 |- ( ( F e. dom S /\ ( ( # ` F ) - 1 ) e. NN ) -> ( ( # ` F ) - 1 ) e. ( 1..^ ( # ` F ) ) )
25	14	simp3bi 1078	. . . 4 |- ( F e. dom S -> A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) )
26	25	adantr 481	. . 3 |- ( ( F e. dom S /\ ( ( # ` F ) - 1 ) e. NN ) -> A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) )
27		fveq2 6191	. . . . 5 |- ( i = ( ( # ` F ) - 1 ) -> ( F ` i ) = ( F ` ( ( # ` F ) - 1 ) ) )
28		oveq1 6657	. . . . . . . 8 |- ( i = ( ( # ` F ) - 1 ) -> ( i - 1 ) = ( ( ( # ` F ) - 1 ) - 1 ) )
29	28	fveq2d 6195	. . . . . . 7 |- ( i = ( ( # ` F ) - 1 ) -> ( F ` ( i - 1 ) ) = ( F ` ( ( ( # ` F ) - 1 ) - 1 ) ) )
30	29	fveq2d 6195	. . . . . 6 |- ( i = ( ( # ` F ) - 1 ) -> ( T ` ( F ` ( i - 1 ) ) ) = ( T ` ( F ` ( ( ( # ` F ) - 1 ) - 1 ) ) ) )
31	30	rneqd 5353	. . . . 5 |- ( i = ( ( # ` F ) - 1 ) -> ran ( T ` ( F ` ( i - 1 ) ) ) = ran ( T ` ( F ` ( ( ( # ` F ) - 1 ) - 1 ) ) ) )
32	27, 31	eleq12d 2695	. . . 4 |- ( i = ( ( # ` F ) - 1 ) -> ( ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) <-> ( F ` ( ( # ` F ) - 1 ) ) e. ran ( T ` ( F ` ( ( ( # ` F ) - 1 ) - 1 ) ) ) ) )
33	32	rspcv 3305	. . 3 |- ( ( ( # ` F ) - 1 ) e. ( 1..^ ( # ` F ) ) -> ( A. i e. ( 1..^ ( # ` F ) ) ( F ` i ) e. ran ( T ` ( F ` ( i - 1 ) ) ) -> ( F ` ( ( # ` F ) - 1 ) ) e. ran ( T ` ( F ` ( ( ( # ` F ) - 1 ) - 1 ) ) ) ) )
34	24, 26, 33	sylc 65	. 2 |- ( ( F e. dom S /\ ( ( # ` F ) - 1 ) e. NN ) -> ( F ` ( ( # ` F ) - 1 ) ) e. ran ( T ` ( F ` ( ( ( # ` F ) - 1 ) - 1 ) ) ) )
35	8, 34	eqeltrd 2701	1 |- ( ( F e. dom S /\ ( ( # ` F ) - 1 ) e. NN ) -> ( S ` F ) e. ran ( T ` ( F ` ( ( ( # ` F ) - 1 ) - 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = 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   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...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:  efgs1b  18149  efgredlemg  18155  efgredlemd  18157  efgredlem  18160