Theorem efgredlema 18153

Description: The reduced word that forms the base of the sequence in efgsval 18144 is uniquely determined, given the ending representation. (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 ) ) )
efgredlem.1	|- ( ph -> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( # ` ( S ` A ) ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) )
efgredlem.2	|- ( ph -> A e. dom S )
efgredlem.3	|- ( ph -> B e. dom S )
efgredlem.4	|- ( ph -> ( S ` A ) = ( S ` B ) )
efgredlem.5	|- ( ph -> -. ( A ` 0 ) = ( B ` 0 ) )

Assertion

Ref	Expression
efgredlema	|- ( ph -> ( ( ( # ` A ) - 1 ) e. NN /\ ( ( # ` B ) - 1 ) e. NN ) )

Distinct variable groups:   a, b, A   y, a, z, b   t, n, v, w, y, z   m, a, n, t, v, w, x, M, b   k, a, T, b, m, t, x   W, a, b   k, n, v, w, y, z, W, m, t, x   .~ , a, b, m, t, x, y, z   B, a, b   S, a, b   I, a, b, m, n, t, v, w, x, y, z   D, a, b, m, t

Allowed substitution hints:   ph( x, y, z, w, v, t, k, m, n, a, b)   A( x, y, z, w, v, t, k, m, n)   B( 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)   I( k)   M( y, z, k)

Proof of Theorem efgredlema

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgredlem.5	. . . . 5 |- ( ph -> -. ( A ` 0 ) = ( B ` 0 ) )
2		efgredlem.3	. . . . . . . . 9 |- ( ph -> B e. dom S )
3		efgval.w	. . . . . . . . . 10 |- W = ( _I ` Word ( I X. 2o ) )
4		efgval.r	. . . . . . . . . 10 |- .~ = ( ~FG ` I )
5		efgval2.m	. . . . . . . . . 10 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
6		efgval2.t	. . . . . . . . . 10 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
7		efgred.d	. . . . . . . . . 10 |- D = ( W \ U_ x e. W ran ( T ` x ) )
8		efgred.s	. . . . . . . . . 10 |- 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 ) ) )
9	3, 4, 5, 6, 7, 8	efgsval 18144	. . . . . . . . 9 |- ( B e. dom S -> ( S ` B ) = ( B ` ( ( # ` B ) - 1 ) ) )
10	2, 9	syl 17	. . . . . . . 8 |- ( ph -> ( S ` B ) = ( B ` ( ( # ` B ) - 1 ) ) )
11		efgredlem.4	. . . . . . . . 9 |- ( ph -> ( S ` A ) = ( S ` B ) )
12		efgredlem.2	. . . . . . . . . 10 |- ( ph -> A e. dom S )
13	3, 4, 5, 6, 7, 8	efgsval 18144	. . . . . . . . . 10 |- ( A e. dom S -> ( S ` A ) = ( A ` ( ( # ` A ) - 1 ) ) )
14	12, 13	syl 17	. . . . . . . . 9 |- ( ph -> ( S ` A ) = ( A ` ( ( # ` A ) - 1 ) ) )
15	11, 14	eqtr3d 2658	. . . . . . . 8 |- ( ph -> ( S ` B ) = ( A ` ( ( # ` A ) - 1 ) ) )
16	10, 15	eqtr3d 2658	. . . . . . 7 |- ( ph -> ( B ` ( ( # ` B ) - 1 ) ) = ( A ` ( ( # ` A ) - 1 ) ) )
17		oveq1 6657	. . . . . . . . 9 |- ( ( # ` A ) = 1 -> ( ( # ` A ) - 1 ) = ( 1 - 1 ) )
18		1m1e0 11089	. . . . . . . . 9 |- ( 1 - 1 ) = 0
19	17, 18	syl6eq 2672	. . . . . . . 8 |- ( ( # ` A ) = 1 -> ( ( # ` A ) - 1 ) = 0 )
20	19	fveq2d 6195	. . . . . . 7 |- ( ( # ` A ) = 1 -> ( A ` ( ( # ` A ) - 1 ) ) = ( A ` 0 ) )
21	16, 20	sylan9eq 2676	. . . . . 6 |- ( ( ph /\ ( # ` A ) = 1 ) -> ( B ` ( ( # ` B ) - 1 ) ) = ( A ` 0 ) )
22	11	eleq1d 2686	. . . . . . . . 9 |- ( ph -> ( ( S ` A ) e. D <-> ( S ` B ) e. D ) )
23	3, 4, 5, 6, 7, 8	efgs1b 18149	. . . . . . . . . 10 |- ( A e. dom S -> ( ( S ` A ) e. D <-> ( # ` A ) = 1 ) )
24	12, 23	syl 17	. . . . . . . . 9 |- ( ph -> ( ( S ` A ) e. D <-> ( # ` A ) = 1 ) )
25	3, 4, 5, 6, 7, 8	efgs1b 18149	. . . . . . . . . 10 |- ( B e. dom S -> ( ( S ` B ) e. D <-> ( # ` B ) = 1 ) )
26	2, 25	syl 17	. . . . . . . . 9 |- ( ph -> ( ( S ` B ) e. D <-> ( # ` B ) = 1 ) )
27	22, 24, 26	3bitr3d 298	. . . . . . . 8 |- ( ph -> ( ( # ` A ) = 1 <-> ( # ` B ) = 1 ) )
28	27	biimpa 501	. . . . . . 7 |- ( ( ph /\ ( # ` A ) = 1 ) -> ( # ` B ) = 1 )
29		oveq1 6657	. . . . . . . . 9 |- ( ( # ` B ) = 1 -> ( ( # ` B ) - 1 ) = ( 1 - 1 ) )
30	29, 18	syl6eq 2672	. . . . . . . 8 |- ( ( # ` B ) = 1 -> ( ( # ` B ) - 1 ) = 0 )
31	30	fveq2d 6195	. . . . . . 7 |- ( ( # ` B ) = 1 -> ( B ` ( ( # ` B ) - 1 ) ) = ( B ` 0 ) )
32	28, 31	syl 17	. . . . . 6 |- ( ( ph /\ ( # ` A ) = 1 ) -> ( B ` ( ( # ` B ) - 1 ) ) = ( B ` 0 ) )
33	21, 32	eqtr3d 2658	. . . . 5 |- ( ( ph /\ ( # ` A ) = 1 ) -> ( A ` 0 ) = ( B ` 0 ) )
34	1, 33	mtand 691	. . . 4 |- ( ph -> -. ( # ` A ) = 1 )
35	3, 4, 5, 6, 7, 8	efgsdm 18143	. . . . . . . 8 |- ( A e. dom S <-> ( A e. (Word W \ { (/) } ) /\ ( A ` 0 ) e. D /\ A. u e. ( 1..^ ( # ` A ) ) ( A ` u ) e. ran ( T ` ( A ` ( u - 1 ) ) ) ) )
36	35	simp1bi 1076	. . . . . . 7 |- ( A e. dom S -> A e. (Word W \ { (/) } ) )
37		eldifsn 4317	. . . . . . . 8 |- ( A e. (Word W \ { (/) } ) <-> ( A e. Word W /\ A =/= (/) ) )
38		lennncl 13325	. . . . . . . 8 |- ( ( A e. Word W /\ A =/= (/) ) -> ( # ` A ) e. NN )
39	37, 38	sylbi 207	. . . . . . 7 |- ( A e. (Word W \ { (/) } ) -> ( # ` A ) e. NN )
40	12, 36, 39	3syl 18	. . . . . 6 |- ( ph -> ( # ` A ) e. NN )
41		elnn1uz2 11765	. . . . . 6 |- ( ( # ` A ) e. NN <-> ( ( # ` A ) = 1 \/ ( # ` A ) e. ( ZZ>= ` 2 ) ) )
42	40, 41	sylib 208	. . . . 5 |- ( ph -> ( ( # ` A ) = 1 \/ ( # ` A ) e. ( ZZ>= ` 2 ) ) )
43	42	ord 392	. . . 4 |- ( ph -> ( -. ( # ` A ) = 1 -> ( # ` A ) e. ( ZZ>= ` 2 ) ) )
44	34, 43	mpd 15	. . 3 |- ( ph -> ( # ` A ) e. ( ZZ>= ` 2 ) )
45		uz2m1nn 11763	. . 3 |- ( ( # ` A ) e. ( ZZ>= ` 2 ) -> ( ( # ` A ) - 1 ) e. NN )
46	44, 45	syl 17	. 2 |- ( ph -> ( ( # ` A ) - 1 ) e. NN )
47	34, 27	mtbid 314	. . . 4 |- ( ph -> -. ( # ` B ) = 1 )
48	3, 4, 5, 6, 7, 8	efgsdm 18143	. . . . . . . 8 |- ( B e. dom S <-> ( B e. (Word W \ { (/) } ) /\ ( B ` 0 ) e. D /\ A. u e. ( 1..^ ( # ` B ) ) ( B ` u ) e. ran ( T ` ( B ` ( u - 1 ) ) ) ) )
49	48	simp1bi 1076	. . . . . . 7 |- ( B e. dom S -> B e. (Word W \ { (/) } ) )
50		eldifsn 4317	. . . . . . . 8 |- ( B e. (Word W \ { (/) } ) <-> ( B e. Word W /\ B =/= (/) ) )
51		lennncl 13325	. . . . . . . 8 |- ( ( B e. Word W /\ B =/= (/) ) -> ( # ` B ) e. NN )
52	50, 51	sylbi 207	. . . . . . 7 |- ( B e. (Word W \ { (/) } ) -> ( # ` B ) e. NN )
53	2, 49, 52	3syl 18	. . . . . 6 |- ( ph -> ( # ` B ) e. NN )
54		elnn1uz2 11765	. . . . . 6 |- ( ( # ` B ) e. NN <-> ( ( # ` B ) = 1 \/ ( # ` B ) e. ( ZZ>= ` 2 ) ) )
55	53, 54	sylib 208	. . . . 5 |- ( ph -> ( ( # ` B ) = 1 \/ ( # ` B ) e. ( ZZ>= ` 2 ) ) )
56	55	ord 392	. . . 4 |- ( ph -> ( -. ( # ` B ) = 1 -> ( # ` B ) e. ( ZZ>= ` 2 ) ) )
57	47, 56	mpd 15	. . 3 |- ( ph -> ( # ` B ) e. ( ZZ>= ` 2 ) )
58		uz2m1nn 11763	. . 3 |- ( ( # ` B ) e. ( ZZ>= ` 2 ) -> ( ( # ` B ) - 1 ) e. NN )
59	57, 58	syl 17	. 2 |- ( ph -> ( ( # ` B ) - 1 ) e. NN )
60	46, 59	jca 554	1 |- ( ph -> ( ( ( # ` A ) - 1 ) e. NN /\ ( ( # ` B ) - 1 ) e. NN ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   \ cdif 3571   (/)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   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   0cc0 9936   1c1 9937   < clt 10074   - cmin 10266   NNcn 11020   2c2 11070   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-2 11079  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:  efgredlemf  18154  efgredlemg  18155  efgredlemd  18157  efgredlemc  18158  efgredlem  18160