Theorem efgredlemb 18159

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, 30-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 ) ) )
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 ) )
efgredlemb.k	|- K = ( ( ( # ` A ) - 1 ) - 1 )
efgredlemb.l	|- L = ( ( ( # ` B ) - 1 ) - 1 )
efgredlemb.p	|- ( ph -> P e. ( 0 ... ( # ` ( A ` K ) ) ) )
efgredlemb.q	|- ( ph -> Q e. ( 0 ... ( # ` ( B ` L ) ) ) )
efgredlemb.u	|- ( ph -> U e. ( I X. 2o ) )
efgredlemb.v	|- ( ph -> V e. ( I X. 2o ) )
efgredlemb.6	|- ( ph -> ( S ` A ) = ( P ( T ` ( A ` K ) ) U ) )
efgredlemb.7	|- ( ph -> ( S ` B ) = ( Q ( T ` ( B ` L ) ) V ) )
efgredlemb.8	|- ( ph -> -. ( A ` K ) = ( B ` L ) )

Assertion

Ref	Expression
efgredlemb	|- -. ph

Distinct variable groups:   a, b, A   y, a, z, b   L, a, b   K, a, b   t, n, v, w, y, z, P   m, a, n, t, v, w, x, M, b   U, n, v, w, y, z   k, a, T, b, m, t, x   n, V, v, w, y, z   Q, n, t, v, w, y, z   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)   P( x, k, m, a, b)   Q( x, k, m, a, b)   .~ ( w, v, k, n)   S( x, y, z, w, v, t, k, m, n)   T( y, z, w, v, n)   U( x, t, k, m, a, b)   I( k)   K( x, y, z, w, v, t, k, m, n)   L( x, y, z, w, v, t, k, m, n)   M( y, z, k)   V( x, t, k, m, a, b)

Proof of Theorem efgredlemb

Step	Hyp	Ref	Expression
1		efgval.w	. . . . 5 |- W = ( _I ` Word ( I X. 2o ) )
2		efgval.r	. . . . 5 |- .~ = ( ~FG ` I )
3		efgval2.m	. . . . 5 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
4		efgval2.t	. . . . 5 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
5		efgred.d	. . . . 5 |- D = ( W \ U_ x e. W ran ( T ` x ) )
6		efgred.s	. . . . 5 |- 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		efgredlem.1	. . . . . 6 |- ( ph -> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( # ` ( S ` A ) ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) )
8		efgredlem.4	. . . . . . 7 |- ( ph -> ( S ` A ) = ( S ` B ) )
9		fveq2 6191	. . . . . . . . . 10 |- ( ( S ` A ) = ( S ` B ) -> ( # ` ( S ` A ) ) = ( # ` ( S ` B ) ) )
10	9	breq2d 4665	. . . . . . . . 9 |- ( ( S ` A ) = ( S ` B ) -> ( ( # ` ( S ` a ) ) < ( # ` ( S ` A ) ) <-> ( # ` ( S ` a ) ) < ( # ` ( S ` B ) ) ) )
11	10	imbi1d 331	. . . . . . . 8 |- ( ( S ` A ) = ( S ` B ) -> ( ( ( # ` ( S ` a ) ) < ( # ` ( S ` A ) ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> ( ( # ` ( S ` a ) ) < ( # ` ( S ` B ) ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
12	11	2ralbidv 2989	. . . . . . 7 |- ( ( S ` A ) = ( S ` B ) -> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( # ` ( S ` A ) ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( # ` ( S ` B ) ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
13	8, 12	syl 17	. . . . . 6 |- ( ph -> ( A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( # ` ( S ` A ) ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) <-> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( # ` ( S ` B ) ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) ) )
14	7, 13	mpbid 222	. . . . 5 |- ( ph -> A. a e. dom S A. b e. dom S ( ( # ` ( S ` a ) ) < ( # ` ( S ` B ) ) -> ( ( S ` a ) = ( S ` b ) -> ( a ` 0 ) = ( b ` 0 ) ) ) )
15		efgredlem.3	. . . . 5 |- ( ph -> B e. dom S )
16		efgredlem.2	. . . . 5 |- ( ph -> A e. dom S )
17	8	eqcomd 2628	. . . . 5 |- ( ph -> ( S ` B ) = ( S ` A ) )
18		efgredlem.5	. . . . . 6 |- ( ph -> -. ( A ` 0 ) = ( B ` 0 ) )
19		eqcom 2629	. . . . . 6 |- ( ( A ` 0 ) = ( B ` 0 ) <-> ( B ` 0 ) = ( A ` 0 ) )
20	18, 19	sylnib 318	. . . . 5 |- ( ph -> -. ( B ` 0 ) = ( A ` 0 ) )
21		efgredlemb.l	. . . . 5 |- L = ( ( ( # ` B ) - 1 ) - 1 )
22		efgredlemb.k	. . . . 5 |- K = ( ( ( # ` A ) - 1 ) - 1 )
23		efgredlemb.q	. . . . 5 |- ( ph -> Q e. ( 0 ... ( # ` ( B ` L ) ) ) )
24		efgredlemb.p	. . . . 5 |- ( ph -> P e. ( 0 ... ( # ` ( A ` K ) ) ) )
25		efgredlemb.v	. . . . 5 |- ( ph -> V e. ( I X. 2o ) )
26		efgredlemb.u	. . . . 5 |- ( ph -> U e. ( I X. 2o ) )
27		efgredlemb.7	. . . . 5 |- ( ph -> ( S ` B ) = ( Q ( T ` ( B ` L ) ) V ) )
28		efgredlemb.6	. . . . 5 |- ( ph -> ( S ` A ) = ( P ( T ` ( A ` K ) ) U ) )
29		efgredlemb.8	. . . . . 6 |- ( ph -> -. ( A ` K ) = ( B ` L ) )
30		eqcom 2629	. . . . . 6 |- ( ( A ` K ) = ( B ` L ) <-> ( B ` L ) = ( A ` K ) )
31	29, 30	sylnib 318	. . . . 5 |- ( ph -> -. ( B ` L ) = ( A ` K ) )
32	1, 2, 3, 4, 5, 6, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 31	efgredlemc 18158	. . . 4 |- ( ph -> ( Q e. ( ZZ>= ` P ) -> ( B ` 0 ) = ( A ` 0 ) ) )
33	32, 19	syl6ibr 242	. . 3 |- ( ph -> ( Q e. ( ZZ>= ` P ) -> ( A ` 0 ) = ( B ` 0 ) ) )
34	1, 2, 3, 4, 5, 6, 7, 16, 15, 8, 18, 22, 21, 24, 23, 26, 25, 28, 27, 29	efgredlemc 18158	. . 3 |- ( ph -> ( P e. ( ZZ>= ` Q ) -> ( A ` 0 ) = ( B ` 0 ) ) )
35		elfzelz 12342	. . . . 5 |- ( P e. ( 0 ... ( # ` ( A ` K ) ) ) -> P e. ZZ )
36	24, 35	syl 17	. . . 4 |- ( ph -> P e. ZZ )
37		elfzelz 12342	. . . . 5 |- ( Q e. ( 0 ... ( # ` ( B ` L ) ) ) -> Q e. ZZ )
38	23, 37	syl 17	. . . 4 |- ( ph -> Q e. ZZ )
39		uztric 11709	. . . 4 |- ( ( P e. ZZ /\ Q e. ZZ ) -> ( Q e. ( ZZ>= ` P ) \/ P e. ( ZZ>= ` Q ) ) )
40	36, 38, 39	syl2anc 693	. . 3 |- ( ph -> ( Q e. ( ZZ>= ` P ) \/ P e. ( ZZ>= ` Q ) ) )
41	33, 34, 40	mpjaod 396	. 2 |- ( ph -> ( A ` 0 ) = ( B ` 0 ) )
42	41, 18	pm2.65i 185	1 |- -. ph

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ 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   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   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-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-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  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:  efgredlem  18160