Theorem pmtrdifwrdel2 17906

Description: A sequence of transpositions of elements of a set without a special element corresponds to a sequence of transpositions of elements of the set not moving the special element. (Contributed by AV, 31-Jan-2019.)

Hypotheses

Ref	Expression
pmtrdifel.t	|- T = ran (pmTrsp ` ( N \ { K } ) )
pmtrdifel.r	|- R = ran (pmTrsp ` N )

Assertion

Ref	Expression
pmtrdifwrdel2	|- ( K e. N -> A. w e. Word T E. u e. Word R ( ( # ` w ) = ( # ` u ) /\ A. i e. ( 0..^ ( # ` w ) ) ( ( ( u ` i ) ` K ) = K /\ A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( u ` i ) ` x ) ) ) )

Distinct variable groups:   x, N   x, T   u, K   i, N, u   T, i   R, i, u   w, i, x, u   i, K, w   w, N

Allowed substitution hints:   R( x, w)   T( w, u)   K( x)

Proof of Theorem pmtrdifwrdel2

Dummy variables j n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmtrdifel.t	. . . . 5 |- T = ran (pmTrsp ` ( N \ { K } ) )
2		pmtrdifel.r	. . . . 5 |- R = ran (pmTrsp ` N )
3		fveq2 6191	. . . . . . . . 9 |- ( j = n -> ( w ` j ) = ( w ` n ) )
4	3	difeq1d 3727	. . . . . . . 8 |- ( j = n -> ( ( w ` j ) \ _I ) = ( ( w ` n ) \ _I ) )
5	4	dmeqd 5326	. . . . . . 7 |- ( j = n -> dom ( ( w ` j ) \ _I ) = dom ( ( w ` n ) \ _I ) )
6	5	fveq2d 6195	. . . . . 6 |- ( j = n -> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) = ( (pmTrsp ` N ) ` dom ( ( w ` n ) \ _I ) ) )
7	6	cbvmptv 4750	. . . . 5 |- ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) = ( n e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` n ) \ _I ) ) )
8	1, 2, 7	pmtrdifwrdellem1 17901	. . . 4 |- ( w e. Word T -> ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) e. Word R )
9	8	adantl 482	. . 3 |- ( ( K e. N /\ w e. Word T ) -> ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) e. Word R )
10	1, 2, 7	pmtrdifwrdellem2 17902	. . . 4 |- ( w e. Word T -> ( # ` w ) = ( # ` ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ) )
11	10	adantl 482	. . 3 |- ( ( K e. N /\ w e. Word T ) -> ( # ` w ) = ( # ` ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ) )
12	1, 2, 7	pmtrdifwrdel2lem1 17904	. . . . 5 |- ( ( w e. Word T /\ K e. N ) -> A. i e. ( 0..^ ( # ` w ) ) ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` K ) = K )
13	12	ancoms 469	. . . 4 |- ( ( K e. N /\ w e. Word T ) -> A. i e. ( 0..^ ( # ` w ) ) ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` K ) = K )
14	1, 2, 7	pmtrdifwrdellem3 17903	. . . . 5 |- ( w e. Word T -> A. i e. ( 0..^ ( # ` w ) ) A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` x ) )
15	14	adantl 482	. . . 4 |- ( ( K e. N /\ w e. Word T ) -> A. i e. ( 0..^ ( # ` w ) ) A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` x ) )
16		r19.26 3064	. . . 4 |- ( A. i e. ( 0..^ ( # ` w ) ) ( ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` K ) = K /\ A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` x ) ) <-> ( A. i e. ( 0..^ ( # ` w ) ) ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` K ) = K /\ A. i e. ( 0..^ ( # ` w ) ) A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` x ) ) )
17	13, 15, 16	sylanbrc 698	. . 3 |- ( ( K e. N /\ w e. Word T ) -> A. i e. ( 0..^ ( # ` w ) ) ( ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` K ) = K /\ A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` x ) ) )
18		fveq2 6191	. . . . . 6 |- ( u = ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) -> ( # ` u ) = ( # ` ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ) )
19	18	eqeq2d 2632	. . . . 5 |- ( u = ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) -> ( ( # ` w ) = ( # ` u ) <-> ( # ` w ) = ( # ` ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ) ) )
20		fveq1 6190	. . . . . . . . 9 |- ( u = ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) -> ( u ` i ) = ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) )
21	20	fveq1d 6193	. . . . . . . 8 |- ( u = ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) -> ( ( u ` i ) ` K ) = ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` K ) )
22	21	eqeq1d 2624	. . . . . . 7 |- ( u = ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) -> ( ( ( u ` i ) ` K ) = K <-> ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` K ) = K ) )
23	20	fveq1d 6193	. . . . . . . . 9 |- ( u = ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) -> ( ( u ` i ) ` x ) = ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` x ) )
24	23	eqeq2d 2632	. . . . . . . 8 |- ( u = ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) -> ( ( ( w ` i ) ` x ) = ( ( u ` i ) ` x ) <-> ( ( w ` i ) ` x ) = ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` x ) ) )
25	24	ralbidv 2986	. . . . . . 7 |- ( u = ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) -> ( A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( u ` i ) ` x ) <-> A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` x ) ) )
26	22, 25	anbi12d 747	. . . . . 6 |- ( u = ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) -> ( ( ( ( u ` i ) ` K ) = K /\ A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( u ` i ) ` x ) ) <-> ( ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` K ) = K /\ A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` x ) ) ) )
27	26	ralbidv 2986	. . . . 5 |- ( u = ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) -> ( A. i e. ( 0..^ ( # ` w ) ) ( ( ( u ` i ) ` K ) = K /\ A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( u ` i ) ` x ) ) <-> A. i e. ( 0..^ ( # ` w ) ) ( ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` K ) = K /\ A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` x ) ) ) )
28	19, 27	anbi12d 747	. . . 4 |- ( u = ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) -> ( ( ( # ` w ) = ( # ` u ) /\ A. i e. ( 0..^ ( # ` w ) ) ( ( ( u ` i ) ` K ) = K /\ A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( u ` i ) ` x ) ) ) <-> ( ( # ` w ) = ( # ` ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ) /\ A. i e. ( 0..^ ( # ` w ) ) ( ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` K ) = K /\ A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` x ) ) ) ) )
29	28	rspcev 3309	. . 3 |- ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) e. Word R /\ ( ( # ` w ) = ( # ` ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ) /\ A. i e. ( 0..^ ( # ` w ) ) ( ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` K ) = K /\ A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( ( j e. ( 0..^ ( # ` w ) ) |-> ( (pmTrsp ` N ) ` dom ( ( w ` j ) \ _I ) ) ) ` i ) ` x ) ) ) ) -> E. u e. Word R ( ( # ` w ) = ( # ` u ) /\ A. i e. ( 0..^ ( # ` w ) ) ( ( ( u ` i ) ` K ) = K /\ A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( u ` i ) ` x ) ) ) )
30	9, 11, 17, 29	syl12anc 1324	. 2 |- ( ( K e. N /\ w e. Word T ) -> E. u e. Word R ( ( # ` w ) = ( # ` u ) /\ A. i e. ( 0..^ ( # ` w ) ) ( ( ( u ` i ) ` K ) = K /\ A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( u ` i ) ` x ) ) ) )
31	30	ralrimiva 2966	1 |- ( K e. N -> A. w e. Word T E. u e. Word R ( ( # ` w ) = ( # ` u ) /\ A. i e. ( 0..^ ( # ` w ) ) ( ( ( u ` i ) ` K ) = K /\ A. x e. ( N \ { K } ) ( ( w ` i ) ` x ) = ( ( u ` i ) ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   \ cdif 3571   {csn 4177   |-> cmpt 4729   _I cid 5023   dom cdm 5114   ran crn 5115   ` cfv 5888  (class class class)co 6650   0cc0 9936  ..^cfzo 12465   #chash 13117  Word cword 13291  pmTrspcpmtr 17861

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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-tset 15960  df-symg 17798  df-pmtr 17862

This theorem is referenced by:  psgndiflemA  19947