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Theorem pmtrdifwrdel 15989
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. (Contributed by AV, 15-Jan-2019.)
Hypotheses
Ref Expression
pmtrdifel.t  |-  T  =  ran  (pmTrsp `  ( N  \  { K }
) )
pmtrdifel.r  |-  R  =  ran  (pmTrsp `  N
)
Assertion
Ref Expression
pmtrdifwrdel  |-  A. w  e. Word  T E. u  e. Word  R ( ( # `  w )  =  (
# `  u )  /\  A. i  e.  ( 0..^ ( # `  w
) ) 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
Allowed substitution hints:    R( x, w)    T( w, u)    K( x, w, i)    N( w)

Proof of Theorem pmtrdifwrdel
Dummy variables  j  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pmtrdifel.t . . . 4  |-  T  =  ran  (pmTrsp `  ( N  \  { K }
) )
2 pmtrdifel.r . . . 4  |-  R  =  ran  (pmTrsp `  N
)
3 fveq2 5689 . . . . . . . 8  |-  ( j  =  n  ->  (
w `  j )  =  ( w `  n ) )
43difeq1d 3471 . . . . . . 7  |-  ( j  =  n  ->  (
( w `  j
)  \  _I  )  =  ( ( w `
 n )  \  _I  ) )
54dmeqd 5040 . . . . . 6  |-  ( j  =  n  ->  dom  ( ( w `  j )  \  _I  )  =  dom  ( ( w `  n ) 
\  _I  ) )
65fveq2d 5693 . . . . 5  |-  ( j  =  n  ->  (
(pmTrsp `  N ) `  dom  ( ( w `
 j )  \  _I  ) )  =  ( (pmTrsp `  N ) `  dom  ( ( w `
 n )  \  _I  ) ) )
76cbvmptv 4381 . . . 4  |-  ( j  e.  ( 0..^ (
# `  w )
)  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) )  =  ( n  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  n )  \  _I  ) ) )
81, 2, 7pmtrdifwrdellem1 15985 . . 3  |-  ( w  e. Word  T  ->  (
j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) )  e. Word  R )
91, 2, 7pmtrdifwrdellem2 15986 . . 3  |-  ( w  e. Word  T  ->  ( # `
 w )  =  ( # `  (
j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) ) ) )
101, 2, 7pmtrdifwrdellem3 15987 . . 3  |-  ( 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
) )
11 fveq2 5689 . . . . . 6  |-  ( u  =  ( j  e.  ( 0..^ ( # `  w ) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `
 j )  \  _I  ) ) )  -> 
( # `  u )  =  ( # `  (
j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) ) ) )
1211eqeq2d 2452 . . . . 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  ) ) ) ) ) )
13 fveq1 5688 . . . . . . . . 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 ) )
1413fveq1d 5691 . . . . . . . 8  |-  ( 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
) )
1514eqeq2d 2452 . . . . . . 7  |-  ( 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
) ) )
1615ralbidv 2733 . . . . . 6  |-  ( 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
) ) )
1716ralbidv 2733 . . . . 5  |-  ( u  =  ( j  e.  ( 0..^ ( # `  w ) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `
 j )  \  _I  ) ) )  -> 
( A. i  e.  ( 0..^ ( # `  w ) ) A. x  e.  ( N  \  { K } ) ( ( w `  i ) `  x
)  =  ( ( u `  i ) `
 x )  <->  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
) ) )
1812, 17anbi12d 710 . . . 4  |-  ( u  =  ( j  e.  ( 0..^ ( # `  w ) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `
 j )  \  _I  ) ) )  -> 
( ( ( # `  w )  =  (
# `  u )  /\  A. i  e.  ( 0..^ ( # `  w
) ) 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 ) ) A. x  e.  ( N  \  { K } ) ( ( w `  i ) `  x
)  =  ( ( ( j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) ) `  i ) `  x
) ) ) )
1918rspcev 3071 . . 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 ) ) 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
) ) A. x  e.  ( N  \  { K } ) ( ( w `  i ) `
 x )  =  ( ( u `  i ) `  x
) ) )
208, 9, 10, 19syl12anc 1216 . 2  |-  ( w  e. Word  T  ->  E. u  e. Word  R ( ( # `  w )  =  (
# `  u )  /\  A. i  e.  ( 0..^ ( # `  w
) ) A. x  e.  ( N  \  { K } ) ( ( w `  i ) `
 x )  =  ( ( u `  i ) `  x
) ) )
2120rgen 2779 1  |-  A. w  e. Word  T E. u  e. Word  R ( ( # `  w )  =  (
# `  u )  /\  A. i  e.  ( 0..^ ( # `  w
) ) A. x  e.  ( N  \  { K } ) ( ( w `  i ) `
 x )  =  ( ( u `  i ) `  x
) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2713   E.wrex 2714    \ cdif 3323   {csn 3875    e. cmpt 4348    _I cid 4629   dom cdm 4838   ran crn 4839   ` cfv 5416  (class class class)co 6089   0cc0 9280  ..^cfzo 11546   #chash 12101  Word cword 12219  pmTrspcpmtr 15945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-hash 12102  df-word 12227  df-pmtr 15946
This theorem is referenced by: (None)
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