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Theorem pmtrdifwrdel 17077
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 5881 . . . . . . . 8  |-  ( j  =  n  ->  (
w `  j )  =  ( w `  n ) )
43difeq1d 3588 . . . . . . 7  |-  ( j  =  n  ->  (
( w `  j
)  \  _I  )  =  ( ( w `
 n )  \  _I  ) )
54dmeqd 5057 . . . . . 6  |-  ( j  =  n  ->  dom  ( ( w `  j )  \  _I  )  =  dom  ( ( w `  n ) 
\  _I  ) )
65fveq2d 5885 . . . . 5  |-  ( j  =  n  ->  (
(pmTrsp `  N ) `  dom  ( ( w `
 j )  \  _I  ) )  =  ( (pmTrsp `  N ) `  dom  ( ( w `
 n )  \  _I  ) ) )
76cbvmptv 4518 . . . 4  |-  ( j  e.  ( 0..^ (
# `  w )
)  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) )  =  ( n  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  n )  \  _I  ) ) )
81, 2, 7pmtrdifwrdellem1 17073 . . 3  |-  ( w  e. Word  T  ->  (
j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) )  e. Word  R )
91, 2, 7pmtrdifwrdellem2 17074 . . 3  |-  ( w  e. Word  T  ->  ( # `
 w )  =  ( # `  (
j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) ) ) )
101, 2, 7pmtrdifwrdellem3 17075 . . 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 5881 . . . . . 6  |-  ( u  =  ( j  e.  ( 0..^ ( # `  w ) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `
 j )  \  _I  ) ) )  -> 
( # `  u )  =  ( # `  (
j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) ) ) )
1211eqeq2d 2443 . . . . 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 5880 . . . . . . . 8  |-  ( 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 5883 . . . . . . 7  |-  ( 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 2443 . . . . . 6  |-  ( 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
) ) )
16152ralbidv 2876 . . . . 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
) ) )
1712, 16anbi12d 715 . . . 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
) ) ) )
1817rspcev 3188 . . 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
) ) )
198, 9, 10, 18syl12anc 1262 . 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
) ) )
2019rgen 2792 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 370    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783    \ cdif 3439   {csn 4002    |-> cmpt 4484    _I cid 4764   dom cdm 4854   ran crn 4855   ` cfv 5601  (class class class)co 6305   0cc0 9538  ..^cfzo 11913   #chash 12512  Word cword 12643  pmTrspcpmtr 17033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651  df-pmtr 17034
This theorem is referenced by: (None)
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