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Theorem pmtrdifwrdel2 16304
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.
StepHypRef Expression
1 pmtrdifel.t . . . . 5  |-  T  =  ran  (pmTrsp `  ( N  \  { K }
) )
2 pmtrdifel.r . . . . 5  |-  R  =  ran  (pmTrsp `  N
)
3 fveq2 5864 . . . . . . . . 9  |-  ( j  =  n  ->  (
w `  j )  =  ( w `  n ) )
43difeq1d 3621 . . . . . . . 8  |-  ( j  =  n  ->  (
( w `  j
)  \  _I  )  =  ( ( w `
 n )  \  _I  ) )
54dmeqd 5203 . . . . . . 7  |-  ( j  =  n  ->  dom  ( ( w `  j )  \  _I  )  =  dom  ( ( w `  n ) 
\  _I  ) )
65fveq2d 5868 . . . . . 6  |-  ( j  =  n  ->  (
(pmTrsp `  N ) `  dom  ( ( w `
 j )  \  _I  ) )  =  ( (pmTrsp `  N ) `  dom  ( ( w `
 n )  \  _I  ) ) )
76cbvmptv 4538 . . . . 5  |-  ( j  e.  ( 0..^ (
# `  w )
)  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) )  =  ( n  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  n )  \  _I  ) ) )
81, 2, 7pmtrdifwrdellem1 16299 . . . 4  |-  ( w  e. Word  T  ->  (
j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) )  e. Word  R )
98adantl 466 . . 3  |-  ( ( K  e.  N  /\  w  e. Word  T )  ->  ( j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) )  e. Word  R )
101, 2, 7pmtrdifwrdellem2 16300 . . . 4  |-  ( w  e. Word  T  ->  ( # `
 w )  =  ( # `  (
j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) ) ) )
1110adantl 466 . . 3  |-  ( ( K  e.  N  /\  w  e. Word  T )  ->  ( # `  w
)  =  ( # `  ( j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) ) ) )
121, 2, 7pmtrdifwrdel2lem1 16302 . . . . 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 )
1312ancoms 453 . . . 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 )
141, 2, 7pmtrdifwrdellem3 16301 . . . . 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
) )
1514adantl 466 . . . 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 2989 . . . 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
) ) )
1713, 15, 16sylanbrc 664 . . 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 5864 . . . . . 6  |-  ( u  =  ( j  e.  ( 0..^ ( # `  w ) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `
 j )  \  _I  ) ) )  -> 
( # `  u )  =  ( # `  (
j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) ) ) )
1918eqeq2d 2481 . . . . 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 5863 . . . . . . . . 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 ) )
2120fveq1d 5866 . . . . . . . 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
) )
2221eqeq1d 2469 . . . . . . 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 ) )
2320fveq1d 5866 . . . . . . . . 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
) )
2423eqeq2d 2481 . . . . . . . 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
) ) )
2524ralbidv 2903 . . . . . . 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
) ) )
2622, 25anbi12d 710 . . . . . 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
) ) ) )
2726ralbidv 2903 . . . . 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
) ) ) )
2819, 27anbi12d 710 . . . 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
) ) ) ) )
2928rspcev 3214 . . 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
) ) ) )
309, 11, 17, 29syl12anc 1226 . 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
) ) ) )
3130ralrimiva 2878 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
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    \ cdif 3473   {csn 4027    |-> cmpt 4505    _I cid 4790   dom cdm 4999   ran crn 5000   ` cfv 5586  (class class class)co 6282   0cc0 9488  ..^cfzo 11788   #chash 12367  Word cword 12494  pmTrspcpmtr 16259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12368  df-word 12502  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-plusg 14561  df-tset 14567  df-symg 16195  df-pmtr 16260
This theorem is referenced by:  psgndiflemA  18401
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