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Theorem pmtrdifwrdel2 15997
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 5696 . . . . . . . . 9  |-  ( j  =  n  ->  (
w `  j )  =  ( w `  n ) )
43difeq1d 3478 . . . . . . . 8  |-  ( j  =  n  ->  (
( w `  j
)  \  _I  )  =  ( ( w `
 n )  \  _I  ) )
54dmeqd 5047 . . . . . . 7  |-  ( j  =  n  ->  dom  ( ( w `  j )  \  _I  )  =  dom  ( ( w `  n ) 
\  _I  ) )
65fveq2d 5700 . . . . . 6  |-  ( j  =  n  ->  (
(pmTrsp `  N ) `  dom  ( ( w `
 j )  \  _I  ) )  =  ( (pmTrsp `  N ) `  dom  ( ( w `
 n )  \  _I  ) ) )
76cbvmptv 4388 . . . . 5  |-  ( j  e.  ( 0..^ (
# `  w )
)  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) )  =  ( n  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  n )  \  _I  ) ) )
81, 2, 7pmtrdifwrdellem1 15992 . . . 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 15993 . . . 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 15995 . . . . 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 15994 . . . . 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 2854 . . . 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 5696 . . . . . 6  |-  ( u  =  ( j  e.  ( 0..^ ( # `  w ) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `
 j )  \  _I  ) ) )  -> 
( # `  u )  =  ( # `  (
j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) ) ) )
1918eqeq2d 2454 . . . . 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 5695 . . . . . . . . 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 5698 . . . . . . . 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 2451 . . . . . . 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 5698 . . . . . . . . 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 2454 . . . . . . . 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 2740 . . . . . . 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 2740 . . . . 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 3078 . . 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 1216 . 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 2804 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 1369    e. wcel 1756   A.wral 2720   E.wrex 2721    \ cdif 3330   {csn 3882    e. cmpt 4355    _I cid 4636   dom cdm 4845   ran crn 4846   ` cfv 5423  (class class class)co 6096   0cc0 9287  ..^cfzo 11553   #chash 12108  Word cword 12226  pmTrspcpmtr 15952
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-hash 12109  df-word 12234  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-tset 14262  df-symg 15888  df-pmtr 15953
This theorem is referenced by:  psgndiflemA  18036
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