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Theorem pmtrdifwrdel 16316
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 5866 . . . . . . . 8  |-  ( j  =  n  ->  (
w `  j )  =  ( w `  n ) )
43difeq1d 3621 . . . . . . 7  |-  ( j  =  n  ->  (
( w `  j
)  \  _I  )  =  ( ( w `
 n )  \  _I  ) )
54dmeqd 5205 . . . . . 6  |-  ( j  =  n  ->  dom  ( ( w `  j )  \  _I  )  =  dom  ( ( w `  n ) 
\  _I  ) )
65fveq2d 5870 . . . . 5  |-  ( j  =  n  ->  (
(pmTrsp `  N ) `  dom  ( ( w `
 j )  \  _I  ) )  =  ( (pmTrsp `  N ) `  dom  ( ( w `
 n )  \  _I  ) ) )
76cbvmptv 4538 . . . 4  |-  ( j  e.  ( 0..^ (
# `  w )
)  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) )  =  ( n  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  n )  \  _I  ) ) )
81, 2, 7pmtrdifwrdellem1 16312 . . 3  |-  ( w  e. Word  T  ->  (
j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) )  e. Word  R )
91, 2, 7pmtrdifwrdellem2 16313 . . 3  |-  ( w  e. Word  T  ->  ( # `
 w )  =  ( # `  (
j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) ) ) )
101, 2, 7pmtrdifwrdellem3 16314 . . 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 5866 . . . . . 6  |-  ( u  =  ( j  e.  ( 0..^ ( # `  w ) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `
 j )  \  _I  ) ) )  -> 
( # `  u )  =  ( # `  (
j  e.  ( 0..^ ( # `  w
) )  |->  ( (pmTrsp `  N ) `  dom  ( ( w `  j )  \  _I  ) ) ) ) )
1211eqeq2d 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  ) ) ) ) ) )
13 fveq1 5865 . . . . . . . . 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 5868 . . . . . . . 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 2481 . . . . . . 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 2903 . . . . . 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 2903 . . . . 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 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 ) ) 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 1226 . 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 2824 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 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 5588  (class class class)co 6284   0cc0 9492  ..^cfzo 11792   #chash 12373  Word cword 12500  pmTrspcpmtr 16272
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 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-pmtr 16273
This theorem is referenced by: (None)
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