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Theorem pmtr3ncom 17171
Description: Transpositions over sets with at least 3 elements are not commutative, see also the remark in [Rotman] p. 28. (Contributed by AV, 21-Mar-2019.)
Hypothesis
Ref Expression
pmtr3ncom.t  |-  T  =  (pmTrsp `  D )
Assertion
Ref Expression
pmtr3ncom  |-  ( ( D  e.  V  /\  3  <_  ( # `  D
) )  ->  E. f  e.  ran  T E. g  e.  ran  T ( g  o.  f )  =/=  ( f  o.  g
) )
Distinct variable groups:    D, f,
g    T, f, g
Allowed substitution hints:    V( f, g)

Proof of Theorem pmtr3ncom
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hashge3el3dif 12681 . 2  |-  ( ( D  e.  V  /\  3  <_  ( # `  D
) )  ->  E. x  e.  D  E. y  e.  D  E. z  e.  D  ( x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )
2 simprl 769 . . . . . . 7  |-  ( ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  /\  ( D  e.  V  /\  3  <_  ( # `  D ) ) )  ->  D  e.  V
)
3 prssi 4141 . . . . . . . . 9  |-  ( ( x  e.  D  /\  y  e.  D )  ->  { x ,  y }  C_  D )
43adantr 471 . . . . . . . 8  |-  ( ( ( x  e.  D  /\  y  e.  D
)  /\  z  e.  D )  ->  { x ,  y }  C_  D )
54ad2antrr 737 . . . . . . 7  |-  ( ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  /\  ( D  e.  V  /\  3  <_  ( # `  D ) ) )  ->  { x ,  y }  C_  D
)
6 simplll 773 . . . . . . . . 9  |-  ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  ->  x  e.  D )
7 simplr 767 . . . . . . . . . 10  |-  ( ( ( x  e.  D  /\  y  e.  D
)  /\  z  e.  D )  ->  y  e.  D )
87adantr 471 . . . . . . . . 9  |-  ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  -> 
y  e.  D )
9 simpr1 1020 . . . . . . . . 9  |-  ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  ->  x  =/=  y )
10 pr2nelem 8466 . . . . . . . . 9  |-  ( ( x  e.  D  /\  y  e.  D  /\  x  =/=  y )  ->  { x ,  y }  ~~  2o )
116, 8, 9, 10syl3anc 1276 . . . . . . . 8  |-  ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  ->  { x ,  y }  ~~  2o )
1211adantr 471 . . . . . . 7  |-  ( ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  /\  ( D  e.  V  /\  3  <_  ( # `  D ) ) )  ->  { x ,  y }  ~~  2o )
13 pmtr3ncom.t . . . . . . . 8  |-  T  =  (pmTrsp `  D )
14 eqid 2462 . . . . . . . 8  |-  ran  T  =  ran  T
1513, 14pmtrrn 17153 . . . . . . 7  |-  ( ( D  e.  V  /\  { x ,  y } 
C_  D  /\  {
x ,  y } 
~~  2o )  -> 
( T `  {
x ,  y } )  e.  ran  T
)
162, 5, 12, 15syl3anc 1276 . . . . . 6  |-  ( ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  /\  ( D  e.  V  /\  3  <_  ( # `  D ) ) )  ->  ( T `  { x ,  y } )  e.  ran  T )
17 prssi 4141 . . . . . . . . 9  |-  ( ( y  e.  D  /\  z  e.  D )  ->  { y ,  z }  C_  D )
1817adantll 725 . . . . . . . 8  |-  ( ( ( x  e.  D  /\  y  e.  D
)  /\  z  e.  D )  ->  { y ,  z }  C_  D )
1918ad2antrr 737 . . . . . . 7  |-  ( ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  /\  ( D  e.  V  /\  3  <_  ( # `  D ) ) )  ->  { y ,  z }  C_  D
)
20 simplr 767 . . . . . . . . 9  |-  ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  -> 
z  e.  D )
21 simpr3 1022 . . . . . . . . 9  |-  ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  -> 
y  =/=  z )
22 pr2nelem 8466 . . . . . . . . 9  |-  ( ( y  e.  D  /\  z  e.  D  /\  y  =/=  z )  ->  { y ,  z }  ~~  2o )
238, 20, 21, 22syl3anc 1276 . . . . . . . 8  |-  ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  ->  { y ,  z }  ~~  2o )
2423adantr 471 . . . . . . 7  |-  ( ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  /\  ( D  e.  V  /\  3  <_  ( # `  D ) ) )  ->  { y ,  z }  ~~  2o )
2513, 14pmtrrn 17153 . . . . . . 7  |-  ( ( D  e.  V  /\  { y ,  z } 
C_  D  /\  {
y ,  z } 
~~  2o )  -> 
( T `  {
y ,  z } )  e.  ran  T
)
262, 19, 24, 25syl3anc 1276 . . . . . 6  |-  ( ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  /\  ( D  e.  V  /\  3  <_  ( # `  D ) ) )  ->  ( T `  { y ,  z } )  e.  ran  T )
27 df-3an 993 . . . . . . . . 9  |-  ( ( x  e.  D  /\  y  e.  D  /\  z  e.  D )  <->  ( ( x  e.  D  /\  y  e.  D
)  /\  z  e.  D ) )
2827biimpri 211 . . . . . . . 8  |-  ( ( ( x  e.  D  /\  y  e.  D
)  /\  z  e.  D )  ->  (
x  e.  D  /\  y  e.  D  /\  z  e.  D )
)
2928ad2antrr 737 . . . . . . 7  |-  ( ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  /\  ( D  e.  V  /\  3  <_  ( # `  D ) ) )  ->  ( x  e.  D  /\  y  e.  D  /\  z  e.  D ) )
30 simplr 767 . . . . . . 7  |-  ( ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  /\  ( D  e.  V  /\  3  <_  ( # `  D ) ) )  ->  ( x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )
31 eqid 2462 . . . . . . . 8  |-  ( T `
 { x ,  y } )  =  ( T `  {
x ,  y } )
32 eqid 2462 . . . . . . . 8  |-  ( T `
 { y ,  z } )  =  ( T `  {
y ,  z } )
3313, 31, 32pmtr3ncomlem2 17170 . . . . . . 7  |-  ( ( D  e.  V  /\  ( x  e.  D  /\  y  e.  D  /\  z  e.  D
)  /\  ( x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  -> 
( ( T `  { y ,  z } )  o.  ( T `  { x ,  y } ) )  =/=  ( ( T `  { x ,  y } )  o.  ( T `  { y ,  z } ) ) )
342, 29, 30, 33syl3anc 1276 . . . . . 6  |-  ( ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  /\  ( D  e.  V  /\  3  <_  ( # `  D ) ) )  ->  ( ( T `
 { y ,  z } )  o.  ( T `  {
x ,  y } ) )  =/=  (
( T `  {
x ,  y } )  o.  ( T `
 { y ,  z } ) ) )
35 coeq2 5015 . . . . . . . 8  |-  ( f  =  ( T `  { x ,  y } )  ->  (
g  o.  f )  =  ( g  o.  ( T `  {
x ,  y } ) ) )
36 coeq1 5014 . . . . . . . 8  |-  ( f  =  ( T `  { x ,  y } )  ->  (
f  o.  g )  =  ( ( T `
 { x ,  y } )  o.  g ) )
3735, 36neeq12d 2697 . . . . . . 7  |-  ( f  =  ( T `  { x ,  y } )  ->  (
( g  o.  f
)  =/=  ( f  o.  g )  <->  ( g  o.  ( T `  {
x ,  y } ) )  =/=  (
( T `  {
x ,  y } )  o.  g ) ) )
38 coeq1 5014 . . . . . . . 8  |-  ( g  =  ( T `  { y ,  z } )  ->  (
g  o.  ( T `
 { x ,  y } ) )  =  ( ( T `
 { y ,  z } )  o.  ( T `  {
x ,  y } ) ) )
39 coeq2 5015 . . . . . . . 8  |-  ( g  =  ( T `  { y ,  z } )  ->  (
( T `  {
x ,  y } )  o.  g )  =  ( ( T `
 { x ,  y } )  o.  ( T `  {
y ,  z } ) ) )
4038, 39neeq12d 2697 . . . . . . 7  |-  ( g  =  ( T `  { y ,  z } )  ->  (
( g  o.  ( T `  { x ,  y } ) )  =/=  ( ( T `  { x ,  y } )  o.  g )  <->  ( ( T `  { y ,  z } )  o.  ( T `  { x ,  y } ) )  =/=  ( ( T `  { x ,  y } )  o.  ( T `  { y ,  z } ) ) ) )
4137, 40rspc2ev 3173 . . . . . 6  |-  ( ( ( T `  {
x ,  y } )  e.  ran  T  /\  ( T `  {
y ,  z } )  e.  ran  T  /\  ( ( T `  { y ,  z } )  o.  ( T `  { x ,  y } ) )  =/=  ( ( T `  { x ,  y } )  o.  ( T `  { y ,  z } ) ) )  ->  E. f  e.  ran  T E. g  e.  ran  T ( g  o.  f
)  =/=  ( f  o.  g ) )
4216, 26, 34, 41syl3anc 1276 . . . . 5  |-  ( ( ( ( ( x  e.  D  /\  y  e.  D )  /\  z  e.  D )  /\  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  /\  ( D  e.  V  /\  3  <_  ( # `  D ) ) )  ->  E. f  e.  ran  T E. g  e.  ran  T ( g  o.  f
)  =/=  ( f  o.  g ) )
4342exp31 613 . . . 4  |-  ( ( ( x  e.  D  /\  y  e.  D
)  /\  z  e.  D )  ->  (
( x  =/=  y  /\  x  =/=  z  /\  y  =/=  z
)  ->  ( ( D  e.  V  /\  3  <_  ( # `  D
) )  ->  E. f  e.  ran  T E. g  e.  ran  T ( g  o.  f )  =/=  ( f  o.  g
) ) ) )
4443rexlimdva 2891 . . 3  |-  ( ( x  e.  D  /\  y  e.  D )  ->  ( E. z  e.  D  ( x  =/=  y  /\  x  =/=  z  /\  y  =/=  z )  ->  (
( D  e.  V  /\  3  <_  ( # `  D ) )  ->  E. f  e.  ran  T E. g  e.  ran  T ( g  o.  f
)  =/=  ( f  o.  g ) ) ) )
4544rexlimivv 2896 . 2  |-  ( E. x  e.  D  E. y  e.  D  E. z  e.  D  (
x  =/=  y  /\  x  =/=  z  /\  y  =/=  z )  ->  (
( D  e.  V  /\  3  <_  ( # `  D ) )  ->  E. f  e.  ran  T E. g  e.  ran  T ( g  o.  f
)  =/=  ( f  o.  g ) ) )
461, 45mpcom 37 1  |-  ( ( D  e.  V  /\  3  <_  ( # `  D
) )  ->  E. f  e.  ran  T E. g  e.  ran  T ( g  o.  f )  =/=  ( f  o.  g
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   E.wrex 2750    C_ wss 3416   {cpr 3982   class class class wbr 4418   ran crn 4857    o. ccom 4860   ` cfv 5605   2oc2o 7207    ~~ cen 7597    <_ cle 9707   3c3 10693   #chash 12553  pmTrspcpmtr 17137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-2o 7214  df-oadd 7217  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-card 8404  df-cda 8629  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-2 10701  df-3 10702  df-n0 10904  df-z 10972  df-uz 11194  df-fz 11820  df-hash 12554  df-pmtr 17138
This theorem is referenced by:  pgrpgt2nabl  40520
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