MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  symgmatr01 Structured version   Unicode version

Theorem symgmatr01 19323
Description: Applying a permutation that does not fix a certain element of a set to a second element to an index of a matrix a row with 0's and a 1. (Contributed by AV, 3-Jan-2019.)
Hypotheses
Ref Expression
symgmatr01.p  |-  P  =  ( Base `  ( SymGrp `
 N ) )
symgmatr01.0  |-  .0.  =  ( 0g `  R )
symgmatr01.1  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
symgmatr01  |-  ( ( K  e.  N  /\  L  e.  N )  ->  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  ->  E. k  e.  N  ( k ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) ( Q `  k ) )  =  .0.  ) )
Distinct variable groups:    k, q, L    k, K, q    k, M    k, N    P, k,
q    Q, k, q    i,
j, k, q, L   
i, K, j    i, M, j    i, N, j    P, i, j    Q, i, j    .1. , i, j,
k    .0. , i, j, k
Allowed substitution hints:    R( i, j, k, q)    .1. ( q)    M( q)    N( q)    .0. ( q)

Proof of Theorem symgmatr01
StepHypRef Expression
1 symgmatr01.p . . . . 5  |-  P  =  ( Base `  ( SymGrp `
 N ) )
21symgmatr01lem 19322 . . . 4  |-  ( ( K  e.  N  /\  L  e.  N )  ->  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  ->  E. k  e.  N  if ( k  =  K ,  if ( ( Q `  k )  =  L ,  .1.  ,  .0.  ) ,  ( k M ( Q `
 k ) ) )  =  .0.  )
)
32imp 427 . . 3  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  E. k  e.  N  if (
k  =  K ,  if ( ( Q `  k )  =  L ,  .1.  ,  .0.  ) ,  ( k M ( Q `  k ) ) )  =  .0.  )
4 eqidd 2455 . . . . . 6  |-  ( ( ( ( K  e.  N  /\  L  e.  N )  /\  Q  e.  ( P  \  {
q  e.  P  | 
( q `  K
)  =  L }
) )  /\  k  e.  N )  ->  (
i  e.  N , 
j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) )  =  ( i  e.  N , 
j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) )
5 eqeq1 2458 . . . . . . . . 9  |-  ( i  =  k  ->  (
i  =  K  <->  k  =  K ) )
65adantr 463 . . . . . . . 8  |-  ( ( i  =  k  /\  j  =  ( Q `  k ) )  -> 
( i  =  K  <-> 
k  =  K ) )
7 eqeq1 2458 . . . . . . . . . 10  |-  ( j  =  ( Q `  k )  ->  (
j  =  L  <->  ( Q `  k )  =  L ) )
87adantl 464 . . . . . . . . 9  |-  ( ( i  =  k  /\  j  =  ( Q `  k ) )  -> 
( j  =  L  <-> 
( Q `  k
)  =  L ) )
98ifbid 3951 . . . . . . . 8  |-  ( ( i  =  k  /\  j  =  ( Q `  k ) )  ->  if ( j  =  L ,  .1.  ,  .0.  )  =  if (
( Q `  k
)  =  L ,  .1.  ,  .0.  ) )
10 oveq12 6279 . . . . . . . 8  |-  ( ( i  =  k  /\  j  =  ( Q `  k ) )  -> 
( i M j )  =  ( k M ( Q `  k ) ) )
116, 9, 10ifbieq12d 3956 . . . . . . 7  |-  ( ( i  =  k  /\  j  =  ( Q `  k ) )  ->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) )  =  if ( k  =  K ,  if ( ( Q `  k )  =  L ,  .1.  ,  .0.  ) ,  ( k M ( Q `  k ) ) ) )
1211adantl 464 . . . . . 6  |-  ( ( ( ( ( K  e.  N  /\  L  e.  N )  /\  Q  e.  ( P  \  {
q  e.  P  | 
( q `  K
)  =  L }
) )  /\  k  e.  N )  /\  (
i  =  k  /\  j  =  ( Q `  k ) ) )  ->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) )  =  if ( k  =  K ,  if ( ( Q `  k )  =  L ,  .1.  ,  .0.  ) ,  ( k M ( Q `  k ) ) ) )
13 simpr 459 . . . . . 6  |-  ( ( ( ( K  e.  N  /\  L  e.  N )  /\  Q  e.  ( P  \  {
q  e.  P  | 
( q `  K
)  =  L }
) )  /\  k  e.  N )  ->  k  e.  N )
14 eldifi 3612 . . . . . . . . 9  |-  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  ->  Q  e.  P )
15 eqid 2454 . . . . . . . . . . 11  |-  ( SymGrp `  N )  =  (
SymGrp `  N )
1615, 1symgfv 16611 . . . . . . . . . 10  |-  ( ( Q  e.  P  /\  k  e.  N )  ->  ( Q `  k
)  e.  N )
1716ex 432 . . . . . . . . 9  |-  ( Q  e.  P  ->  (
k  e.  N  -> 
( Q `  k
)  e.  N ) )
1814, 17syl 16 . . . . . . . 8  |-  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  ->  (
k  e.  N  -> 
( Q `  k
)  e.  N ) )
1918adantl 464 . . . . . . 7  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  ( k  e.  N  ->  ( Q `
 k )  e.  N ) )
2019imp 427 . . . . . 6  |-  ( ( ( ( K  e.  N  /\  L  e.  N )  /\  Q  e.  ( P  \  {
q  e.  P  | 
( q `  K
)  =  L }
) )  /\  k  e.  N )  ->  ( Q `  k )  e.  N )
21 symgmatr01.1 . . . . . . . . . 10  |-  .1.  =  ( 1r `  R )
22 fvex 5858 . . . . . . . . . 10  |-  ( 1r
`  R )  e. 
_V
2321, 22eqeltri 2538 . . . . . . . . 9  |-  .1.  e.  _V
24 symgmatr01.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  R )
25 fvex 5858 . . . . . . . . . 10  |-  ( 0g
`  R )  e. 
_V
2624, 25eqeltri 2538 . . . . . . . . 9  |-  .0.  e.  _V
2723, 26ifex 3997 . . . . . . . 8  |-  if ( ( Q `  k
)  =  L ,  .1.  ,  .0.  )  e. 
_V
28 ovex 6298 . . . . . . . 8  |-  ( k M ( Q `  k ) )  e. 
_V
2927, 28ifex 3997 . . . . . . 7  |-  if ( k  =  K ,  if ( ( Q `  k )  =  L ,  .1.  ,  .0.  ) ,  ( k M ( Q `  k ) ) )  e.  _V
3029a1i 11 . . . . . 6  |-  ( ( ( ( K  e.  N  /\  L  e.  N )  /\  Q  e.  ( P  \  {
q  e.  P  | 
( q `  K
)  =  L }
) )  /\  k  e.  N )  ->  if ( k  =  K ,  if ( ( Q `  k )  =  L ,  .1.  ,  .0.  ) ,  ( k M ( Q `
 k ) ) )  e.  _V )
314, 12, 13, 20, 30ovmpt2d 6403 . . . . 5  |-  ( ( ( ( K  e.  N  /\  L  e.  N )  /\  Q  e.  ( P  \  {
q  e.  P  | 
( q `  K
)  =  L }
) )  /\  k  e.  N )  ->  (
k ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) ( Q `  k ) )  =  if ( k  =  K ,  if ( ( Q `  k
)  =  L ,  .1.  ,  .0.  ) ,  ( k M ( Q `  k ) ) ) )
3231eqeq1d 2456 . . . 4  |-  ( ( ( ( K  e.  N  /\  L  e.  N )  /\  Q  e.  ( P  \  {
q  e.  P  | 
( q `  K
)  =  L }
) )  /\  k  e.  N )  ->  (
( k ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) ( Q `  k ) )  =  .0.  <->  if ( k  =  K ,  if ( ( Q `  k
)  =  L ,  .1.  ,  .0.  ) ,  ( k M ( Q `  k ) ) )  =  .0.  ) )
3332rexbidva 2962 . . 3  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  ( E. k  e.  N  (
k ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) ( Q `  k ) )  =  .0.  <->  E. k  e.  N  if ( k  =  K ,  if ( ( Q `  k )  =  L ,  .1.  ,  .0.  ) ,  ( k M ( Q `
 k ) ) )  =  .0.  )
)
343, 33mpbird 232 . 2  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  E. k  e.  N  ( k
( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) ( Q `  k ) )  =  .0.  )
3534ex 432 1  |-  ( ( K  e.  N  /\  L  e.  N )  ->  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  ->  E. k  e.  N  ( k ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) ( Q `  k ) )  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805   {crab 2808   _Vcvv 3106    \ cdif 3458   ifcif 3929   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   Basecbs 14716   0gc0g 14929   SymGrpcsymg 16601   1rcur 17348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  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 11083  df-fz 11676  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-plusg 14797  df-tset 14803  df-symg 16602
This theorem is referenced by:  smadiadetlem0  19330
  Copyright terms: Public domain W3C validator