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Theorem symgmatr01lem 18464
Description: Lemma for symgmatr01 18465. (Contributed by AV, 3-Jan-2019.)
Hypothesis
Ref Expression
symgmatr01.p  |-  P  =  ( Base `  ( SymGrp `
 N ) )
Assertion
Ref Expression
symgmatr01lem  |-  ( ( 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 ,  A ,  B ) ,  ( k M ( Q `
 k ) ) )  =  B ) )
Distinct variable groups:    A, k    B, k    k, q, L   
k, K, q    k, M    k, N    P, k,
q    Q, k, q
Allowed substitution hints:    A( q)    B( q)    M( q)    N( q)

Proof of Theorem symgmatr01lem
StepHypRef Expression
1 eqidd 2444 . . . . 5  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  K  =  K )
2 iftrue 3802 . . . . 5  |-  ( K  =  K  ->  if ( K  =  K ,  if ( ( Q `
 K )  =  L ,  A ,  B ) ,  ( K M ( Q `
 K ) ) )  =  if ( ( Q `  K
)  =  L ,  A ,  B )
)
31, 2syl 16 . . . 4  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  if ( K  =  K ,  if ( ( Q `  K )  =  L ,  A ,  B
) ,  ( K M ( Q `  K ) ) )  =  if ( ( Q `  K )  =  L ,  A ,  B ) )
4 eldif 3343 . . . . . . 7  |-  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  <->  ( Q  e.  P  /\  -.  Q  e.  { q  e.  P  |  ( q `  K )  =  L } ) )
5 ianor 488 . . . . . . . . . 10  |-  ( -.  ( Q  e.  P  /\  ( Q `  K
)  =  L )  <-> 
( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L ) )
6 fveq1 5695 . . . . . . . . . . . 12  |-  ( q  =  Q  ->  (
q `  K )  =  ( Q `  K ) )
76eqeq1d 2451 . . . . . . . . . . 11  |-  ( q  =  Q  ->  (
( q `  K
)  =  L  <->  ( Q `  K )  =  L ) )
87elrab 3122 . . . . . . . . . 10  |-  ( Q  e.  { q  e.  P  |  ( q `
 K )  =  L }  <->  ( Q  e.  P  /\  ( Q `  K )  =  L ) )
95, 8xchnxbir 309 . . . . . . . . 9  |-  ( -.  Q  e.  { q  e.  P  |  ( q `  K )  =  L }  <->  ( -.  Q  e.  P  \/  -.  ( Q `  K
)  =  L ) )
10 pm2.21 108 . . . . . . . . . 10  |-  ( -.  Q  e.  P  -> 
( Q  e.  P  ->  -.  ( Q `  K )  =  L ) )
11 ax-1 6 . . . . . . . . . 10  |-  ( -.  ( Q `  K
)  =  L  -> 
( Q  e.  P  ->  -.  ( Q `  K )  =  L ) )
1210, 11jaoi 379 . . . . . . . . 9  |-  ( ( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L )  ->  ( Q  e.  P  ->  -.  ( Q `  K )  =  L ) )
139, 12sylbi 195 . . . . . . . 8  |-  ( -.  Q  e.  { q  e.  P  |  ( q `  K )  =  L }  ->  ( Q  e.  P  ->  -.  ( Q `  K
)  =  L ) )
1413impcom 430 . . . . . . 7  |-  ( ( Q  e.  P  /\  -.  Q  e.  { q  e.  P  |  ( q `  K )  =  L } )  ->  -.  ( Q `  K )  =  L )
154, 14sylbi 195 . . . . . 6  |-  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  ->  -.  ( Q `  K )  =  L )
1615adantl 466 . . . . 5  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  -.  ( Q `  K )  =  L )
17 iffalse 3804 . . . . 5  |-  ( -.  ( Q `  K
)  =  L  ->  if ( ( Q `  K )  =  L ,  A ,  B
)  =  B )
1816, 17syl 16 . . . 4  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  if (
( Q `  K
)  =  L ,  A ,  B )  =  B )
193, 18eqtrd 2475 . . 3  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  if ( K  =  K ,  if ( ( Q `  K )  =  L ,  A ,  B
) ,  ( K M ( Q `  K ) ) )  =  B )
20 simpll 753 . . . 4  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  K  e.  N )
21 eqeq1 2449 . . . . . . 7  |-  ( k  =  K  ->  (
k  =  K  <->  K  =  K ) )
22 fveq2 5696 . . . . . . . . 9  |-  ( k  =  K  ->  ( Q `  k )  =  ( Q `  K ) )
2322eqeq1d 2451 . . . . . . . 8  |-  ( k  =  K  ->  (
( Q `  k
)  =  L  <->  ( Q `  K )  =  L ) )
2423ifbid 3816 . . . . . . 7  |-  ( k  =  K  ->  if ( ( Q `  k )  =  L ,  A ,  B
)  =  if ( ( Q `  K
)  =  L ,  A ,  B )
)
25 id 22 . . . . . . . 8  |-  ( k  =  K  ->  k  =  K )
2625, 22oveq12d 6114 . . . . . . 7  |-  ( k  =  K  ->  (
k M ( Q `
 k ) )  =  ( K M ( Q `  K
) ) )
2721, 24, 26ifbieq12d 3821 . . . . . 6  |-  ( k  =  K  ->  if ( k  =  K ,  if ( ( Q `  k )  =  L ,  A ,  B ) ,  ( k M ( Q `
 k ) ) )  =  if ( K  =  K ,  if ( ( Q `  K )  =  L ,  A ,  B
) ,  ( K M ( Q `  K ) ) ) )
2827eqeq1d 2451 . . . . 5  |-  ( k  =  K  ->  ( if ( k  =  K ,  if ( ( Q `  k )  =  L ,  A ,  B ) ,  ( k M ( Q `
 k ) ) )  =  B  <->  if ( K  =  K ,  if ( ( Q `  K )  =  L ,  A ,  B
) ,  ( K M ( Q `  K ) ) )  =  B ) )
2928adantl 466 . . . 4  |-  ( ( ( ( K  e.  N  /\  L  e.  N )  /\  Q  e.  ( P  \  {
q  e.  P  | 
( q `  K
)  =  L }
) )  /\  k  =  K )  ->  ( if ( k  =  K ,  if ( ( Q `  k )  =  L ,  A ,  B ) ,  ( k M ( Q `
 k ) ) )  =  B  <->  if ( K  =  K ,  if ( ( Q `  K )  =  L ,  A ,  B
) ,  ( K M ( Q `  K ) ) )  =  B ) )
3020, 29rspcedv 3082 . . 3  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  ( if ( K  =  K ,  if ( ( Q `
 K )  =  L ,  A ,  B ) ,  ( K M ( Q `
 K ) ) )  =  B  ->  E. k  e.  N  if ( k  =  K ,  if ( ( Q `  k )  =  L ,  A ,  B ) ,  ( k M ( Q `
 k ) ) )  =  B ) )
3119, 30mpd 15 . 2  |-  ( ( ( 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 ,  A ,  B
) ,  ( k M ( Q `  k ) ) )  =  B )
3231ex 434 1  |-  ( ( 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 ,  A ,  B ) ,  ( k M ( Q `
 k ) ) )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2721   {crab 2724    \ cdif 3330   ifcif 3796   ` cfv 5423  (class class class)co 6096   Basecbs 14179   SymGrpcsymg 15887
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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-iota 5386  df-fv 5431  df-ov 6099
This theorem is referenced by:  symgmatr01  18465
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