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Theorem symgmatr01lem 19620
Description: Lemma for symgmatr01 19621. (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 simpll 758 . . 3  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  K  e.  N )
2 eqeq1 2432 . . . . . 6  |-  ( k  =  K  ->  (
k  =  K  <->  K  =  K ) )
3 fveq2 5825 . . . . . . . 8  |-  ( k  =  K  ->  ( Q `  k )  =  ( Q `  K ) )
43eqeq1d 2430 . . . . . . 7  |-  ( k  =  K  ->  (
( Q `  k
)  =  L  <->  ( Q `  K )  =  L ) )
54ifbid 3876 . . . . . 6  |-  ( k  =  K  ->  if ( ( Q `  k )  =  L ,  A ,  B
)  =  if ( ( Q `  K
)  =  L ,  A ,  B )
)
6 id 22 . . . . . . 7  |-  ( k  =  K  ->  k  =  K )
76, 3oveq12d 6267 . . . . . 6  |-  ( k  =  K  ->  (
k M ( Q `
 k ) )  =  ( K M ( Q `  K
) ) )
82, 5, 7ifbieq12d 3881 . . . . 5  |-  ( 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 ) ) ) )
98eqeq1d 2430 . . . 4  |-  ( 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 ) )
109adantl 467 . . 3  |-  ( ( ( ( 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 ) )
11 eqidd 2429 . . . . 5  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  K  =  K )
1211iftrued 3862 . . . 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 ) )
13 eldif 3389 . . . . . . 7  |-  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  <->  ( Q  e.  P  /\  -.  Q  e.  { q  e.  P  |  ( q `  K )  =  L } ) )
14 ianor 490 . . . . . . . . . 10  |-  ( -.  ( Q  e.  P  /\  ( Q `  K
)  =  L )  <-> 
( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L ) )
15 fveq1 5824 . . . . . . . . . . . 12  |-  ( q  =  Q  ->  (
q `  K )  =  ( Q `  K ) )
1615eqeq1d 2430 . . . . . . . . . . 11  |-  ( q  =  Q  ->  (
( q `  K
)  =  L  <->  ( Q `  K )  =  L ) )
1716elrab 3171 . . . . . . . . . 10  |-  ( Q  e.  { q  e.  P  |  ( q `
 K )  =  L }  <->  ( Q  e.  P  /\  ( Q `  K )  =  L ) )
1814, 17xchnxbir 310 . . . . . . . . 9  |-  ( -.  Q  e.  { q  e.  P  |  ( q `  K )  =  L }  <->  ( -.  Q  e.  P  \/  -.  ( Q `  K
)  =  L ) )
19 pm2.21 111 . . . . . . . . . 10  |-  ( -.  Q  e.  P  -> 
( Q  e.  P  ->  -.  ( Q `  K )  =  L ) )
20 ax-1 6 . . . . . . . . . 10  |-  ( -.  ( Q `  K
)  =  L  -> 
( Q  e.  P  ->  -.  ( Q `  K )  =  L ) )
2119, 20jaoi 380 . . . . . . . . 9  |-  ( ( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L )  ->  ( Q  e.  P  ->  -.  ( Q `  K )  =  L ) )
2218, 21sylbi 198 . . . . . . . 8  |-  ( -.  Q  e.  { q  e.  P  |  ( q `  K )  =  L }  ->  ( Q  e.  P  ->  -.  ( Q `  K
)  =  L ) )
2322impcom 431 . . . . . . 7  |-  ( ( Q  e.  P  /\  -.  Q  e.  { q  e.  P  |  ( q `  K )  =  L } )  ->  -.  ( Q `  K )  =  L )
2413, 23sylbi 198 . . . . . 6  |-  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  ->  -.  ( Q `  K )  =  L )
2524adantl 467 . . . . 5  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  -.  ( Q `  K )  =  L )
2625iffalsed 3865 . . . 4  |-  ( ( ( K  e.  N  /\  L  e.  N
)  /\  Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } ) )  ->  if (
( Q `  K
)  =  L ,  A ,  B )  =  B )
2712, 26eqtrd 2462 . . 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 )
281, 10, 27rspcedvd 3130 . 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 )
2928ex 435 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 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872   E.wrex 2715   {crab 2718    \ cdif 3376   ifcif 3854   ` cfv 5544  (class class class)co 6249   Basecbs 15064   SymGrpcsymg 16961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-iota 5508  df-fv 5552  df-ov 6252
This theorem is referenced by:  symgmatr01  19621
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