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Theorem symgfix2 15926
Description: If a permutation does not move a certain element of a set to a second element, there is a third element which is moved to the second element. (Contributed by AV, 2-Jan-2019.)
Hypothesis
Ref Expression
symgfix2.p  |-  P  =  ( Base `  ( SymGrp `
 N ) )
Assertion
Ref Expression
symgfix2  |-  ( L  e.  N  ->  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  ->  E. k  e.  ( N  \  { K } ) ( Q `
 k )  =  L ) )
Distinct variable groups:    k, N    Q, k    k, K, q   
k, L, q    P, q    Q, q
Allowed substitution hints:    P( k)    N( q)

Proof of Theorem symgfix2
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 eldif 3343 . . 3  |-  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  <->  ( Q  e.  P  /\  -.  Q  e.  { q  e.  P  |  ( q `  K )  =  L } ) )
2 ianor 488 . . . . 5  |-  ( -.  ( Q  e.  P  /\  ( Q `  K
)  =  L )  <-> 
( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L ) )
3 fveq1 5695 . . . . . . 7  |-  ( q  =  Q  ->  (
q `  K )  =  ( Q `  K ) )
43eqeq1d 2451 . . . . . 6  |-  ( q  =  Q  ->  (
( q `  K
)  =  L  <->  ( Q `  K )  =  L ) )
54elrab 3122 . . . . 5  |-  ( Q  e.  { q  e.  P  |  ( q `
 K )  =  L }  <->  ( Q  e.  P  /\  ( Q `  K )  =  L ) )
62, 5xchnxbir 309 . . . 4  |-  ( -.  Q  e.  { q  e.  P  |  ( q `  K )  =  L }  <->  ( -.  Q  e.  P  \/  -.  ( Q `  K
)  =  L ) )
76anbi2i 694 . . 3  |-  ( ( Q  e.  P  /\  -.  Q  e.  { q  e.  P  |  ( q `  K )  =  L } )  <-> 
( Q  e.  P  /\  ( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L ) ) )
81, 7bitri 249 . 2  |-  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  <->  ( Q  e.  P  /\  ( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L ) ) )
9 pm2.21 108 . . . . 5  |-  ( -.  Q  e.  P  -> 
( Q  e.  P  ->  ( L  e.  N  ->  E. k  e.  ( N  \  { K } ) ( Q `
 k )  =  L ) ) )
10 symgfix2.p . . . . . . 7  |-  P  =  ( Base `  ( SymGrp `
 N ) )
1110symgmov2 15903 . . . . . 6  |-  ( Q  e.  P  ->  A. l  e.  N  E. k  e.  N  ( Q `  k )  =  l )
12 eqeq2 2452 . . . . . . . . . . 11  |-  ( l  =  L  ->  (
( Q `  k
)  =  l  <->  ( Q `  k )  =  L ) )
1312rexbidv 2741 . . . . . . . . . 10  |-  ( l  =  L  ->  ( E. k  e.  N  ( Q `  k )  =  l  <->  E. k  e.  N  ( Q `  k )  =  L ) )
1413rspcva 3076 . . . . . . . . 9  |-  ( ( L  e.  N  /\  A. l  e.  N  E. k  e.  N  ( Q `  k )  =  l )  ->  E. k  e.  N  ( Q `  k )  =  L )
15 eqeq2 2452 . . . . . . . . . . . . . . . 16  |-  ( L  =  ( Q `  k )  ->  (
( Q `  K
)  =  L  <->  ( Q `  K )  =  ( Q `  k ) ) )
1615eqcoms 2446 . . . . . . . . . . . . . . 15  |-  ( ( Q `  k )  =  L  ->  (
( Q `  K
)  =  L  <->  ( Q `  K )  =  ( Q `  k ) ) )
1716notbid 294 . . . . . . . . . . . . . 14  |-  ( ( Q `  k )  =  L  ->  ( -.  ( Q `  K
)  =  L  <->  -.  ( Q `  K )  =  ( Q `  k ) ) )
18 fveq2 5696 . . . . . . . . . . . . . . . 16  |-  ( K  =  k  ->  ( Q `  K )  =  ( Q `  k ) )
1918eqcoms 2446 . . . . . . . . . . . . . . 15  |-  ( k  =  K  ->  ( Q `  K )  =  ( Q `  k ) )
2019necon3bi 2657 . . . . . . . . . . . . . 14  |-  ( -.  ( Q `  K
)  =  ( Q `
 k )  -> 
k  =/=  K )
2117, 20syl6bi 228 . . . . . . . . . . . . 13  |-  ( ( Q `  k )  =  L  ->  ( -.  ( Q `  K
)  =  L  -> 
k  =/=  K ) )
2221com12 31 . . . . . . . . . . . 12  |-  ( -.  ( Q `  K
)  =  L  -> 
( ( Q `  k )  =  L  ->  k  =/=  K
) )
2322pm4.71rd 635 . . . . . . . . . . 11  |-  ( -.  ( Q `  K
)  =  L  -> 
( ( Q `  k )  =  L  <-> 
( k  =/=  K  /\  ( Q `  k
)  =  L ) ) )
2423rexbidv 2741 . . . . . . . . . 10  |-  ( -.  ( Q `  K
)  =  L  -> 
( E. k  e.  N  ( Q `  k )  =  L  <->  E. k  e.  N  ( k  =/=  K  /\  ( Q `  k
)  =  L ) ) )
25 rexdifsn 4009 . . . . . . . . . 10  |-  ( E. k  e.  ( N 
\  { K }
) ( Q `  k )  =  L  <->  E. k  e.  N  ( k  =/=  K  /\  ( Q `  k
)  =  L ) )
2624, 25syl6bbr 263 . . . . . . . . 9  |-  ( -.  ( Q `  K
)  =  L  -> 
( E. k  e.  N  ( Q `  k )  =  L  <->  E. k  e.  ( N  \  { K }
) ( Q `  k )  =  L ) )
2714, 26syl5ibcom 220 . . . . . . . 8  |-  ( ( L  e.  N  /\  A. l  e.  N  E. k  e.  N  ( Q `  k )  =  l )  -> 
( -.  ( Q `
 K )  =  L  ->  E. k  e.  ( N  \  { K } ) ( Q `
 k )  =  L ) )
2827ex 434 . . . . . . 7  |-  ( L  e.  N  ->  ( A. l  e.  N  E. k  e.  N  ( Q `  k )  =  l  ->  ( -.  ( Q `  K
)  =  L  ->  E. k  e.  ( N  \  { K }
) ( Q `  k )  =  L ) ) )
2928com13 80 . . . . . 6  |-  ( -.  ( Q `  K
)  =  L  -> 
( A. l  e.  N  E. k  e.  N  ( Q `  k )  =  l  ->  ( L  e.  N  ->  E. k  e.  ( N  \  { K } ) ( Q `
 k )  =  L ) ) )
3011, 29syl5 32 . . . . 5  |-  ( -.  ( Q `  K
)  =  L  -> 
( Q  e.  P  ->  ( L  e.  N  ->  E. k  e.  ( N  \  { K } ) ( Q `
 k )  =  L ) ) )
319, 30jaoi 379 . . . 4  |-  ( ( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L )  ->  ( Q  e.  P  ->  ( L  e.  N  ->  E. k  e.  ( N  \  { K } ) ( Q `
 k )  =  L ) ) )
3231com13 80 . . 3  |-  ( L  e.  N  ->  ( Q  e.  P  ->  ( ( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L )  ->  E. k  e.  ( N  \  { K } ) ( Q `
 k )  =  L ) ) )
3332impd 431 . 2  |-  ( L  e.  N  ->  (
( Q  e.  P  /\  ( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L ) )  ->  E. k  e.  ( N  \  { K }
) ( Q `  k )  =  L ) )
348, 33syl5bi 217 1  |-  ( L  e.  N  ->  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  ->  E. k  e.  ( N  \  { K } ) ( Q `
 k )  =  L ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   {crab 2724    \ cdif 3330   {csn 3882   ` cfv 5423   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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-tset 14262  df-symg 15888
This theorem is referenced by: (None)
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