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Theorem symgfix2 16567
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 3481 . . 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 5871 . . . . . . 7  |-  ( q  =  Q  ->  (
q `  K )  =  ( Q `  K ) )
43eqeq1d 2459 . . . . . 6  |-  ( q  =  Q  ->  (
( q `  K
)  =  L  <->  ( Q `  K )  =  L ) )
54elrab 3257 . . . . 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 16544 . . . . . 6  |-  ( Q  e.  P  ->  A. l  e.  N  E. k  e.  N  ( Q `  k )  =  l )
12 eqeq2 2472 . . . . . . . . . . 11  |-  ( l  =  L  ->  (
( Q `  k
)  =  l  <->  ( Q `  k )  =  L ) )
1312rexbidv 2968 . . . . . . . . . 10  |-  ( l  =  L  ->  ( E. k  e.  N  ( Q `  k )  =  l  <->  E. k  e.  N  ( Q `  k )  =  L ) )
1413rspcva 3208 . . . . . . . . 9  |-  ( ( L  e.  N  /\  A. l  e.  N  E. k  e.  N  ( Q `  k )  =  l )  ->  E. k  e.  N  ( Q `  k )  =  L )
15 eqeq2 2472 . . . . . . . . . . . . . . . 16  |-  ( L  =  ( Q `  k )  ->  (
( Q `  K
)  =  L  <->  ( Q `  K )  =  ( Q `  k ) ) )
1615eqcoms 2469 . . . . . . . . . . . . . . 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 5872 . . . . . . . . . . . . . . . 16  |-  ( K  =  k  ->  ( Q `  K )  =  ( Q `  k ) )
1918eqcoms 2469 . . . . . . . . . . . . . . 15  |-  ( k  =  K  ->  ( Q `  K )  =  ( Q `  k ) )
2019necon3bi 2686 . . . . . . . . . . . . . 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 2968 . . . . . . . . . 10  |-  ( -.  ( Q `  K
)  =  L  -> 
( E. k  e.  N  ( Q `  k )  =  L  <->  E. k  e.  N  ( k  =/=  K  /\  ( Q `  k
)  =  L ) ) )
25 rexdifsn 4161 . . . . . . . . . 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811    \ cdif 3468   {csn 4032   ` cfv 5594   Basecbs 14643   SymGrpcsymg 16528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-plusg 14724  df-tset 14730  df-symg 16529
This theorem is referenced by: (None)
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