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Theorem symgfix2 17057
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 3446 . . 3  |-  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  <->  ( Q  e.  P  /\  -.  Q  e.  { q  e.  P  |  ( q `  K )  =  L } ) )
2 ianor 490 . . . . 5  |-  ( -.  ( Q  e.  P  /\  ( Q `  K
)  =  L )  <-> 
( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L ) )
3 fveq1 5881 . . . . . . 7  |-  ( q  =  Q  ->  (
q `  K )  =  ( Q `  K ) )
43eqeq1d 2424 . . . . . 6  |-  ( q  =  Q  ->  (
( q `  K
)  =  L  <->  ( Q `  K )  =  L ) )
54elrab 3228 . . . . 5  |-  ( Q  e.  { q  e.  P  |  ( q `
 K )  =  L }  <->  ( Q  e.  P  /\  ( Q `  K )  =  L ) )
62, 5xchnxbir 310 . . . 4  |-  ( -.  Q  e.  { q  e.  P  |  ( q `  K )  =  L }  <->  ( -.  Q  e.  P  \/  -.  ( Q `  K
)  =  L ) )
76anbi2i 698 . . 3  |-  ( ( Q  e.  P  /\  -.  Q  e.  { q  e.  P  |  ( q `  K )  =  L } )  <-> 
( Q  e.  P  /\  ( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L ) ) )
81, 7bitri 252 . 2  |-  ( Q  e.  ( P  \  { q  e.  P  |  ( q `  K )  =  L } )  <->  ( Q  e.  P  /\  ( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L ) ) )
9 pm2.21 111 . . . . 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 17034 . . . . . 6  |-  ( Q  e.  P  ->  A. l  e.  N  E. k  e.  N  ( Q `  k )  =  l )
12 eqeq2 2437 . . . . . . . . . . 11  |-  ( l  =  L  ->  (
( Q `  k
)  =  l  <->  ( Q `  k )  =  L ) )
1312rexbidv 2936 . . . . . . . . . 10  |-  ( l  =  L  ->  ( E. k  e.  N  ( Q `  k )  =  l  <->  E. k  e.  N  ( Q `  k )  =  L ) )
1413rspcva 3180 . . . . . . . . 9  |-  ( ( L  e.  N  /\  A. l  e.  N  E. k  e.  N  ( Q `  k )  =  l )  ->  E. k  e.  N  ( Q `  k )  =  L )
15 eqeq2 2437 . . . . . . . . . . . . . . . 16  |-  ( L  =  ( Q `  k )  ->  (
( Q `  K
)  =  L  <->  ( Q `  K )  =  ( Q `  k ) ) )
1615eqcoms 2434 . . . . . . . . . . . . . . 15  |-  ( ( Q `  k )  =  L  ->  (
( Q `  K
)  =  L  <->  ( Q `  K )  =  ( Q `  k ) ) )
1716notbid 295 . . . . . . . . . . . . . 14  |-  ( ( Q `  k )  =  L  ->  ( -.  ( Q `  K
)  =  L  <->  -.  ( Q `  K )  =  ( Q `  k ) ) )
18 fveq2 5882 . . . . . . . . . . . . . . . 16  |-  ( K  =  k  ->  ( Q `  K )  =  ( Q `  k ) )
1918eqcoms 2434 . . . . . . . . . . . . . . 15  |-  ( k  =  K  ->  ( Q `  K )  =  ( Q `  k ) )
2019necon3bi 2649 . . . . . . . . . . . . . 14  |-  ( -.  ( Q `  K
)  =  ( Q `
 k )  -> 
k  =/=  K )
2117, 20syl6bi 231 . . . . . . . . . . . . 13  |-  ( ( Q `  k )  =  L  ->  ( -.  ( Q `  K
)  =  L  -> 
k  =/=  K ) )
2221com12 32 . . . . . . . . . . . 12  |-  ( -.  ( Q `  K
)  =  L  -> 
( ( Q `  k )  =  L  ->  k  =/=  K
) )
2322pm4.71rd 639 . . . . . . . . . . 11  |-  ( -.  ( Q `  K
)  =  L  -> 
( ( Q `  k )  =  L  <-> 
( k  =/=  K  /\  ( Q `  k
)  =  L ) ) )
2423rexbidv 2936 . . . . . . . . . 10  |-  ( -.  ( Q `  K
)  =  L  -> 
( E. k  e.  N  ( Q `  k )  =  L  <->  E. k  e.  N  ( k  =/=  K  /\  ( Q `  k
)  =  L ) ) )
25 rexdifsn 4129 . . . . . . . . . 10  |-  ( E. k  e.  ( N 
\  { K }
) ( Q `  k )  =  L  <->  E. k  e.  N  ( k  =/=  K  /\  ( Q `  k
)  =  L ) )
2624, 25syl6bbr 266 . . . . . . . . 9  |-  ( -.  ( Q `  K
)  =  L  -> 
( E. k  e.  N  ( Q `  k )  =  L  <->  E. k  e.  ( N  \  { K }
) ( Q `  k )  =  L ) )
2714, 26syl5ibcom 223 . . . . . . . 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 435 . . . . . . 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 83 . . . . . 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 33 . . . . 5  |-  ( -.  ( Q `  K
)  =  L  -> 
( Q  e.  P  ->  ( L  e.  N  ->  E. k  e.  ( N  \  { K } ) ( Q `
 k )  =  L ) ) )
319, 30jaoi 380 . . . 4  |-  ( ( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L )  ->  ( Q  e.  P  ->  ( L  e.  N  ->  E. k  e.  ( N  \  { K } ) ( Q `
 k )  =  L ) ) )
3231com13 83 . . 3  |-  ( L  e.  N  ->  ( Q  e.  P  ->  ( ( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L )  ->  E. k  e.  ( N  \  { K } ) ( Q `
 k )  =  L ) ) )
3332impd 432 . 2  |-  ( L  e.  N  ->  (
( Q  e.  P  /\  ( -.  Q  e.  P  \/  -.  ( Q `  K )  =  L ) )  ->  E. k  e.  ( N  \  { K }
) ( Q `  k )  =  L ) )
348, 33syl5bi 220 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 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772   {crab 2775    \ cdif 3433   {csn 3998   ` cfv 5601   Basecbs 15121   SymGrpcsymg 17018
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-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-oadd 7198  df-er 7375  df-map 7486  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-nn 10618  df-2 10676  df-3 10677  df-4 10678  df-5 10679  df-6 10680  df-7 10681  df-8 10682  df-9 10683  df-n0 10878  df-z 10946  df-uz 11168  df-fz 11793  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-plusg 15203  df-tset 15209  df-symg 17019
This theorem is referenced by: (None)
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