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Theorem axlowdimlem9 24578
Description: Lemma for axlowdim 24589. Calulate the value of  P away from three. (Contributed by Scott Fenton, 21-Apr-2013.)
Hypothesis
Ref Expression
axlowdimlem7.1  |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N )  \  { 3 } )  X.  { 0 } ) )
Assertion
Ref Expression
axlowdimlem9  |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  3 )  -> 
( P `  K
)  =  0 )

Proof of Theorem axlowdimlem9
StepHypRef Expression
1 axlowdimlem7.1 . . 3  |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N )  \  { 3 } )  X.  { 0 } ) )
21fveq1i 5526 . 2  |-  ( P `
 K )  =  ( ( { <. 3 ,  -u 1 >. }  u.  ( (
( 1 ... N
)  \  { 3 } )  X.  {
0 } ) ) `
 K )
3 eldifsn 3749 . . 3  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  <-> 
( K  e.  ( 1 ... N )  /\  K  =/=  3
) )
4 disjdif 3526 . . . . 5  |-  ( { 3 }  i^i  (
( 1 ... N
)  \  { 3 } ) )  =  (/)
5 3re 9817 . . . . . . . 8  |-  3  e.  RR
65elexi 2797 . . . . . . 7  |-  3  e.  _V
7 negex 9050 . . . . . . 7  |-  -u 1  e.  _V
86, 7fnsn 5304 . . . . . 6  |-  { <. 3 ,  -u 1 >. }  Fn  { 3 }
9 c0ex 8832 . . . . . . . 8  |-  0  e.  _V
109fconst 5427 . . . . . . 7  |-  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) : ( ( 1 ... N )  \  {
3 } ) --> { 0 }
11 ffn 5389 . . . . . . 7  |-  ( ( ( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) : ( ( 1 ... N )  \  { 3 } ) --> { 0 }  ->  ( ( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } )  Fn  ( ( 1 ... N )  \  { 3 } ) )
1210, 11ax-mp 8 . . . . . 6  |-  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } )  Fn  ( ( 1 ... N )  \  {
3 } )
13 fvun2 5591 . . . . . 6  |-  ( ( { <. 3 ,  -u
1 >. }  Fn  {
3 }  /\  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } )  Fn  ( ( 1 ... N )  \  { 3 } )  /\  ( ( { 3 }  i^i  (
( 1 ... N
)  \  { 3 } ) )  =  (/)  /\  K  e.  ( ( 1 ... N
)  \  { 3 } ) ) )  ->  ( ( {
<. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) ) `  K )  =  ( ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) `  K ) )
148, 12, 13mp3an12 1267 . . . . 5  |-  ( ( ( { 3 }  i^i  ( ( 1 ... N )  \  { 3 } ) )  =  (/)  /\  K  e.  ( ( 1 ... N )  \  {
3 } ) )  ->  ( ( {
<. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) ) `  K )  =  ( ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) `  K ) )
154, 14mpan 651 . . . 4  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( {
<. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) ) `  K )  =  ( ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) `  K ) )
169fvconst2 5729 . . . 4  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) `  K )  =  0 )
1715, 16eqtrd 2315 . . 3  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( {
<. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) ) `  K )  =  0 )
183, 17sylbir 204 . 2  |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  3 )  -> 
( ( { <. 3 ,  -u 1 >. }  u.  ( (
( 1 ... N
)  \  { 3 } )  X.  {
0 } ) ) `
 K )  =  0 )
192, 18syl5eq 2327 1  |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  3 )  -> 
( P `  K
)  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640   <.cop 3643    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738   -ucneg 9038   3c3 9796   ...cfz 10782
This theorem is referenced by:  axlowdimlem16  24585  axlowdimlem17  24586
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-neg 9040  df-2 9804  df-3 9805
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