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Theorem axlowdimlem9 25801
Description: Lemma for axlowdim 25812. 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 5696 . 2  |-  ( P `
 K )  =  ( ( { <. 3 ,  -u 1 >. }  u.  ( (
( 1 ... N
)  \  { 3 } )  X.  {
0 } ) ) `
 K )
3 eldifsn 3895 . . 3  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  <-> 
( K  e.  ( 1 ... N )  /\  K  =/=  3
) )
4 disjdif 3668 . . . . 5  |-  ( { 3 }  i^i  (
( 1 ... N
)  \  { 3 } ) )  =  (/)
5 3re 10035 . . . . . . . 8  |-  3  e.  RR
65elexi 2933 . . . . . . 7  |-  3  e.  _V
7 negex 9268 . . . . . . 7  |-  -u 1  e.  _V
86, 7fnsn 5471 . . . . . 6  |-  { <. 3 ,  -u 1 >. }  Fn  { 3 }
9 c0ex 9049 . . . . . . . 8  |-  0  e.  _V
109fconst 5596 . . . . . . 7  |-  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) : ( ( 1 ... N )  \  {
3 } ) --> { 0 }
11 ffn 5558 . . . . . . 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 5762 . . . . . 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 1269 . . . . 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 652 . . . 4  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( {
<. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) ) `  K )  =  ( ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) `  K ) )
169fvconst2 5914 . . . 4  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) `  K )  =  0 )
1715, 16eqtrd 2444 . . 3  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( {
<. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) ) `  K )  =  0 )
183, 17sylbir 205 . 2  |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  3 )  -> 
( ( { <. 3 ,  -u 1 >. }  u.  ( (
( 1 ... N
)  \  { 3 } )  X.  {
0 } ) ) `
 K )  =  0 )
192, 18syl5eq 2456 1  |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  3 )  -> 
( P `  K
)  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575    \ cdif 3285    u. cun 3286    i^i cin 3287   (/)c0 3596   {csn 3782   <.cop 3785    X. cxp 4843    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048   RRcr 8953   0cc0 8954   1c1 8955   -ucneg 9256   3c3 10014   ...cfz 11007
This theorem is referenced by:  axlowdimlem16  25808  axlowdimlem17  25809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-i2m1 9022  ax-1ne0 9023  ax-rrecex 9026  ax-cnre 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-neg 9258  df-2 10022  df-3 10023
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