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Theorem axlowdimlem9 24455
Description: Lemma for axlowdim 24466. Calculate 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 5849 . 2  |-  ( P `
 K )  =  ( ( { <. 3 ,  -u 1 >. }  u.  ( (
( 1 ... N
)  \  { 3 } )  X.  {
0 } ) ) `
 K )
3 eldifsn 4141 . . 3  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  <-> 
( K  e.  ( 1 ... N )  /\  K  =/=  3
) )
4 disjdif 3888 . . . . 5  |-  ( { 3 }  i^i  (
( 1 ... N
)  \  { 3 } ) )  =  (/)
5 3re 10605 . . . . . . . 8  |-  3  e.  RR
65elexi 3116 . . . . . . 7  |-  3  e.  _V
7 negex 9809 . . . . . . 7  |-  -u 1  e.  _V
86, 7fnsn 5623 . . . . . 6  |-  { <. 3 ,  -u 1 >. }  Fn  { 3 }
9 c0ex 9579 . . . . . . . 8  |-  0  e.  _V
109fconst 5753 . . . . . . 7  |-  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) : ( ( 1 ... N )  \  {
3 } ) --> { 0 }
11 ffn 5713 . . . . . . 7  |-  ( ( ( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) : ( ( 1 ... N )  \  { 3 } ) --> { 0 }  ->  ( ( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } )  Fn  ( ( 1 ... N )  \  { 3 } ) )
1210, 11ax-mp 5 . . . . . 6  |-  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } )  Fn  ( ( 1 ... N )  \  {
3 } )
13 fvun2 5920 . . . . . 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 1312 . . . . 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 668 . . . 4  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( {
<. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) ) `  K )  =  ( ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) `  K ) )
169fvconst2 6103 . . . 4  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) `  K )  =  0 )
1715, 16eqtrd 2495 . . 3  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( {
<. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) ) `  K )  =  0 )
183, 17sylbir 213 . 2  |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  3 )  -> 
( ( { <. 3 ,  -u 1 >. }  u.  ( (
( 1 ... N
)  \  { 3 } )  X.  {
0 } ) ) `
 K )  =  0 )
192, 18syl5eq 2507 1  |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  3 )  -> 
( P `  K
)  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649    \ cdif 3458    u. cun 3459    i^i cin 3460   (/)c0 3783   {csn 4016   <.cop 4022    X. cxp 4986    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481   1c1 9482   -ucneg 9797   3c3 10582   ...cfz 11675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rrecex 9553  ax-cnre 9554
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-neg 9799  df-2 10590  df-3 10591
This theorem is referenced by:  axlowdimlem16  24462  axlowdimlem17  24463
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