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Theorem axlowdimlem9 24650
Description: Lemma for axlowdim 24661. 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 5542 . 2  |-  ( P `
 K )  =  ( ( { <. 3 ,  -u 1 >. }  u.  ( (
( 1 ... N
)  \  { 3 } )  X.  {
0 } ) ) `
 K )
3 eldifsn 3762 . . 3  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  <-> 
( K  e.  ( 1 ... N )  /\  K  =/=  3
) )
4 disjdif 3539 . . . . 5  |-  ( { 3 }  i^i  (
( 1 ... N
)  \  { 3 } ) )  =  (/)
5 3re 9833 . . . . . . . 8  |-  3  e.  RR
65elexi 2810 . . . . . . 7  |-  3  e.  _V
7 negex 9066 . . . . . . 7  |-  -u 1  e.  _V
86, 7fnsn 5320 . . . . . 6  |-  { <. 3 ,  -u 1 >. }  Fn  { 3 }
9 c0ex 8848 . . . . . . . 8  |-  0  e.  _V
109fconst 5443 . . . . . . 7  |-  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) : ( ( 1 ... N )  \  {
3 } ) --> { 0 }
11 ffn 5405 . . . . . . 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 5607 . . . . . 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 5745 . . . 4  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) `  K )  =  0 )
1715, 16eqtrd 2328 . . 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 2340 1  |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  3 )  -> 
( P `  K
)  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   <.cop 3656    X. cxp 4703    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754   -ucneg 9054   3c3 9812   ...cfz 10798
This theorem is referenced by:  axlowdimlem16  24657  axlowdimlem17  24658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-neg 9056  df-2 9820  df-3 9821
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