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Theorem axlowdimlem9 23341
Description: Lemma for axlowdim 23352. 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 5793 . 2  |-  ( P `
 K )  =  ( ( { <. 3 ,  -u 1 >. }  u.  ( (
( 1 ... N
)  \  { 3 } )  X.  {
0 } ) ) `
 K )
3 eldifsn 4101 . . 3  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  <-> 
( K  e.  ( 1 ... N )  /\  K  =/=  3
) )
4 disjdif 3852 . . . . 5  |-  ( { 3 }  i^i  (
( 1 ... N
)  \  { 3 } ) )  =  (/)
5 3re 10499 . . . . . . . 8  |-  3  e.  RR
65elexi 3081 . . . . . . 7  |-  3  e.  _V
7 negex 9712 . . . . . . 7  |-  -u 1  e.  _V
86, 7fnsn 5572 . . . . . 6  |-  { <. 3 ,  -u 1 >. }  Fn  { 3 }
9 c0ex 9484 . . . . . . . 8  |-  0  e.  _V
109fconst 5697 . . . . . . 7  |-  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) : ( ( 1 ... N )  \  {
3 } ) --> { 0 }
11 ffn 5660 . . . . . . 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 5865 . . . . . 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 1305 . . . . 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 670 . . . 4  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( {
<. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) ) `  K )  =  ( ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) `  K ) )
169fvconst2 6035 . . . 4  |-  ( K  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) `  K )  =  0 )
1715, 16eqtrd 2492 . . 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 2504 1  |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  3 )  -> 
( P `  K
)  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644    \ cdif 3426    u. cun 3427    i^i cin 3428   (/)c0 3738   {csn 3978   <.cop 3984    X. cxp 4939    Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193   RRcr 9385   0cc0 9386   1c1 9387   -ucneg 9700   3c3 10476   ...cfz 11547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-i2m1 9454  ax-1ne0 9455  ax-rrecex 9458  ax-cnre 9459
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196  df-neg 9702  df-2 10484  df-3 10485
This theorem is referenced by:  axlowdimlem16  23348  axlowdimlem17  23349
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