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Theorem axlowdimlem12 25892
Description: Lemma for axlowdim 25900. Calculate the value of  Q away from its distunguished point. (Contributed by Scott Fenton, 21-Apr-2013.)
Hypothesis
Ref Expression
axlowdimlem10.1  |-  Q  =  ( { <. (
I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) )
Assertion
Ref Expression
axlowdimlem12  |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  ( I  + 
1 ) )  -> 
( Q `  K
)  =  0 )

Proof of Theorem axlowdimlem12
StepHypRef Expression
1 axlowdimlem10.1 . . 3  |-  Q  =  ( { <. (
I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) )
21fveq1i 5729 . 2  |-  ( Q `
 K )  =  ( ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) `  K
)
3 eldifsn 3927 . . 3  |-  ( K  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  <-> 
( K  e.  ( 1 ... N )  /\  K  =/=  (
I  +  1 ) ) )
4 disjdif 3700 . . . . 5  |-  ( { ( I  +  1 ) }  i^i  (
( 1 ... N
)  \  { (
I  +  1 ) } ) )  =  (/)
5 ovex 6106 . . . . . . 7  |-  ( I  +  1 )  e. 
_V
6 1ex 9086 . . . . . . 7  |-  1  e.  _V
75, 6fnsn 5504 . . . . . 6  |-  { <. ( I  +  1 ) ,  1 >. }  Fn  { ( I  +  1 ) }
8 c0ex 9085 . . . . . . . 8  |-  0  e.  _V
98fconst 5629 . . . . . . 7  |-  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) : ( ( 1 ... N )  \  {
( I  +  1 ) } ) --> { 0 }
10 ffn 5591 . . . . . . 7  |-  ( ( ( ( 1 ... N )  \  {
( I  +  1 ) } )  X. 
{ 0 } ) : ( ( 1 ... N )  \  { ( I  + 
1 ) } ) --> { 0 }  ->  ( ( ( 1 ... N )  \  {
( I  +  1 ) } )  X. 
{ 0 } )  Fn  ( ( 1 ... N )  \  { ( I  + 
1 ) } ) )
119, 10ax-mp 8 . . . . . 6  |-  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } )  Fn  ( ( 1 ... N )  \  {
( I  +  1 ) } )
12 fvun2 5795 . . . . . 6  |-  ( ( { <. ( I  + 
1 ) ,  1
>. }  Fn  { ( I  +  1 ) }  /\  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } )  Fn  ( ( 1 ... N )  \  {
( I  +  1 ) } )  /\  ( ( { ( I  +  1 ) }  i^i  ( ( 1 ... N ) 
\  { ( I  +  1 ) } ) )  =  (/)  /\  K  e.  ( ( 1 ... N ) 
\  { ( I  +  1 ) } ) ) )  -> 
( ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) `  K
)  =  ( ( ( ( 1 ... N )  \  {
( I  +  1 ) } )  X. 
{ 0 } ) `
 K ) )
137, 11, 12mp3an12 1269 . . . . 5  |-  ( ( ( { ( I  +  1 ) }  i^i  ( ( 1 ... N )  \  { ( I  + 
1 ) } ) )  =  (/)  /\  K  e.  ( ( 1 ... N )  \  {
( I  +  1 ) } ) )  ->  ( ( {
<. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) ) `
 K )  =  ( ( ( ( 1 ... N ) 
\  { ( I  +  1 ) } )  X.  { 0 } ) `  K
) )
144, 13mpan 652 . . . 4  |-  ( K  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  ->  ( ( {
<. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) ) `
 K )  =  ( ( ( ( 1 ... N ) 
\  { ( I  +  1 ) } )  X.  { 0 } ) `  K
) )
158fvconst2 5947 . . . 4  |-  ( K  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  ->  ( ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) `  K )  =  0 )
1614, 15eqtrd 2468 . . 3  |-  ( K  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  ->  ( ( {
<. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) ) `
 K )  =  0 )
173, 16sylbir 205 . 2  |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  ( I  + 
1 ) )  -> 
( ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) `  K
)  =  0 )
182, 17syl5eq 2480 1  |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  ( I  + 
1 ) )  -> 
( Q `  K
)  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317    u. cun 3318    i^i cin 3319   (/)c0 3628   {csn 3814   <.cop 3817    X. cxp 4876    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991    + caddc 8993   ...cfz 11043
This theorem is referenced by:  axlowdimlem14  25894  axlowdimlem16  25896  axlowdimlem17  25897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-mulcl 9052  ax-i2m1 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084
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