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Theorem axlowdimlem12 23932
Description: Lemma for axlowdim 23940. Calculate the value of  Q away from its distinguished 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 5865 . 2  |-  ( Q `
 K )  =  ( ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) `  K
)
3 eldifsn 4152 . . 3  |-  ( K  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  <-> 
( K  e.  ( 1 ... N )  /\  K  =/=  (
I  +  1 ) ) )
4 disjdif 3899 . . . . 5  |-  ( { ( I  +  1 ) }  i^i  (
( 1 ... N
)  \  { (
I  +  1 ) } ) )  =  (/)
5 ovex 6307 . . . . . . 7  |-  ( I  +  1 )  e. 
_V
6 1ex 9587 . . . . . . 7  |-  1  e.  _V
75, 6fnsn 5639 . . . . . 6  |-  { <. ( I  +  1 ) ,  1 >. }  Fn  { ( I  +  1 ) }
8 c0ex 9586 . . . . . . . 8  |-  0  e.  _V
98fconst 5769 . . . . . . 7  |-  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) : ( ( 1 ... N )  \  {
( I  +  1 ) } ) --> { 0 }
10 ffn 5729 . . . . . . 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 5 . . . . . 6  |-  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } )  Fn  ( ( 1 ... N )  \  {
( I  +  1 ) } )
12 fvun2 5937 . . . . . 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 1314 . . . . 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 670 . . . 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 6114 . . . 4  |-  ( K  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  ->  ( ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) `  K )  =  0 )
1614, 15eqtrd 2508 . . 3  |-  ( K  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  ->  ( ( {
<. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) ) `
 K )  =  0 )
173, 16sylbir 213 . 2  |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  ( I  + 
1 ) )  -> 
( ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) `  K
)  =  0 )
182, 17syl5eq 2520 1  |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  ( I  + 
1 ) )  -> 
( Q `  K
)  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473    u. cun 3474    i^i cin 3475   (/)c0 3785   {csn 4027   <.cop 4033    X. cxp 4997    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489    + caddc 9491   ...cfz 11668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-mulcl 9550  ax-i2m1 9556
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285
This theorem is referenced by:  axlowdimlem14  23934  axlowdimlem16  23936  axlowdimlem17  23937
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