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Theorem axlowdimlem11 25598
Description: Lemma for axlowdim 25607. Calculate the value of  Q at 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
axlowdimlem11  |-  ( Q `
 ( I  + 
1 ) )  =  1

Proof of Theorem axlowdimlem11
StepHypRef Expression
1 axlowdimlem10.1 . . 3  |-  Q  =  ( { <. (
I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) )
21fveq1i 5662 . 2  |-  ( Q `
 ( I  + 
1 ) )  =  ( ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) `  (
I  +  1 ) )
3 ovex 6038 . . . 4  |-  ( I  +  1 )  e. 
_V
4 1ex 9012 . . . 4  |-  1  e.  _V
53, 4fnsn 5437 . . 3  |-  { <. ( I  +  1 ) ,  1 >. }  Fn  { ( I  +  1 ) }
6 c0ex 9011 . . . . 5  |-  0  e.  _V
76fconst 5562 . . . 4  |-  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) : ( ( 1 ... N )  \  {
( I  +  1 ) } ) --> { 0 }
8 ffn 5524 . . . 4  |-  ( ( ( ( 1 ... N )  \  {
( I  +  1 ) } )  X. 
{ 0 } ) : ( ( 1 ... N )  \  { ( I  + 
1 ) } ) --> { 0 }  ->  ( ( ( 1 ... N )  \  {
( I  +  1 ) } )  X. 
{ 0 } )  Fn  ( ( 1 ... N )  \  { ( I  + 
1 ) } ) )
97, 8ax-mp 8 . . 3  |-  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } )  Fn  ( ( 1 ... N )  \  {
( I  +  1 ) } )
10 disjdif 3636 . . . 4  |-  ( { ( I  +  1 ) }  i^i  (
( 1 ... N
)  \  { (
I  +  1 ) } ) )  =  (/)
113snid 3777 . . . 4  |-  ( I  +  1 )  e. 
{ ( I  + 
1 ) }
1210, 11pm3.2i 442 . . 3  |-  ( ( { ( I  + 
1 ) }  i^i  ( ( 1 ... N )  \  {
( I  +  1 ) } ) )  =  (/)  /\  (
I  +  1 )  e.  { ( I  +  1 ) } )
13 fvun1 5726 . . 3  |-  ( ( { <. ( 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 ) } ) )  =  (/)  /\  ( I  +  1 )  e.  { ( I  +  1 ) } ) )  -> 
( ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) `  (
I  +  1 ) )  =  ( {
<. ( I  +  1 ) ,  1 >. } `  ( I  +  1 ) ) )
145, 9, 12, 13mp3an 1279 . 2  |-  ( ( { <. ( I  + 
1 ) ,  1
>. }  u.  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) ) `
 ( I  + 
1 ) )  =  ( { <. (
I  +  1 ) ,  1 >. } `  ( I  +  1
) )
153, 4fvsn 5858 . 2  |-  ( {
<. ( I  +  1 ) ,  1 >. } `  ( I  +  1 ) )  =  1
162, 14, 153eqtri 2404 1  |-  ( Q `
 ( I  + 
1 ) )  =  1
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717    \ cdif 3253    u. cun 3254    i^i cin 3255   (/)c0 3564   {csn 3750   <.cop 3753    X. cxp 4809    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013   0cc0 8916   1c1 8917    + caddc 8919   ...cfz 10968
This theorem is referenced by:  axlowdimlem14  25601  axlowdimlem16  25603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-mulcl 8978  ax-i2m1 8984
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016
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