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Theorem axlowdimlem11 24382
Description: Lemma for axlowdim 24391. 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 5873 . 2  |-  ( Q `
 ( I  + 
1 ) )  =  ( ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) `  (
I  +  1 ) )
3 ovex 6324 . . . 4  |-  ( I  +  1 )  e. 
_V
4 1ex 9608 . . . 4  |-  1  e.  _V
53, 4fnsn 5647 . . 3  |-  { <. ( I  +  1 ) ,  1 >. }  Fn  { ( I  +  1 ) }
6 c0ex 9607 . . . . 5  |-  0  e.  _V
76fconst 5777 . . . 4  |-  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) : ( ( 1 ... N )  \  {
( I  +  1 ) } ) --> { 0 }
8 ffn 5737 . . . 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 5 . . 3  |-  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } )  Fn  ( ( 1 ... N )  \  {
( I  +  1 ) } )
10 disjdif 3903 . . . 4  |-  ( { ( I  +  1 ) }  i^i  (
( 1 ... N
)  \  { (
I  +  1 ) } ) )  =  (/)
113snid 4060 . . . 4  |-  ( I  +  1 )  e. 
{ ( I  + 
1 ) }
1210, 11pm3.2i 455 . . 3  |-  ( ( { ( I  + 
1 ) }  i^i  ( ( 1 ... N )  \  {
( I  +  1 ) } ) )  =  (/)  /\  (
I  +  1 )  e.  { ( I  +  1 ) } )
13 fvun1 5944 . . 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 1324 . 2  |-  ( ( { <. ( I  + 
1 ) ,  1
>. }  u.  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) ) `
 ( I  + 
1 ) )  =  ( { <. (
I  +  1 ) ,  1 >. } `  ( I  +  1
) )
153, 4fvsn 6105 . 2  |-  ( {
<. ( I  +  1 ) ,  1 >. } `  ( I  +  1 ) )  =  1
162, 14, 153eqtri 2490 1  |-  ( Q `
 ( I  + 
1 ) )  =  1
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1395    e. wcel 1819    \ cdif 3468    u. cun 3469    i^i cin 3470   (/)c0 3793   {csn 4032   <.cop 4038    X. cxp 5006    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512   ...cfz 11697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-mulcl 9571  ax-i2m1 9577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299
This theorem is referenced by:  axlowdimlem14  24385  axlowdimlem16  24387
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