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Theorem axlowdimlem11 23377
Description: Lemma for axlowdim 23386. 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 5803 . 2  |-  ( Q `
 ( I  + 
1 ) )  =  ( ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) `  (
I  +  1 ) )
3 ovex 6228 . . . 4  |-  ( I  +  1 )  e. 
_V
4 1ex 9496 . . . 4  |-  1  e.  _V
53, 4fnsn 5582 . . 3  |-  { <. ( I  +  1 ) ,  1 >. }  Fn  { ( I  +  1 ) }
6 c0ex 9495 . . . . 5  |-  0  e.  _V
76fconst 5707 . . . 4  |-  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) : ( ( 1 ... N )  \  {
( I  +  1 ) } ) --> { 0 }
8 ffn 5670 . . . 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 3862 . . . 4  |-  ( { ( I  +  1 ) }  i^i  (
( 1 ... N
)  \  { (
I  +  1 ) } ) )  =  (/)
113snid 4016 . . . 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 5874 . . 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 1315 . 2  |-  ( ( { <. ( I  + 
1 ) ,  1
>. }  u.  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) ) `
 ( I  + 
1 ) )  =  ( { <. (
I  +  1 ) ,  1 >. } `  ( I  +  1
) )
153, 4fvsn 6023 . 2  |-  ( {
<. ( I  +  1 ) ,  1 >. } `  ( I  +  1 ) )  =  1
162, 14, 153eqtri 2487 1  |-  ( Q `
 ( I  + 
1 ) )  =  1
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758    \ cdif 3436    u. cun 3437    i^i cin 3438   (/)c0 3748   {csn 3988   <.cop 3994    X. cxp 4949    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203   0cc0 9397   1c1 9398    + caddc 9400   ...cfz 11558
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-mulcl 9459  ax-i2m1 9465
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206
This theorem is referenced by:  axlowdimlem14  23380  axlowdimlem16  23382
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