Theorem axlowdimlem12 25833

Description: Lemma for axlowdim 25841. 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

Step	Hyp	Ref	Expression
1		axlowdimlem10.1	. . 3 |- Q = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) )
2	1	fveq1i 6192	. 2 |- ( Q ` K ) = ( ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) ` K )
3		eldifsn 4317	. . 3 |- ( K e. ( ( 1 ... N ) \ { ( I + 1 ) } ) <-> ( K e. ( 1 ... N ) /\ K =/= ( I + 1 ) ) )
4		disjdif 4040	. . . . 5 |- ( { ( I + 1 ) } i^i ( ( 1 ... N ) \ { ( I + 1 ) } ) ) = (/)
5		ovex 6678	. . . . . . 7 |- ( I + 1 ) e. _V
6		1ex 10035	. . . . . . 7 |- 1 e. _V
7	5, 6	fnsn 5946	. . . . . 6 |- { <. ( I + 1 ) , 1 >. } Fn { ( I + 1 ) }
8		c0ex 10034	. . . . . . . 8 |- 0 e. _V
9	8	fconst 6091	. . . . . . 7 |- ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) : ( ( 1 ... N ) \ { ( I + 1 ) } ) --> { 0 }
10		ffn 6045	. . . . . . 7 |- ( ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) : ( ( 1 ... N ) \ { ( I + 1 ) } ) --> { 0 } -> ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) Fn ( ( 1 ... N ) \ { ( I + 1 ) } ) )
11	9, 10	ax-mp 5	. . . . . 6 |- ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) Fn ( ( 1 ... N ) \ { ( I + 1 ) } )
12		fvun2 6270	. . . . . 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 ) )
13	7, 11, 12	mp3an12 1414	. . . . 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 ) )
14	4, 13	mpan 706	. . . 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 ) )
15	8	fvconst2 6469	. . . 4 |- ( K e. ( ( 1 ... N ) \ { ( I + 1 ) } ) -> ( ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ` K ) = 0 )
16	14, 15	eqtrd 2656	. . 3 |- ( K e. ( ( 1 ... N ) \ { ( I + 1 ) } ) -> ( ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) ` K ) = 0 )
17	3, 16	sylbir 225	. 2 |- ( ( K e. ( 1 ... N ) /\ K =/= ( I + 1 ) ) -> ( ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) ` K ) = 0 )
18	2, 17	syl5eq 2668	1 |- ( ( K e. ( 1 ... N ) /\ K =/= ( I + 1 ) ) -> ( Q ` K ) = 0 )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   <.cop 4183   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653

This theorem is referenced by:  axlowdimlem14  25835  axlowdimlem16  25837  axlowdimlem17  25838