Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemeiota Structured version   Visualization version   Unicode version

Theorem cdlemeiota 34152
Description: A translation is uniquely determined by one of its values. (Contributed by NM, 18-Apr-2013.)
Hypotheses
Ref Expression
cdlemg1c.l  |-  .<_  =  ( le `  K )
cdlemg1c.a  |-  A  =  ( Atoms `  K )
cdlemg1c.h  |-  H  =  ( LHyp `  K
)
cdlemg1c.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemeiota  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T )  ->  F  =  ( iota_ f  e.  T  ( f `  P )  =  ( F `  P ) ) )
Distinct variable groups:    A, f    f, F    f, H    f, K   
.<_ , f    P, f    T, f   
f, W

Proof of Theorem cdlemeiota
StepHypRef Expression
1 eqidd 2452 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T )  ->  ( F `  P )  =  ( F `  P ) )
2 simp3 1010 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T )  ->  F  e.  T )
3 cdlemg1c.l . . . . . . 7  |-  .<_  =  ( le `  K )
4 cdlemg1c.a . . . . . . 7  |-  A  =  ( Atoms `  K )
5 cdlemg1c.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
6 cdlemg1c.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
73, 4, 5, 6ltrnel 33704 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
873com23 1214 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T )  ->  (
( F `  P
)  e.  A  /\  -.  ( F `  P
)  .<_  W ) )
93, 4, 5, 6cdleme 34127 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  e.  A  /\  -.  ( F `  P
)  .<_  W ) )  ->  E! f  e.  T  ( f `  P )  =  ( F `  P ) )
108, 9syld3an3 1313 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T )  ->  E! f  e.  T  (
f `  P )  =  ( F `  P ) )
11 fveq1 5864 . . . . . 6  |-  ( f  =  F  ->  (
f `  P )  =  ( F `  P ) )
1211eqeq1d 2453 . . . . 5  |-  ( f  =  F  ->  (
( f `  P
)  =  ( F `
 P )  <->  ( F `  P )  =  ( F `  P ) ) )
1312riota2 6274 . . . 4  |-  ( ( F  e.  T  /\  E! f  e.  T  ( f `  P
)  =  ( F `
 P ) )  ->  ( ( F `
 P )  =  ( F `  P
)  <->  ( iota_ f  e.  T  ( f `  P )  =  ( F `  P ) )  =  F ) )
142, 10, 13syl2anc 667 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T )  ->  (
( F `  P
)  =  ( F `
 P )  <->  ( iota_ f  e.  T  ( f `
 P )  =  ( F `  P
) )  =  F ) )
151, 14mpbid 214 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T )  ->  ( iota_ f  e.  T  ( f `  P )  =  ( F `  P ) )  =  F )
1615eqcomd 2457 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T )  ->  F  =  ( iota_ f  e.  T  ( f `  P )  =  ( F `  P ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   E!wreu 2739   class class class wbr 4402   ` cfv 5582   iota_crio 6251   lecple 15197   Atomscatm 32829   HLchlt 32916   LHypclh 33549   LTrncltrn 33666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-riotaBAD 32525
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-undef 7020  df-map 7474  df-preset 16173  df-poset 16191  df-plt 16204  df-lub 16220  df-glb 16221  df-join 16222  df-meet 16223  df-p0 16285  df-p1 16286  df-lat 16292  df-clat 16354  df-oposet 32742  df-ol 32744  df-oml 32745  df-covers 32832  df-ats 32833  df-atl 32864  df-cvlat 32888  df-hlat 32917  df-llines 33063  df-lplanes 33064  df-lvols 33065  df-lines 33066  df-psubsp 33068  df-pmap 33069  df-padd 33361  df-lhyp 33553  df-laut 33554  df-ldil 33669  df-ltrn 33670  df-trl 33725
This theorem is referenced by:  cdlemg1cN  34154  cdlemg1cex  34155  cdlemm10N  34686
  Copyright terms: Public domain W3C validator