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Theorem cdlemeiota 31319
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 2436 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T )  ->  ( F `  P )  =  ( F `  P ) )
2 simp3 959 . . . 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 30873 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
873com23 1159 . . . . 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 31294 . . . . 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 1229 . . . 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 5719 . . . . . 6  |-  ( f  =  F  ->  (
f `  P )  =  ( F `  P ) )
1211eqeq1d 2443 . . . . 5  |-  ( f  =  F  ->  (
( f `  P
)  =  ( F `
 P )  <->  ( F `  P )  =  ( F `  P ) ) )
1312riota2 6564 . . . 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 643 . . 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 202 . 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 2440 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E!wreu 2699   class class class wbr 4204   ` cfv 5446   iota_crio 6534   lecple 13528   Atomscatm 29998   HLchlt 30085   LHypclh 30718   LTrncltrn 30835
This theorem is referenced by:  cdlemg1cN  31321  cdlemg1cex  31322  cdlemm10N  31853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893
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