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Theorem ltrniotaval 35377
Description: Value of the unique translation specified by a value. (Contributed by NM, 21-Feb-2014.)
Hypotheses
Ref Expression
ltrniotaval.l  |-  .<_  =  ( le `  K )
ltrniotaval.a  |-  A  =  ( Atoms `  K )
ltrniotaval.h  |-  H  =  ( LHyp `  K
)
ltrniotaval.t  |-  T  =  ( ( LTrn `  K
) `  W )
ltrniotaval.f  |-  F  =  ( iota_ f  e.  T  ( f `  P
)  =  Q )
Assertion
Ref Expression
ltrniotaval  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( F `  P )  =  Q )
Distinct variable groups:    A, f    f, H    f, K    .<_ , f    P, f    Q, f    T, f   
f, W
Allowed substitution hint:    F( f)

Proof of Theorem ltrniotaval
StepHypRef Expression
1 ltrniotaval.l . . 3  |-  .<_  =  ( le `  K )
2 ltrniotaval.a . . 3  |-  A  =  ( Atoms `  K )
3 ltrniotaval.h . . 3  |-  H  =  ( LHyp `  K
)
4 ltrniotaval.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
51, 2, 3, 4cdleme 35356 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  E! f  e.  T  ( f `  P )  =  Q )
6 ltrniotaval.f . . . . . . 7  |-  F  =  ( iota_ f  e.  T  ( f `  P
)  =  Q )
7 nfriota1 6250 . . . . . . 7  |-  F/_ f
( iota_ f  e.  T  ( f `  P
)  =  Q )
86, 7nfcxfr 2627 . . . . . 6  |-  F/_ f F
9 nfcv 2629 . . . . . 6  |-  F/_ f P
108, 9nffv 5871 . . . . 5  |-  F/_ f
( F `  P
)
1110nfeq1 2644 . . . 4  |-  F/ f ( F `  P
)  =  Q
12 fveq1 5863 . . . . 5  |-  ( f  =  F  ->  (
f `  P )  =  ( F `  P ) )
1312eqeq1d 2469 . . . 4  |-  ( f  =  F  ->  (
( f `  P
)  =  Q  <->  ( F `  P )  =  Q ) )
1411, 6, 13riotaprop 6267 . . 3  |-  ( E! f  e.  T  ( f `  P )  =  Q  ->  ( F  e.  T  /\  ( F `  P )  =  Q ) )
1514simprd 463 . 2  |-  ( E! f  e.  T  ( f `  P )  =  Q  ->  ( F `  P )  =  Q )
165, 15syl 16 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( F `  P )  =  Q )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E!wreu 2816   class class class wbr 4447   ` cfv 5586   iota_crio 6242   lecple 14558   Atomscatm 34060   HLchlt 34147   LHypclh 34780   LTrncltrn 34897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-undef 6999  df-map 7419  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955
This theorem is referenced by:  ltrniotacnvval  35378  ltrniotaidvalN  35379  ltrniotavalbN  35380  cdlemm10N  35915  cdlemn2  35992  cdlemn3  35994  cdlemn9  36002  dihmeetlem13N  36116  dih1dimatlem0  36125  dihjatcclem3  36217
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