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Theorem ltrniotaval 36704
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 36683 . 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 6239 . . . . . . 7  |-  F/_ f
( iota_ f  e.  T  ( f `  P
)  =  Q )
86, 7nfcxfr 2614 . . . . . 6  |-  F/_ f F
9 nfcv 2616 . . . . . 6  |-  F/_ f P
108, 9nffv 5855 . . . . 5  |-  F/_ f
( F `  P
)
1110nfeq1 2631 . . . 4  |-  F/ f ( F `  P
)  =  Q
12 fveq1 5847 . . . . 5  |-  ( f  =  F  ->  (
f `  P )  =  ( F `  P ) )
1312eqeq1d 2456 . . . 4  |-  ( f  =  F  ->  (
( f `  P
)  =  Q  <->  ( F `  P )  =  Q ) )
1411, 6, 13riotaprop 6255 . . 3  |-  ( E! f  e.  T  ( f `  P )  =  Q  ->  ( F  e.  T  /\  ( F `  P )  =  Q ) )
1514simprd 461 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E!wreu 2806   class class class wbr 4439   ` cfv 5570   iota_crio 6231   lecple 14791   Atomscatm 35385   HLchlt 35472   LHypclh 36105   LTrncltrn 36222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-riotaBAD 35081
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-undef 6994  df-map 7414  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619  df-lplanes 35620  df-lvols 35621  df-lines 35622  df-psubsp 35624  df-pmap 35625  df-padd 35917  df-lhyp 36109  df-laut 36110  df-ldil 36225  df-ltrn 36226  df-trl 36281
This theorem is referenced by:  ltrniotacnvval  36705  ltrniotaidvalN  36706  ltrniotavalbN  36707  cdlemm10N  37242  cdlemn2  37319  cdlemn3  37321  cdlemn9  37329  dihmeetlem13N  37443  dih1dimatlem0  37452  dihjatcclem3  37544
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