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

Theorem ltrniotaval 31392
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 31371 . 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 6328 . . . . . . 7  |-  F/_ f
( iota_ f  e.  T
( f `  P
)  =  Q )
86, 7nfcxfr 2429 . . . . . 6  |-  F/_ f F
9 nfcv 2432 . . . . . 6  |-  F/_ f P
108, 9nffv 5548 . . . . 5  |-  F/_ f
( F `  P
)
1110nfeq1 2441 . . . 4  |-  F/ f ( F `  P
)  =  Q
12 fveq1 5540 . . . . 5  |-  ( f  =  F  ->  (
f `  P )  =  ( F `  P ) )
1312eqeq1d 2304 . . . 4  |-  ( f  =  F  ->  (
( f `  P
)  =  Q  <->  ( F `  P )  =  Q ) )
1411, 6, 13riotaprop 6344 . . 3  |-  ( E! f  e.  T  ( f `  P )  =  Q  ->  ( F  e.  T  /\  ( F `  P )  =  Q ) )
1514simprd 449 . 2  |-  ( E! f  e.  T  ( f `  P )  =  Q  ->  ( F `  P )  =  Q )
165, 15syl 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( F `  P )  =  Q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E!wreu 2558   class class class wbr 4039   ` cfv 5271   iota_crio 6313   lecple 13231   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912
This theorem is referenced by:  ltrniotacnvval  31393  ltrniotaidvalN  31394  ltrniotavalbN  31395  cdlemm10N  31930  cdlemn2  32007  cdlemn3  32009  cdlemn9  32017  dihmeetlem13N  32131  dih1dimatlem0  32140  dihjatcclem3  32232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970
  Copyright terms: Public domain W3C validator