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Theorem ltrneq 30946
Description: The equality of two translations is determined by their equality at atoms not under co-atom  W. (Contributed by NM, 20-Jun-2013.)
Hypotheses
Ref Expression
ltrne.l  |-  .<_  =  ( le `  K )
ltrne.a  |-  A  =  ( Atoms `  K )
ltrne.h  |-  H  =  ( LHyp `  K
)
ltrne.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrneq  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( A. p  e.  A  ( -.  p  .<_  W  -> 
( F `  p
)  =  ( G `
 p ) )  <-> 
F  =  G ) )
Distinct variable groups:    A, p    F, p    G, p    H, p    K, p    T, p    W, p
Allowed substitution hint:    .<_ ( p)

Proof of Theorem ltrneq
StepHypRef Expression
1 simp11 987 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A  /\  p  .<_  W )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp12 988 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A  /\  p  .<_  W )  ->  F  e.  T )
3 eqid 2436 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
4 ltrne.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
53, 4atbase 30087 . . . . . . . . 9  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
653ad2ant2 979 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A  /\  p  .<_  W )  ->  p  e.  ( Base `  K
) )
7 simp3 959 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A  /\  p  .<_  W )  ->  p  .<_  W )
8 ltrne.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
9 ltrne.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
10 ltrne.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
113, 8, 9, 10ltrnval1 30931 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Base `  K )  /\  p  .<_  W ) )  ->  ( F `  p )  =  p )
121, 2, 6, 7, 11syl112anc 1188 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A  /\  p  .<_  W )  ->  ( F `  p )  =  p )
13 simp13 989 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A  /\  p  .<_  W )  ->  G  e.  T )
143, 8, 9, 10ltrnval1 30931 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( p  e.  (
Base `  K )  /\  p  .<_  W ) )  ->  ( G `  p )  =  p )
151, 13, 6, 7, 14syl112anc 1188 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A  /\  p  .<_  W )  ->  ( G `  p )  =  p )
1612, 15eqtr4d 2471 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A  /\  p  .<_  W )  ->  ( F `  p )  =  ( G `  p ) )
17163expia 1155 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A )  ->  (
p  .<_  W  ->  ( F `  p )  =  ( G `  p ) ) )
18 pm2.61 165 . . . . 5  |-  ( ( p  .<_  W  ->  ( F `  p )  =  ( G `  p ) )  -> 
( ( -.  p  .<_  W  ->  ( F `  p )  =  ( G `  p ) )  ->  ( F `  p )  =  ( G `  p ) ) )
1917, 18syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A )  ->  (
( -.  p  .<_  W  ->  ( F `  p )  =  ( G `  p ) )  ->  ( F `  p )  =  ( G `  p ) ) )
20 re1tbw2 1520 . . . 4  |-  ( ( F `  p )  =  ( G `  p )  ->  ( -.  p  .<_  W  -> 
( F `  p
)  =  ( G `
 p ) ) )
2119, 20impbid1 195 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A )  ->  (
( -.  p  .<_  W  ->  ( F `  p )  =  ( G `  p ) )  <->  ( F `  p )  =  ( G `  p ) ) )
2221ralbidva 2721 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( A. p  e.  A  ( -.  p  .<_  W  -> 
( F `  p
)  =  ( G `
 p ) )  <->  A. p  e.  A  ( F `  p )  =  ( G `  p ) ) )
234, 9, 10ltrneq2 30945 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( A. p  e.  A  ( F `  p )  =  ( G `  p )  <->  F  =  G ) )
2422, 23bitrd 245 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( A. p  e.  A  ( -.  p  .<_  W  -> 
( F `  p
)  =  ( G `
 p ) )  <-> 
F  =  G ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   Atomscatm 30061   HLchlt 30148   LHypclh 30781   LTrncltrn 30898
This theorem is referenced by:  cdlemj2  31619
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902
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