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Theorem ltrneq 30960
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 985 . . . . . . . 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 986 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A  /\  p  .<_  W )  ->  F  e.  T )
3 eqid 2296 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
4 ltrne.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
53, 4atbase 30101 . . . . . . . . 9  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
653ad2ant2 977 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A  /\  p  .<_  W )  ->  p  e.  ( Base `  K
) )
7 simp3 957 . . . . . . . 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 30945 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Base `  K )  /\  p  .<_  W ) )  ->  ( F `  p )  =  p )
121, 2, 6, 7, 11syl112anc 1186 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A  /\  p  .<_  W )  ->  ( F `  p )  =  p )
13 simp13 987 . . . . . . . 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 30945 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( p  e.  (
Base `  K )  /\  p  .<_  W ) )  ->  ( G `  p )  =  p )
151, 13, 6, 7, 14syl112anc 1186 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A  /\  p  .<_  W )  ->  ( G `  p )  =  p )
1612, 15eqtr4d 2331 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A  /\  p  .<_  W )  ->  ( F `  p )  =  ( G `  p ) )
17163expia 1153 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  p  e.  A )  ->  (
p  .<_  W  ->  ( F `  p )  =  ( G `  p ) ) )
18 pm2.61 163 . . . . 5  |-  ( ( p  .<_  W  ->  ( F `  p )  =  ( G `  p ) )  -> 
( ( -.  p  .<_  W  ->  ( F `  p )  =  ( G `  p ) )  ->  ( F `  p )  =  ( G `  p ) ) )
1917, 18syl 15 . . . 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 1501 . . . 4  |-  ( ( F `  p )  =  ( G `  p )  ->  ( -.  p  .<_  W  -> 
( F `  p
)  =  ( G `
 p ) ) )
2119, 20impbid1 194 . . 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 2572 . 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 30959 . 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 244 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912
This theorem is referenced by:  cdlemj2  31633
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-3an 936  df-tru 1310  df-fal 1311  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-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-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-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-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916
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