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Theorem ltrncnvatb 34934
Description: The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.)
Hypotheses
Ref Expression
ltrnatb.b  |-  B  =  ( Base `  K
)
ltrnatb.a  |-  A  =  ( Atoms `  K )
ltrnatb.h  |-  H  =  ( LHyp `  K
)
ltrnatb.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrncnvatb  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  A  <->  ( `' F `  P )  e.  A
) )

Proof of Theorem ltrncnvatb
StepHypRef Expression
1 ltrnatb.b . . . . . 6  |-  B  =  ( Base `  K
)
2 ltrnatb.h . . . . . 6  |-  H  =  ( LHyp `  K
)
3 ltrnatb.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
41, 2, 3ltrn1o 34920 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
543adant3 1016 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  F : B
-1-1-onto-> B )
6 simp3 998 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  P  e.  B )
7 f1ocnvdm 6174 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  P  e.  B )  ->  ( `' F `  P )  e.  B
)
85, 6, 7syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( `' F `  P )  e.  B )
9 ltrnatb.a . . . 4  |-  A  =  ( Atoms `  K )
101, 9, 2, 3ltrnatb 34933 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( `' F `  P )  e.  B
)  ->  ( ( `' F `  P )  e.  A  <->  ( F `  ( `' F `  P ) )  e.  A ) )
118, 10syld3an3 1273 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( `' F `  P )  e.  A  <->  ( F `  ( `' F `  P ) )  e.  A ) )
12 f1ocnvfv2 6169 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  P  e.  B )  ->  ( F `  ( `' F `  P ) )  =  P )
135, 6, 12syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( F `  ( `' F `  P ) )  =  P )
1413eleq1d 2536 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( F `  ( `' F `  P )
)  e.  A  <->  P  e.  A ) )
1511, 14bitr2d 254 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  A  <->  ( `' F `  P )  e.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   `'ccnv 4998   -1-1-onto->wf1o 5585   ` cfv 5586   Basecbs 14486   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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2819  df-rex 2820  df-reu 2821  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-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-map 7419  df-plt 15441  df-glb 15458  df-p0 15522  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-hlat 34148  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901
This theorem is referenced by:  ltrncnvat  34937
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