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Theorem ldilcnv 33112
Description: The converse of a lattice dilation is a lattice dilation. (Contributed by NM, 10-May-2013.)
Hypotheses
Ref Expression
ldilcnv.h  |-  H  =  ( LHyp `  K
)
ldilcnv.d  |-  D  =  ( ( LDil `  K
) `  W )
Assertion
Ref Expression
ldilcnv  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  `' F  e.  D )

Proof of Theorem ldilcnv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpll 752 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  K  e.  HL )
2 ldilcnv.h . . . 4  |-  H  =  ( LHyp `  K
)
3 eqid 2402 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
4 ldilcnv.d . . . 4  |-  D  =  ( ( LDil `  K
) `  W )
52, 3, 4ldillaut 33108 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  F  e.  ( LAut `  K )
)
63lautcnv 33087 . . 3  |-  ( ( K  e.  HL  /\  F  e.  ( LAut `  K ) )  ->  `' F  e.  ( LAut `  K ) )
71, 5, 6syl2anc 659 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  `' F  e.  ( LAut `  K
) )
8 eqid 2402 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
9 eqid 2402 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
108, 9, 2, 4ldilval 33110 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D  /\  ( x  e.  (
Base `  K )  /\  x ( le `  K ) W ) )  ->  ( F `  x )  =  x )
11103expa 1197 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  (
x  e.  ( Base `  K )  /\  x
( le `  K
) W ) )  ->  ( F `  x )  =  x )
12113impb 1193 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( F `  x )  =  x )
1312fveq2d 5852 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( `' F `  ( F `  x ) )  =  ( `' F `  x ) )
148, 2, 4ldil1o 33109 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
15143ad2ant1 1018 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
16 simp2 998 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  x  e.  ( Base `  K )
)
17 f1ocnvfv1 6162 . . . . . 6  |-  ( ( F : ( Base `  K ) -1-1-onto-> ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( `' F `  ( F `  x ) )  =  x )
1815, 16, 17syl2anc 659 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( `' F `  ( F `  x ) )  =  x )
1913, 18eqtr3d 2445 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( `' F `  x )  =  x )
20193exp 1196 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  ( x  e.  ( Base `  K
)  ->  ( x
( le `  K
) W  ->  ( `' F `  x )  =  x ) ) )
2120ralrimiv 2815 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  A. x  e.  ( Base `  K
) ( x ( le `  K ) W  ->  ( `' F `  x )  =  x ) )
228, 9, 2, 3, 4isldil 33107 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( `' F  e.  D  <->  ( `' F  e.  ( LAut `  K
)  /\  A. x  e.  ( Base `  K
) ( x ( le `  K ) W  ->  ( `' F `  x )  =  x ) ) ) )
2322adantr 463 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  ( `' F  e.  D  <->  ( `' F  e.  ( LAut `  K )  /\  A. x  e.  ( Base `  K ) ( x ( le `  K
) W  ->  ( `' F `  x )  =  x ) ) ) )
247, 21, 23mpbir2and 923 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  `' F  e.  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   class class class wbr 4394   `'ccnv 4821   -1-1-onto->wf1o 5567   ` cfv 5568   Basecbs 14839   lecple 14914   HLchlt 32348   LHypclh 32981   LAutclaut 32982   LDilcldil 33097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7458  df-laut 32986  df-ldil 33101
This theorem is referenced by:  ltrncnv  33143
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