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Theorem ldilcnv 30356
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 730 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  K  e.  HL )
2 ldilcnv.h . . . 4  |-  H  =  ( LHyp `  K
)
3 eqid 2358 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
4 ldilcnv.d . . . 4  |-  D  =  ( ( LDil `  K
) `  W )
52, 3, 4ldillaut 30352 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  F  e.  ( LAut `  K )
)
63lautcnv 30331 . . 3  |-  ( ( K  e.  HL  /\  F  e.  ( LAut `  K ) )  ->  `' F  e.  ( LAut `  K ) )
71, 5, 6syl2anc 642 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  `' F  e.  ( LAut `  K
) )
8 eqid 2358 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
9 eqid 2358 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
108, 9, 2, 4ldilval 30354 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D  /\  ( x  e.  (
Base `  K )  /\  x ( le `  K ) W ) )  ->  ( F `  x )  =  x )
11103expa 1151 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  (
x  e.  ( Base `  K )  /\  x
( le `  K
) W ) )  ->  ( F `  x )  =  x )
12113impb 1147 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( F `  x )  =  x )
1312fveq2d 5609 . . . . 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 30353 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
15143ad2ant1 976 . . . . . 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 956 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  x  e.  ( Base `  K )
)
17 f1ocnvfv1 5876 . . . . . 6  |-  ( ( F : ( Base `  K ) -1-1-onto-> ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( `' F `  ( F `  x ) )  =  x )
1815, 16, 17syl2anc 642 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( `' F `  ( F `  x ) )  =  x )
1913, 18eqtr3d 2392 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( `' F `  x )  =  x )
20193exp 1150 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  ( x  e.  ( Base `  K
)  ->  ( x
( le `  K
) W  ->  ( `' F `  x )  =  x ) ) )
2120ralrimiv 2701 . 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 30351 . . 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 451 . 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 888 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  `' F  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619   class class class wbr 4102   `'ccnv 4767   -1-1-onto->wf1o 5333   ` cfv 5334   Basecbs 13239   lecple 13306   HLchlt 29592   LHypclh 30225   LAutclaut 30226   LDilcldil 30341
This theorem is referenced by:  ltrncnv  30387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-map 6859  df-laut 30230  df-ldil 30345
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