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Theorem ldilcnv 30304
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 2283 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
4 ldilcnv.d . . . 4  |-  D  =  ( ( LDil `  K
) `  W )
52, 3, 4ldillaut 30300 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  F  e.  ( LAut `  K )
)
63lautcnv 30279 . . 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 2283 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
9 eqid 2283 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
108, 9, 2, 4ldilval 30302 . . . . . . . 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 5529 . . . . 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 30301 . . . . . . 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 5792 . . . . . 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 2317 . . . 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 2625 . 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 30299 . . 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 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   `'ccnv 4688   -1-1-onto->wf1o 5254   ` cfv 5255   Basecbs 13148   lecple 13215   HLchlt 29540   LHypclh 30173   LAutclaut 30174   LDilcldil 30289
This theorem is referenced by:  ltrncnv  30335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-laut 30178  df-ldil 30293
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