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Theorem ldilcnv 30609
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 731 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  K  e.  HL )
2 ldilcnv.h . . . 4  |-  H  =  ( LHyp `  K
)
3 eqid 2412 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
4 ldilcnv.d . . . 4  |-  D  =  ( ( LDil `  K
) `  W )
52, 3, 4ldillaut 30605 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  F  e.  ( LAut `  K )
)
63lautcnv 30584 . . 3  |-  ( ( K  e.  HL  /\  F  e.  ( LAut `  K ) )  ->  `' F  e.  ( LAut `  K ) )
71, 5, 6syl2anc 643 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  `' F  e.  ( LAut `  K
) )
8 eqid 2412 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
9 eqid 2412 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
108, 9, 2, 4ldilval 30607 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D  /\  ( x  e.  (
Base `  K )  /\  x ( le `  K ) W ) )  ->  ( F `  x )  =  x )
11103expa 1153 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  (
x  e.  ( Base `  K )  /\  x
( le `  K
) W ) )  ->  ( F `  x )  =  x )
12113impb 1149 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( F `  x )  =  x )
1312fveq2d 5699 . . . . 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 30606 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
15143ad2ant1 978 . . . . . 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 958 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  x  e.  ( Base `  K )
)
17 f1ocnvfv1 5981 . . . . . 6  |-  ( ( F : ( Base `  K ) -1-1-onto-> ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( `' F `  ( F `  x ) )  =  x )
1815, 16, 17syl2anc 643 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( `' F `  ( F `  x ) )  =  x )
1913, 18eqtr3d 2446 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( `' F `  x )  =  x )
20193exp 1152 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  ( x  e.  ( Base `  K
)  ->  ( x
( le `  K
) W  ->  ( `' F `  x )  =  x ) ) )
2120ralrimiv 2756 . 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 30604 . . 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 452 . 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 889 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  `' F  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674   class class class wbr 4180   `'ccnv 4844   -1-1-onto->wf1o 5420   ` cfv 5421   Basecbs 13432   lecple 13499   HLchlt 29845   LHypclh 30478   LAutclaut 30479   LDilcldil 30594
This theorem is referenced by:  ltrncnv  30640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-laut 30483  df-ldil 30598
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