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Theorem ldilcnv 34068
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 753 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  K  e.  HL )
2 ldilcnv.h . . . 4  |-  H  =  ( LHyp `  K
)
3 eqid 2451 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
4 ldilcnv.d . . . 4  |-  D  =  ( ( LDil `  K
) `  W )
52, 3, 4ldillaut 34064 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  F  e.  ( LAut `  K )
)
63lautcnv 34043 . . 3  |-  ( ( K  e.  HL  /\  F  e.  ( LAut `  K ) )  ->  `' F  e.  ( LAut `  K ) )
71, 5, 6syl2anc 661 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  `' F  e.  ( LAut `  K
) )
8 eqid 2451 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
9 eqid 2451 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
108, 9, 2, 4ldilval 34066 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D  /\  ( x  e.  (
Base `  K )  /\  x ( le `  K ) W ) )  ->  ( F `  x )  =  x )
11103expa 1188 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  (
x  e.  ( Base `  K )  /\  x
( le `  K
) W ) )  ->  ( F `  x )  =  x )
12113impb 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( F `  x )  =  x )
1312fveq2d 5796 . . . . 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 34065 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
15143ad2ant1 1009 . . . . . 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 989 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  x  e.  ( Base `  K )
)
17 f1ocnvfv1 6085 . . . . . 6  |-  ( ( F : ( Base `  K ) -1-1-onto-> ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( `' F `  ( F `  x ) )  =  x )
1815, 16, 17syl2anc 661 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( `' F `  ( F `  x ) )  =  x )
1913, 18eqtr3d 2494 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D )  /\  x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( `' F `  x )  =  x )
20193exp 1187 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D
)  ->  ( x  e.  ( Base `  K
)  ->  ( x
( le `  K
) W  ->  ( `' F `  x )  =  x ) ) )
2120ralrimiv 2823 . 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 34063 . . 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 465 . 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 913 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   class class class wbr 4393   `'ccnv 4940   -1-1-onto->wf1o 5518   ` cfv 5519   Basecbs 14285   lecple 14356   HLchlt 33304   LHypclh 33937   LAutclaut 33938   LDilcldil 34053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-map 7319  df-laut 33942  df-ldil 34057
This theorem is referenced by:  ltrncnv  34099
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