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Theorem ldil1o 30971
Description: A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013.)
Hypotheses
Ref Expression
ldil1o.b  |-  B  =  ( Base `  K
)
ldil1o.h  |-  H  =  ( LHyp `  K
)
ldil1o.d  |-  D  =  ( ( LDil `  K
) `  W )
Assertion
Ref Expression
ldil1o  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  D )  ->  F : B -1-1-onto-> B )

Proof of Theorem ldil1o
StepHypRef Expression
1 simpll 732 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  D )  ->  K  e.  V )
2 ldil1o.h . . 3  |-  H  =  ( LHyp `  K
)
3 eqid 2438 . . 3  |-  ( LAut `  K )  =  (
LAut `  K )
4 ldil1o.d . . 3  |-  D  =  ( ( LDil `  K
) `  W )
52, 3, 4ldillaut 30970 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  D )  ->  F  e.  ( LAut `  K
) )
6 ldil1o.b . . 3  |-  B  =  ( Base `  K
)
76, 3laut1o 30944 . 2  |-  ( ( K  e.  V  /\  F  e.  ( LAut `  K ) )  ->  F : B -1-1-onto-> B )
81, 5, 7syl2anc 644 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  D )  ->  F : B -1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   -1-1-onto->wf1o 5455   ` cfv 5456   Basecbs 13471   LHypclh 30843   LAutclaut 30844   LDilcldil 30959
This theorem is referenced by:  ldilcnv  30974  ldilco  30975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-laut 30848  df-ldil 30963
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