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Theorem diael 34410
Description: A member of the value of the partial isomorphism A is a translation i.e. a vector. (Contributed by NM, 17-Jan-2014.)
Hypotheses
Ref Expression
diass.b  |-  B  =  ( Base `  K
)
diass.l  |-  .<_  =  ( le `  K )
diass.h  |-  H  =  ( LHyp `  K
)
diass.t  |-  T  =  ( ( LTrn `  K
) `  W )
diass.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
diael  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  F  e.  ( I `  X
) )  ->  F  e.  T )

Proof of Theorem diael
StepHypRef Expression
1 diass.b . . . 4  |-  B  =  ( Base `  K
)
2 diass.l . . . 4  |-  .<_  =  ( le `  K )
3 diass.h . . . 4  |-  H  =  ( LHyp `  K
)
4 diass.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
5 diass.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
61, 2, 3, 4, 5diass 34409 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  C_  T )
76sseld 3352 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F  e.  ( I `  X )  ->  F  e.  T ) )
873impia 1179 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  F  e.  ( I `  X
) )  ->  F  e.  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   class class class wbr 4289   ` cfv 5415   Basecbs 14170   lecple 14241   LHypclh 33350   LTrncltrn 33467   DIsoAcdia 34395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-disoa 34396
This theorem is referenced by:  dialss  34413  dibelval1st1  34517  diblsmopel  34538
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