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Theorem diass 36056
Description: The value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.)
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
diass  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  C_  T )

Proof of Theorem diass
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 diass.b . . 3  |-  B  =  ( Base `  K
)
2 diass.l . . 3  |-  .<_  =  ( le `  K )
3 diass.h . . 3  |-  H  =  ( LHyp `  K
)
4 diass.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
5 eqid 2467 . . 3  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
6 diass.i . . 3  |-  I  =  ( ( DIsoA `  K
) `  W )
71, 2, 3, 4, 5, 6diaval 36046 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  { f  e.  T  |  ( ( ( trL `  K ) `
 W ) `  f )  .<_  X }
)
8 ssrab2 3585 . 2  |-  { f  e.  T  |  ( ( ( trL `  K
) `  W ) `  f )  .<_  X }  C_  T
97, 8syl6eqss 3554 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  C_  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818    C_ wss 3476   class class class wbr 4447   ` cfv 5588   Basecbs 14493   lecple 14565   LHypclh 34997   LTrncltrn 35114   trLctrl 35171   DIsoAcdia 36042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-disoa 36043
This theorem is referenced by:  diael  36057  diaelrnN  36059  dialss  36060  dia2dimlem12  36089  diaocN  36139  dibss  36183
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