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Theorem diass 34363
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 2420 . . 3  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
6 diass.i . . 3  |-  I  =  ( ( DIsoA `  K
) `  W )
71, 2, 3, 4, 5, 6diaval 34353 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  { f  e.  T  |  ( ( ( trL `  K ) `
 W ) `  f )  .<_  X }
)
8 ssrab2 3543 . 2  |-  { f  e.  T  |  ( ( ( trL `  K
) `  W ) `  f )  .<_  X }  C_  T
97, 8syl6eqss 3511 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 370    = wceq 1437    e. wcel 1867   {crab 2777    C_ wss 3433   class class class wbr 4417   ` cfv 5592   Basecbs 15081   lecple 15157   LHypclh 33302   LTrncltrn 33419   trLctrl 33477   DIsoAcdia 34349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  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 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-disoa 34350
This theorem is referenced by:  diael  34364  diaelrnN  34366  dialss  34367  dia2dimlem12  34396  diaocN  34446  dibss  34490
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