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Theorem diass 31158
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 2388 . . 3  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
6 diass.i . . 3  |-  I  =  ( ( DIsoA `  K
) `  W )
71, 2, 3, 4, 5, 6diaval 31148 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  { f  e.  T  |  ( ( ( trL `  K ) `
 W ) `  f )  .<_  X }
)
8 ssrab2 3372 . 2  |-  { f  e.  T  |  ( ( ( trL `  K
) `  W ) `  f )  .<_  X }  C_  T
97, 8syl6eqss 3342 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  C_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2654    C_ wss 3264   class class class wbr 4154   ` cfv 5395   Basecbs 13397   lecple 13464   LHypclh 30099   LTrncltrn 30216   trLctrl 30273   DIsoAcdia 31144
This theorem is referenced by:  diael  31159  diaelrnN  31161  dialss  31162  dia2dimlem12  31191  diaocN  31241  dibss  31285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-disoa 31145
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