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Theorem diass 39082
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 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ 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 2733 . . 3 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
6 diass.i . . 3 𝐼 = ((DIsoA‘𝐾)‘𝑊)
71, 2, 3, 4, 5, 6diaval 39072 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑋})
8 ssrab2 4016 . 2 {𝑓𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑋} ⊆ 𝑇
97, 8eqsstrdi 3977 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2101  {crab 3221  wss 3889   class class class wbr 5077  cfv 6447  Basecbs 16940  lecple 16997  LHypclh 38024  LTrncltrn 38141  trLctrl 38198  DIsoAcdia 39068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pr 5355
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-disoa 39069
This theorem is referenced by:  diael  39083  diaelrnN  39085  dialss  39086  dia2dimlem12  39115  diaocN  39165  dibss  39209
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