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Theorem diass 40645
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 2725 . . 3 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
6 diass.i . . 3 𝐼 = ((DIsoA‘𝐾)‘𝑊)
71, 2, 3, 4, 5, 6diaval 40635 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑋})
8 ssrab2 4073 . 2 {𝑓𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑋} ⊆ 𝑇
97, 8eqsstrdi 4031 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {crab 3418  wss 3944   class class class wbr 5149  cfv 6549  Basecbs 17183  lecple 17243  LHypclh 39587  LTrncltrn 39704  trLctrl 39761  DIsoAcdia 40631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-disoa 40632
This theorem is referenced by:  diael  40646  diaelrnN  40648  dialss  40649  dia2dimlem12  40678  diaocN  40728  dibss  40772
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