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Theorem diass 35808
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 2621 . . 3 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
6 diass.i . . 3 𝐼 = ((DIsoA‘𝐾)‘𝑊)
71, 2, 3, 4, 5, 6diaval 35798 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑋})
8 ssrab2 3666 . 2 {𝑓𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑋} ⊆ 𝑇
97, 8syl6eqss 3634 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2911  wss 3555   class class class wbr 4613  cfv 5847  Basecbs 15781  lecple 15869  LHypclh 34747  LTrncltrn 34864  trLctrl 34922  DIsoAcdia 35794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-disoa 35795
This theorem is referenced by:  diael  35809  diaelrnN  35811  dialss  35812  dia2dimlem12  35841  diaocN  35891  dibss  35935
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