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Theorem diass 38297
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 2822 . . 3 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
6 diass.i . . 3 𝐼 = ((DIsoA‘𝐾)‘𝑊)
71, 2, 3, 4, 5, 6diaval 38287 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑋})
8 ssrab2 4031 . 2 {𝑓𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑋} ⊆ 𝑇
97, 8eqsstrdi 3996 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  {crab 3134  wss 3908   class class class wbr 5042  cfv 6334  Basecbs 16474  lecple 16563  LHypclh 37239  LTrncltrn 37356  trLctrl 37413  DIsoAcdia 38283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-disoa 38284
This theorem is referenced by:  diael  38298  diaelrnN  38300  dialss  38301  dia2dimlem12  38330  diaocN  38380  dibss  38424
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