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Theorem diass 37117
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 2825 . . 3 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
6 diass.i . . 3 𝐼 = ((DIsoA‘𝐾)‘𝑊)
71, 2, 3, 4, 5, 6diaval 37107 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑋})
8 ssrab2 3912 . 2 {𝑓𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑋} ⊆ 𝑇
97, 8syl6eqss 3880 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166  {crab 3121  wss 3798   class class class wbr 4873  cfv 6123  Basecbs 16222  lecple 16312  LHypclh 36059  LTrncltrn 36176  trLctrl 36233  DIsoAcdia 37103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-disoa 37104
This theorem is referenced by:  diael  37118  diaelrnN  37120  dialss  37121  dia2dimlem12  37150  diaocN  37200  dibss  37244
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