Theorem diass 38172

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	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ 𝑇)

Proof of Theorem diass

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	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 2821	. . 3 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
6		diass.i	. . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
7	1, 2, 3, 4, 5, 6	diaval 38162	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑋})
8		ssrab2 4055	. 2 ⊢ {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑋} ⊆ 𝑇
9	7, 8	eqsstrdi 4020	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ 𝑇)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {crab 3142   ⊆ wss 3935   class class class wbr 5058  ‘cfv 6349  Basecbs 16477  lecple 16566  LHypclh 37114  LTrncltrn 37231  trLctrl 37288  DIsoAcdia 38158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-disoa 38159

This theorem is referenced by:  diael  38173  diaelrnN  38175  dialss  38176  dia2dimlem12  38205  diaocN  38255  dibss  38299