Theorem diass 41902

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 2762	. . 3 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
6		diass.i	. . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
7	1, 2, 3, 4, 5, 6	diaval 41892	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑋})
8		ssrab2 4031	. 2 ⊢ {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑋} ⊆ 𝑇
9	7, 8	eqsstrdi 3978	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ 𝑇)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3414   ⊆ wss 3902   class class class wbr 5107  ‘cfv 6537  Basecbs 17305  lecple 17353  LHypclh 40844  LTrncltrn 40961  trLctrl 41018  DIsoAcdia 41888

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-disoa 41889

This theorem is used by:  diael  41903  diaelrnN  41905  dialss  41906  dia2dimlem12  41935  diaocN  41985  dibss  42029