Theorem diael 38173

Description: A member of the value of the partial isomorphism A is a translation i.e. a vector. (Contributed by NM, 17-Jan-2014.)

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

Proof of Theorem diael

Step	Hyp	Ref	Expression
1		diass.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		diass.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		diass.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4		diass.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5		diass.i	. . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	diass 38172	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ 𝑇)
7	6	sseld 3965	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (𝐼‘𝑋) → 𝐹 ∈ 𝑇))
8	7	3impia 1113	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ (𝐼‘𝑋)) → 𝐹 ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   class class class wbr 5058  ‘cfv 6349  Basecbs 16477  lecple 16566  LHypclh 37114  LTrncltrn 37231  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:  dialss  38176  dibelval1st1  38280  diblsmopel  38301