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Theorem diael 39909
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 𝐡 = (Baseβ€˜πΎ)
diass.l ≀ = (leβ€˜πΎ)
diass.h 𝐻 = (LHypβ€˜πΎ)
diass.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
diass.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
diael (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ 𝐹 ∈ (πΌβ€˜π‘‹)) β†’ 𝐹 ∈ 𝑇)

Proof of Theorem diael
StepHypRef Expression
1 diass.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 diass.l . . . 4 ≀ = (leβ€˜πΎ)
3 diass.h . . . 4 𝐻 = (LHypβ€˜πΎ)
4 diass.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 diass.i . . . 4 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
61, 2, 3, 4, 5diass 39908 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) βŠ† 𝑇)
76sseld 3981 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝐹 ∈ (πΌβ€˜π‘‹) β†’ 𝐹 ∈ 𝑇))
873impia 1117 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ 𝐹 ∈ (πΌβ€˜π‘‹)) β†’ 𝐹 ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  β€˜cfv 6543  Basecbs 17143  lecple 17203  LHypclh 38850  LTrncltrn 38967  DIsoAcdia 39894
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-disoa 39895
This theorem is referenced by:  dialss  39912  dibelval1st1  40016  diblsmopel  40037
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