Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diatrl Structured version   Visualization version   GIF version

Theorem diatrl 39910
Description: Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014.)
Hypotheses
Ref Expression
diatrl.b 𝐡 = (Baseβ€˜πΎ)
diatrl.l ≀ = (leβ€˜πΎ)
diatrl.h 𝐻 = (LHypβ€˜πΎ)
diatrl.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
diatrl.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
diatrl.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
diatrl (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ 𝐹 ∈ (πΌβ€˜π‘‹)) β†’ (π‘…β€˜πΉ) ≀ 𝑋)

Proof of Theorem diatrl
StepHypRef Expression
1 diatrl.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 diatrl.l . . . 4 ≀ = (leβ€˜πΎ)
3 diatrl.h . . . 4 𝐻 = (LHypβ€˜πΎ)
4 diatrl.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 diatrl.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
6 diatrl.i . . . 4 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
71, 2, 3, 4, 5, 6diaelval 39899 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝐹 ∈ (πΌβ€˜π‘‹) ↔ (𝐹 ∈ 𝑇 ∧ (π‘…β€˜πΉ) ≀ 𝑋)))
8 simpr 485 . . 3 ((𝐹 ∈ 𝑇 ∧ (π‘…β€˜πΉ) ≀ 𝑋) β†’ (π‘…β€˜πΉ) ≀ 𝑋)
97, 8syl6bi 252 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝐹 ∈ (πΌβ€˜π‘‹) β†’ (π‘…β€˜πΉ) ≀ 𝑋))
1093impia 1117 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ 𝐹 ∈ (πΌβ€˜π‘‹)) β†’ (π‘…β€˜πΉ) ≀ 𝑋)
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  trLctrl 39024  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  dibelval1st2N  40017  diblss  40036
  Copyright terms: Public domain W3C validator