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 40512
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 40501 . . 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 484 . . 3 ((𝐹 ∈ 𝑇 ∧ (π‘…β€˜πΉ) ≀ 𝑋) β†’ (π‘…β€˜πΉ) ≀ 𝑋)
97, 8syl6bi 253 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝐹 ∈ (πΌβ€˜π‘‹) β†’ (π‘…β€˜πΉ) ≀ 𝑋))
1093impia 1115 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   class class class wbr 5143  β€˜cfv 6543  Basecbs 17174  lecple 17234  LHypclh 39452  LTrncltrn 39569  trLctrl 39626  DIsoAcdia 40496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  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 40497
This theorem is referenced by:  dialss  40514  dibelval1st2N  40619  diblss  40638
  Copyright terms: Public domain W3C validator