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Theorem diatrl 40409
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 40398 . . 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 1114 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 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5139  β€˜cfv 6534  Basecbs 17145  lecple 17205  LHypclh 39349  LTrncltrn 39466  trLctrl 39523  DIsoAcdia 40393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-disoa 40394
This theorem is referenced by:  dialss  40411  dibelval1st2N  40516  diblss  40535
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