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Theorem diadmleN 40543
Description: A member of domain of the partial isomorphism A is under the fiducial hyperplane. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
diadmle.l ≀ = (leβ€˜πΎ)
diadmle.h 𝐻 = (LHypβ€˜πΎ)
diadmle.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
diadmleN (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ 𝑋 ≀ π‘Š)

Proof of Theorem diadmleN
StepHypRef Expression
1 eqid 2728 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 diadmle.l . . 3 ≀ = (leβ€˜πΎ)
3 diadmle.h . . 3 𝐻 = (LHypβ€˜πΎ)
4 diadmle.i . . 3 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
51, 2, 3, 4diaeldm 40541 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ (Baseβ€˜πΎ) ∧ 𝑋 ≀ π‘Š)))
65simplbda 498 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ 𝑋 ≀ π‘Š)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   class class class wbr 5152  dom cdm 5682  β€˜cfv 6553  Basecbs 17187  lecple 17247  LHypclh 39489  DIsoAcdia 40533
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-disoa 40534
This theorem is referenced by:  diaocN  40630  doca2N  40631  djajN  40642
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