Theorem diadmleN 39909

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	⊢ ≤ = (le‘𝐾)
diadmle.h	⊢ 𝐻 = (LHyp‘𝐾)
diadmle.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)

Assertion

Ref	Expression
diadmleN	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ≤ 𝑊)

Proof of Theorem diadmleN

Step	Hyp	Ref	Expression
1		eqid 2733	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		diadmle.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		diadmle.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		diadmle.i	. . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
5	1, 2, 3, 4	diaeldm 39907	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ (Base‘𝐾) ∧ 𝑋 ≤ 𝑊)))
6	5	simplbda 501	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ≤ 𝑊)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5149  dom cdm 5677  ‘cfv 6544  Basecbs 17144  lecple 17204  LHypclh 38855  DIsoAcdia 39899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-disoa 39900

This theorem is referenced by:  diaocN  39996  doca2N  39997  djajN  40008