Theorem diadmleN 41147

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 2731	. . 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 41145	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ (Base‘𝐾) ∧ 𝑋 ≤ 𝑊)))
6	5	simplbda 499	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ≤ 𝑊)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   class class class wbr 5089  dom cdm 5614  ‘cfv 6481  Basecbs 17120  lecple 17168  LHypclh 40093  DIsoAcdia 41137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-disoa 41138

This theorem is referenced by:  diaocN  41234  doca2N  41235  djajN  41246