Theorem diadmleN 41539

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 2739	. . 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 41537	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ (Base‘𝐾) ∧ 𝑋 ≤ 𝑊)))
6	5	simplbda 500	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ≤ 𝑊)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   class class class wbr 5073  dom cdm 5619  ‘cfv 6486  Basecbs 17171  lecple 17219  LHypclh 40485  DIsoAcdia 41529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-disoa 41530

This theorem is referenced by:  diaocN  41626  doca2N  41627  djajN  41638