Theorem diadmleN 40421

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 2726	. . 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 40419	. 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 1533   ∈ wcel 2098   class class class wbr 5141  dom cdm 5669  ‘cfv 6536  Basecbs 17150  lecple 17210  LHypclh 39367  DIsoAcdia 40411

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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420

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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-disoa 40412

This theorem is referenced by:  diaocN  40508  doca2N  40509  djajN  40520