Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diadmleN Structured version   Visualization version   GIF version

Theorem diadmleN 39530
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 2737 . . 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 39528 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ (Baseβ€˜πΎ) ∧ 𝑋 ≀ π‘Š)))
65simplbda 501 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 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5110  dom cdm 5638  β€˜cfv 6501  Basecbs 17090  lecple 17147  LHypclh 38476  DIsoAcdia 39520
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-disoa 39521
This theorem is referenced by:  diaocN  39617  doca2N  39618  djajN  39629
  Copyright terms: Public domain W3C validator