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

Theorem ispsubsp 36999
Description: The predicate "is a projective subspace". (Contributed by NM, 2-Oct-2011.)
Hypotheses
Ref Expression
psubspset.l = (le‘𝐾)
psubspset.j = (join‘𝐾)
psubspset.a 𝐴 = (Atoms‘𝐾)
psubspset.s 𝑆 = (PSubSp‘𝐾)
Assertion
Ref Expression
ispsubsp (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋))))
Distinct variable groups:   𝐴,𝑟   𝑞,𝑝,𝑟,𝐾   𝑋,𝑝,𝑞,𝑟
Allowed substitution hints:   𝐴(𝑞,𝑝)   𝐷(𝑟,𝑞,𝑝)   𝑆(𝑟,𝑞,𝑝)   (𝑟,𝑞,𝑝)   (𝑟,𝑞,𝑝)

Proof of Theorem ispsubsp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 psubspset.l . . . 4 = (le‘𝐾)
2 psubspset.j . . . 4 = (join‘𝐾)
3 psubspset.a . . . 4 𝐴 = (Atoms‘𝐾)
4 psubspset.s . . . 4 𝑆 = (PSubSp‘𝐾)
51, 2, 3, 4psubspset 36998 . . 3 (𝐾𝐷𝑆 = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥))})
65eleq2d 2899 . 2 (𝐾𝐷 → (𝑋𝑆𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥))}))
73fvexi 6666 . . . . 5 𝐴 ∈ V
87ssex 5201 . . . 4 (𝑋𝐴𝑋 ∈ V)
98adantr 484 . . 3 ((𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)) → 𝑋 ∈ V)
10 sseq1 3967 . . . 4 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
11 eleq2 2902 . . . . . . . 8 (𝑥 = 𝑋 → (𝑟𝑥𝑟𝑋))
1211imbi2d 344 . . . . . . 7 (𝑥 = 𝑋 → ((𝑟 (𝑝 𝑞) → 𝑟𝑥) ↔ (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
1312ralbidv 3187 . . . . . 6 (𝑥 = 𝑋 → (∀𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥) ↔ ∀𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
1413raleqbi1dv 3384 . . . . 5 (𝑥 = 𝑋 → (∀𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥) ↔ ∀𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
1514raleqbi1dv 3384 . . . 4 (𝑥 = 𝑋 → (∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥) ↔ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
1610, 15anbi12d 633 . . 3 (𝑥 = 𝑋 → ((𝑥𝐴 ∧ ∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥)) ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋))))
179, 16elab3 3649 . 2 (𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥))} ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
186, 17syl6bb 290 1 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  {cab 2800  wral 3130  Vcvv 3469  wss 3908   class class class wbr 5042  cfv 6334  (class class class)co 7140  lecple 16563  joincjn 17545  Atomscatm 36517  PSubSpcpsubsp 36750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-psubsp 36757
This theorem is referenced by:  ispsubsp2  37000  0psubN  37003  snatpsubN  37004  linepsubN  37006  atpsubN  37007  psubssat  37008  pmapsub  37022  pclclN  37145  pclfinN  37154
  Copyright terms: Public domain W3C validator