Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pol0N Structured version   Visualization version   GIF version

Theorem pol0N 35717
Description: The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
polssat.a 𝐴 = (Atoms‘𝐾)
polssat.p = (⊥𝑃𝐾)
Assertion
Ref Expression
pol0N (𝐾𝐵 → ( ‘∅) = 𝐴)

Proof of Theorem pol0N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 0ss 4113 . . 3 ∅ ⊆ 𝐴
2 eqid 2769 . . . 4 (oc‘𝐾) = (oc‘𝐾)
3 polssat.a . . . 4 𝐴 = (Atoms‘𝐾)
4 eqid 2769 . . . 4 (pmap‘𝐾) = (pmap‘𝐾)
5 polssat.p . . . 4 = (⊥𝑃𝐾)
62, 3, 4, 5polvalN 35713 . . 3 ((𝐾𝐵 ∧ ∅ ⊆ 𝐴) → ( ‘∅) = (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
71, 6mpan2 706 . 2 (𝐾𝐵 → ( ‘∅) = (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
8 0iin 4709 . . . 4 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)) = V
98ineq2i 3959 . . 3 (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = (𝐴 ∩ V)
10 inv1 4111 . . 3 (𝐴 ∩ V) = 𝐴
119, 10eqtri 2791 . 2 (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = 𝐴
127, 11syl6eq 2819 1 (𝐾𝐵 → ( ‘∅) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1629  wcel 2143  Vcvv 3348  cin 3719  wss 3720  c0 4060   ciin 4652  cfv 6030  occoc 16163  Atomscatm 35072  pmapcpmap 35305  𝑃cpolN 35710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-reu 3066  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-iun 4653  df-iin 4654  df-br 4784  df-opab 4844  df-mpt 4861  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-polarityN 35711
This theorem is referenced by:  2pol0N  35719  1psubclN  35752  osumcllem9N  35772  pexmidN  35777  pexmidlem6N  35783  pexmidALTN  35786
  Copyright terms: Public domain W3C validator