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

Theorem lvolset 39271
Description: The set of 3-dim lattice volumes in a Hilbert lattice. (Contributed by NM, 1-Jul-2012.)
Hypotheses
Ref Expression
lvolset.b 𝐵 = (Base‘𝐾)
lvolset.c 𝐶 = ( ⋖ ‘𝐾)
lvolset.p 𝑃 = (LPlanes‘𝐾)
lvolset.v 𝑉 = (LVols‘𝐾)
Assertion
Ref Expression
lvolset (𝐾𝐴𝑉 = {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥})
Distinct variable groups:   𝑦,𝑃   𝑥,𝐵   𝑥,𝑦,𝐾
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝐶(𝑥,𝑦)   𝑃(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem lvolset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3482 . 2 (𝐾𝐴𝐾 ∈ V)
2 lvolset.v . . 3 𝑉 = (LVols‘𝐾)
3 fveq2 6901 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lvolset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4eqtr4di 2784 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6901 . . . . . . 7 (𝑘 = 𝐾 → (LPlanes‘𝑘) = (LPlanes‘𝐾))
7 lvolset.p . . . . . . 7 𝑃 = (LPlanes‘𝐾)
86, 7eqtr4di 2784 . . . . . 6 (𝑘 = 𝐾 → (LPlanes‘𝑘) = 𝑃)
9 fveq2 6901 . . . . . . . 8 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10 lvolset.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
119, 10eqtr4di 2784 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
1211breqd 5164 . . . . . 6 (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥𝑦𝐶𝑥))
138, 12rexeqbidv 3331 . . . . 5 (𝑘 = 𝐾 → (∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦𝑃 𝑦𝐶𝑥))
145, 13rabeqbidv 3437 . . . 4 (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥})
15 df-lvols 39199 . . . 4 LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
164fvexi 6915 . . . . 5 𝐵 ∈ V
1716rabex 5339 . . . 4 {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥} ∈ V
1814, 15, 17fvmpt 7009 . . 3 (𝐾 ∈ V → (LVols‘𝐾) = {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥})
192, 18eqtrid 2778 . 2 (𝐾 ∈ V → 𝑉 = {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥})
201, 19syl 17 1 (𝐾𝐴𝑉 = {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wrex 3060  {crab 3419  Vcvv 3462   class class class wbr 5153  cfv 6554  Basecbs 17213  ccvr 38960  LPlanesclpl 39191  LVolsclvol 39192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6506  df-fun 6556  df-fv 6562  df-lvols 39199
This theorem is referenced by:  islvol  39272
  Copyright terms: Public domain W3C validator