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Theorem lvolset 37170
 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 3428 . 2 (𝐾𝐴𝐾 ∈ V)
2 lvolset.v . . 3 𝑉 = (LVols‘𝐾)
3 fveq2 6658 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lvolset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4eqtr4di 2811 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6658 . . . . . . 7 (𝑘 = 𝐾 → (LPlanes‘𝑘) = (LPlanes‘𝐾))
7 lvolset.p . . . . . . 7 𝑃 = (LPlanes‘𝐾)
86, 7eqtr4di 2811 . . . . . 6 (𝑘 = 𝐾 → (LPlanes‘𝑘) = 𝑃)
9 fveq2 6658 . . . . . . . 8 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10 lvolset.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
119, 10eqtr4di 2811 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
1211breqd 5043 . . . . . 6 (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥𝑦𝐶𝑥))
138, 12rexeqbidv 3320 . . . . 5 (𝑘 = 𝐾 → (∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦𝑃 𝑦𝐶𝑥))
145, 13rabeqbidv 3398 . . . 4 (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥})
15 df-lvols 37098 . . . 4 LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
164fvexi 6672 . . . . 5 𝐵 ∈ V
1716rabex 5202 . . . 4 {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥} ∈ V
1814, 15, 17fvmpt 6759 . . 3 (𝐾 ∈ V → (LVols‘𝐾) = {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥})
192, 18syl5eq 2805 . 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 1538   ∈ wcel 2111  ∃wrex 3071  {crab 3074  Vcvv 3409   class class class wbr 5032  ‘cfv 6335  Basecbs 16541   ⋖ ccvr 36860  LPlanesclpl 37090  LVolsclvol 37091 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-lvols 37098 This theorem is referenced by:  islvol  37171
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