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Theorem lvolset 38025
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 3463 . 2 (𝐾𝐴𝐾 ∈ V)
2 lvolset.v . . 3 𝑉 = (LVols‘𝐾)
3 fveq2 6842 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lvolset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4eqtr4di 2794 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6842 . . . . . . 7 (𝑘 = 𝐾 → (LPlanes‘𝑘) = (LPlanes‘𝐾))
7 lvolset.p . . . . . . 7 𝑃 = (LPlanes‘𝐾)
86, 7eqtr4di 2794 . . . . . 6 (𝑘 = 𝐾 → (LPlanes‘𝑘) = 𝑃)
9 fveq2 6842 . . . . . . . 8 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10 lvolset.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
119, 10eqtr4di 2794 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
1211breqd 5116 . . . . . 6 (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥𝑦𝐶𝑥))
138, 12rexeqbidv 3320 . . . . 5 (𝑘 = 𝐾 → (∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦𝑃 𝑦𝐶𝑥))
145, 13rabeqbidv 3424 . . . 4 (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥})
15 df-lvols 37953 . . . 4 LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
164fvexi 6856 . . . . 5 𝐵 ∈ V
1716rabex 5289 . . . 4 {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥} ∈ V
1814, 15, 17fvmpt 6948 . . 3 (𝐾 ∈ V → (LVols‘𝐾) = {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥})
192, 18eqtrid 2788 . 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 1541  wcel 2106  wrex 3073  {crab 3407  Vcvv 3445   class class class wbr 5105  cfv 6496  Basecbs 17082  ccvr 37714  LPlanesclpl 37945  LVolsclvol 37946
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-lvols 37953
This theorem is referenced by:  islvol  38026
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