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Theorem lvolset 38085
Description: The set of 3-dim lattice volumes in a Hilbert lattice. (Contributed by NM, 1-Jul-2012.)
Hypotheses
Ref Expression
lvolset.b 𝐡 = (Baseβ€˜πΎ)
lvolset.c 𝐢 = ( β‹– β€˜πΎ)
lvolset.p 𝑃 = (LPlanesβ€˜πΎ)
lvolset.v 𝑉 = (LVolsβ€˜πΎ)
Assertion
Ref Expression
lvolset (𝐾 ∈ 𝐴 β†’ 𝑉 = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑃 𝑦𝐢π‘₯})
Distinct variable groups:   𝑦,𝑃   π‘₯,𝐡   π‘₯,𝑦,𝐾
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐡(𝑦)   𝐢(π‘₯,𝑦)   𝑃(π‘₯)   𝑉(π‘₯,𝑦)

Proof of Theorem lvolset
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3465 . 2 (𝐾 ∈ 𝐴 β†’ 𝐾 ∈ V)
2 lvolset.v . . 3 𝑉 = (LVolsβ€˜πΎ)
3 fveq2 6846 . . . . . 6 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = (Baseβ€˜πΎ))
4 lvolset.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
53, 4eqtr4di 2791 . . . . 5 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = 𝐡)
6 fveq2 6846 . . . . . . 7 (π‘˜ = 𝐾 β†’ (LPlanesβ€˜π‘˜) = (LPlanesβ€˜πΎ))
7 lvolset.p . . . . . . 7 𝑃 = (LPlanesβ€˜πΎ)
86, 7eqtr4di 2791 . . . . . 6 (π‘˜ = 𝐾 β†’ (LPlanesβ€˜π‘˜) = 𝑃)
9 fveq2 6846 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ( β‹– β€˜π‘˜) = ( β‹– β€˜πΎ))
10 lvolset.c . . . . . . . 8 𝐢 = ( β‹– β€˜πΎ)
119, 10eqtr4di 2791 . . . . . . 7 (π‘˜ = 𝐾 β†’ ( β‹– β€˜π‘˜) = 𝐢)
1211breqd 5120 . . . . . 6 (π‘˜ = 𝐾 β†’ (𝑦( β‹– β€˜π‘˜)π‘₯ ↔ 𝑦𝐢π‘₯))
138, 12rexeqbidv 3319 . . . . 5 (π‘˜ = 𝐾 β†’ (βˆƒπ‘¦ ∈ (LPlanesβ€˜π‘˜)𝑦( β‹– β€˜π‘˜)π‘₯ ↔ βˆƒπ‘¦ ∈ 𝑃 𝑦𝐢π‘₯))
145, 13rabeqbidv 3423 . . . 4 (π‘˜ = 𝐾 β†’ {π‘₯ ∈ (Baseβ€˜π‘˜) ∣ βˆƒπ‘¦ ∈ (LPlanesβ€˜π‘˜)𝑦( β‹– β€˜π‘˜)π‘₯} = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑃 𝑦𝐢π‘₯})
15 df-lvols 38013 . . . 4 LVols = (π‘˜ ∈ V ↦ {π‘₯ ∈ (Baseβ€˜π‘˜) ∣ βˆƒπ‘¦ ∈ (LPlanesβ€˜π‘˜)𝑦( β‹– β€˜π‘˜)π‘₯})
164fvexi 6860 . . . . 5 𝐡 ∈ V
1716rabex 5293 . . . 4 {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑃 𝑦𝐢π‘₯} ∈ V
1814, 15, 17fvmpt 6952 . . 3 (𝐾 ∈ V β†’ (LVolsβ€˜πΎ) = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑃 𝑦𝐢π‘₯})
192, 18eqtrid 2785 . 2 (𝐾 ∈ V β†’ 𝑉 = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑃 𝑦𝐢π‘₯})
201, 19syl 17 1 (𝐾 ∈ 𝐴 β†’ 𝑉 = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑃 𝑦𝐢π‘₯})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070  {crab 3406  Vcvv 3447   class class class wbr 5109  β€˜cfv 6500  Basecbs 17091   β‹– ccvr 37774  LPlanesclpl 38005  LVolsclvol 38006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-lvols 38013
This theorem is referenced by:  islvol  38086
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