Theorem lvolset 39045

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	⊢ (𝐾 ∈ 𝐴 → 𝑉 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 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.

Step	Hyp	Ref	Expression
1		elex 3490	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		lvolset.v	. . 3 ⊢ 𝑉 = (LVols‘𝐾)
3		fveq2 6897	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		lvolset.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2786	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6		fveq2 6897	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LPlanes‘𝑘) = (LPlanes‘𝐾))
7		lvolset.p	. . . . . . 7 ⊢ 𝑃 = (LPlanes‘𝐾)
8	6, 7	eqtr4di 2786	. . . . . 6 ⊢ (𝑘 = 𝐾 → (LPlanes‘𝑘) = 𝑃)
9		fveq2 6897	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10		lvolset.c	. . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾)
11	9, 10	eqtr4di 2786	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
12	11	breqd 5159	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥 ↔ 𝑦𝐶𝑥))
13	8, 12	rexeqbidv 3340	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥))
14	5, 13	rabeqbidv 3446	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥})
15		df-lvols 38973	. . . 4 ⊢ LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
16	4	fvexi 6911	. . . . 5 ⊢ 𝐵 ∈ V
17	16	rabex 5334	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥} ∈ V
18	14, 15, 17	fvmpt 7005	. . 3 ⊢ (𝐾 ∈ V → (LVols‘𝐾) = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥})
19	2, 18	eqtrid 2780	. 2 ⊢ (𝐾 ∈ V → 𝑉 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥})
20	1, 19	syl 17	1 ⊢ (𝐾 ∈ 𝐴 → 𝑉 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥})

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ∃wrex 3067  {crab 3429  Vcvv 3471   class class class wbr 5148  ‘cfv 6548  Basecbs 17180   ⋖ ccvr 38734  LPlanesclpl 38965  LVolsclvol 38966

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 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-lvols 38973

This theorem is referenced by:  islvol  39046

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