Theorem lvolset 39691

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 3458	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		lvolset.v	. . 3 ⊢ 𝑉 = (LVols‘𝐾)
3		fveq2 6828	. . . . . 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 6828	. . . . . . 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 6828	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10		lvolset.c	. . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾)
11	9, 10	eqtr4di 2786	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
12	11	breqd 5104	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥 ↔ 𝑦𝐶𝑥))
13	8, 12	rexeqbidv 3314	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥))
14	5, 13	rabeqbidv 3414	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥})
15		df-lvols 39619	. . . 4 ⊢ LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
16	4	fvexi 6842	. . . . 5 ⊢ 𝐵 ∈ V
17	16	rabex 5279	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥} ∈ V
18	14, 15, 17	fvmpt 6935	. . 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 1541   ∈ wcel 2113  ∃wrex 3057  {crab 3396  Vcvv 3437   class class class wbr 5093  ‘cfv 6486  Basecbs 17122   ⋖ ccvr 39381  LPlanesclpl 39611  LVolsclvol 39612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494  df-lvols 39619

This theorem is referenced by:  islvol  39692