Theorem lvolset 36710

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 3514	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		lvolset.v	. . 3 ⊢ 𝑉 = (LVols‘𝐾)
3		fveq2 6672	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		lvolset.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	syl6eqr 2876	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6		fveq2 6672	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LPlanes‘𝑘) = (LPlanes‘𝐾))
7		lvolset.p	. . . . . . 7 ⊢ 𝑃 = (LPlanes‘𝐾)
8	6, 7	syl6eqr 2876	. . . . . 6 ⊢ (𝑘 = 𝐾 → (LPlanes‘𝑘) = 𝑃)
9		fveq2 6672	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10		lvolset.c	. . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾)
11	9, 10	syl6eqr 2876	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
12	11	breqd 5079	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥 ↔ 𝑦𝐶𝑥))
13	8, 12	rexeqbidv 3404	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥))
14	5, 13	rabeqbidv 3487	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥})
15		df-lvols 36638	. . . 4 ⊢ LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LPlanes‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
16	4	fvexi 6686	. . . . 5 ⊢ 𝐵 ∈ V
17	16	rabex 5237	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥} ∈ V
18	14, 15, 17	fvmpt 6770	. . 3 ⊢ (𝐾 ∈ V → (LVols‘𝐾) = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥})
19	2, 18	syl5eq 2870	. 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 1537   ∈ wcel 2114  ∃wrex 3141  {crab 3144  Vcvv 3496   class class class wbr 5068  ‘cfv 6357  Basecbs 16485   ⋖ ccvr 36400  LPlanesclpl 36630  LVolsclvol 36631

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-lvols 36638

This theorem is referenced by:  islvol  36711