Theorem islvol 40019

Description: The predicate "is a 3-dim lattice volume". (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
islvol	⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑋)))

Distinct variable groups:   𝑦,𝑃   𝑦,𝐾   𝑦,𝑋

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦)   𝐶(𝑦)   𝑉(𝑦)

Proof of Theorem islvol

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lvolset.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		lvolset.c	. . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾)
3		lvolset.p	. . . 4 ⊢ 𝑃 = (LPlanes‘𝐾)
4		lvolset.v	. . . 4 ⊢ 𝑉 = (LVols‘𝐾)
5	1, 2, 3, 4	lvolset 40018	. . 3 ⊢ (𝐾 ∈ 𝐴 → 𝑉 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥})
6	5	eleq2d 2822	. 2 ⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑉 ↔ 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥}))
7		breq2 5089	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑦𝐶𝑥 ↔ 𝑦𝐶𝑋))
8	7	rexbidv 3161	. . 3 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝑃 𝑦𝐶𝑥 ↔ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑋))
9	8	elrab 3634	. 2 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑥} ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑋))
10	6, 9	bitrdi 287	1 ⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  {crab 3389   class class class wbr 5085  ‘cfv 6498  Basecbs 17179   ⋖ ccvr 39708  LPlanesclpl 39938  LVolsclvol 39939

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-lvols 39946

This theorem is referenced by:  islvol4  40020  lvoli  40021  lvolbase  40024  lvolnle3at  40028