Theorem lplnset 36659

Description: The set of lattice planes in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)

Hypotheses

Ref	Expression
lplnset.b	⊢ 𝐵 = (Base‘𝐾)
lplnset.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
lplnset.n	⊢ 𝑁 = (LLines‘𝐾)
lplnset.p	⊢ 𝑃 = (LPlanes‘𝐾)

Assertion

Ref	Expression
lplnset	⊢ (𝐾 ∈ 𝐴 → 𝑃 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})

Distinct variable groups:   𝑦,𝑁   𝑥,𝐵   𝑥,𝑦,𝐾

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝐶(𝑥,𝑦)   𝑃(𝑥,𝑦)   𝑁(𝑥)

Proof of Theorem lplnset

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3512	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		lplnset.p	. . 3 ⊢ 𝑃 = (LPlanes‘𝐾)
3		fveq2 6664	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		lplnset.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	syl6eqr 2874	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6		fveq2 6664	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LLines‘𝑘) = (LLines‘𝐾))
7		lplnset.n	. . . . . . 7 ⊢ 𝑁 = (LLines‘𝐾)
8	6, 7	syl6eqr 2874	. . . . . 6 ⊢ (𝑘 = 𝐾 → (LLines‘𝑘) = 𝑁)
9		fveq2 6664	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10		lplnset.c	. . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾)
11	9, 10	syl6eqr 2874	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
12	11	breqd 5069	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥 ↔ 𝑦𝐶𝑥))
13	8, 12	rexeqbidv 3402	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥))
14	5, 13	rabeqbidv 3485	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})
15		df-lplanes 36629	. . . 4 ⊢ LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
16	4	fvexi 6678	. . . . 5 ⊢ 𝐵 ∈ V
17	16	rabex 5227	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥} ∈ V
18	14, 15, 17	fvmpt 6762	. . 3 ⊢ (𝐾 ∈ V → (LPlanes‘𝐾) = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})
19	2, 18	syl5eq 2868	. 2 ⊢ (𝐾 ∈ V → 𝑃 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})
20	1, 19	syl 17	1 ⊢ (𝐾 ∈ 𝐴 → 𝑃 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  {crab 3142  Vcvv 3494   class class class wbr 5058  ‘cfv 6349  Basecbs 16477   ⋖ ccvr 36392  LLinesclln 36621  LPlanesclpl 36622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-lplanes 36629

This theorem is referenced by:  islpln  36660