Theorem llnset 38364

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

Hypotheses

Ref	Expression
llnset.b	⊢ 𝐵 = (Base‘𝐾)
llnset.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
llnset.a	⊢ 𝐴 = (Atoms‘𝐾)
llnset.n	⊢ 𝑁 = (LLines‘𝐾)

Assertion

Ref	Expression
llnset	⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥})

Distinct variable groups:   𝐴,𝑝   𝑥,𝐵   𝑥,𝑝,𝐾

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑝)   𝐶(𝑥,𝑝)   𝐷(𝑥,𝑝)   𝑁(𝑥,𝑝)

Proof of Theorem llnset

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3492	. 2 ⊢ (𝐾 ∈ 𝐷 → 𝐾 ∈ V)
2		llnset.n	. . 3 ⊢ 𝑁 = (LLines‘𝐾)
3		fveq2 6888	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		llnset.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2790	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6		fveq2 6888	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
7		llnset.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
8	6, 7	eqtr4di 2790	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
9		fveq2 6888	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10		llnset.c	. . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾)
11	9, 10	eqtr4di 2790	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
12	11	breqd 5158	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑝( ⋖ ‘𝑘)𝑥 ↔ 𝑝𝐶𝑥))
13	8, 12	rexeqbidv 3343	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥 ↔ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥))
14	5, 13	rabeqbidv 3449	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥} = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥})
15		df-llines 38357	. . . 4 ⊢ LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
16	4	fvexi 6902	. . . . 5 ⊢ 𝐵 ∈ V
17	16	rabex 5331	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥} ∈ V
18	14, 15, 17	fvmpt 6995	. . 3 ⊢ (𝐾 ∈ V → (LLines‘𝐾) = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥})
19	2, 18	eqtrid 2784	. 2 ⊢ (𝐾 ∈ V → 𝑁 = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥})
20	1, 19	syl 17	1 ⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥})

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  {crab 3432  Vcvv 3474   class class class wbr 5147  ‘cfv 6540  Basecbs 17140   ⋖ ccvr 38120  Atomscatm 38121  LLinesclln 38350

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-llines 38357

This theorem is referenced by:  islln  38365