Theorem llnset 40828

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 3475	. 2 ⊢ (𝐾 ∈ 𝐷 → 𝐾 ∈ V)
2		llnset.n	. . 3 ⊢ 𝑁 = (LLines‘𝐾)
3		fveq2 6916	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		llnset.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2816	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6		fveq2 6916	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
7		llnset.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
8	6, 7	eqtr4di 2816	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
9		fveq2 6916	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10		llnset.c	. . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾)
11	9, 10	eqtr4di 2816	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
12	11	breqd 5119	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑝( ⋖ ‘𝑘)𝑥 ↔ 𝑝𝐶𝑥))
13	8, 12	rexeqbidv 3339	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥 ↔ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥))
14	5, 13	rabeqbidv 3433	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥} = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥})
15		df-llines 40821	. . . 4 ⊢ LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
16	4	fvexi 6930	. . . . 5 ⊢ 𝐵 ∈ V
17	16	rabex 5307	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥} ∈ V
18	14, 15, 17	fvmpt 7024	. . 3 ⊢ (𝐾 ∈ V → (LLines‘𝐾) = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥})
19	2, 18	eqtrid 2810	. 2 ⊢ (𝐾 ∈ V → 𝑁 = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥})
20	1, 19	syl 18	1 ⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416  Vcvv 3454   class class class wbr 5108  ‘cfv 6566  Basecbs 17431   ⋖ ccvr 40585  Atomscatm 40586  LLinesclln 40814

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-llines 40821

This theorem is used by:  islln  40829