Theorem lhpset 40654

Description: The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.)

Hypotheses

Ref	Expression
lhpset.b	⊢ 𝐵 = (Base‘𝐾)
lhpset.u	⊢ 1 = (1.‘𝐾)
lhpset.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
lhpset.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
lhpset	⊢ (𝐾 ∈ 𝐴 → 𝐻 = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })

Distinct variable groups:   𝑤,𝐵   𝑤,𝐶   𝑤,𝐾   𝑤, 1

Allowed substitution hints:   𝐴(𝑤)   𝐻(𝑤)

Proof of Theorem lhpset

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3484	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		lhpset.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
3		fveq2 6879	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		lhpset.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2822	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6		eqidd 2770	. . . . . 6 ⊢ (𝑘 = 𝐾 → 𝑤 = 𝑤)
7		fveq2 6879	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
8		lhpset.c	. . . . . . 7 ⊢ 𝐶 = ( ⋖ ‘𝐾)
9	7, 8	eqtr4di 2822	. . . . . 6 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
10		fveq2 6879	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
11		lhpset.u	. . . . . . 7 ⊢ 1 = (1.‘𝐾)
12	10, 11	eqtr4di 2822	. . . . . 6 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
13	6, 9, 12	breq123d 5124	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑤( ⋖ ‘𝑘)(1.‘𝑘) ↔ 𝑤𝐶 1 ))
14	5, 13	rabeqbidv 3441	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)} = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })
15		df-lhyp 40647	. . . 4 ⊢ LHyp = (𝑘 ∈ V ↦ {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)})
16	4	fvexi 6893	. . . . 5 ⊢ 𝐵 ∈ V
17	16	rabex 5307	. . . 4 ⊢ {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 } ∈ V
18	14, 15, 17	fvmpt 6987	. . 3 ⊢ (𝐾 ∈ V → (LHyp‘𝐾) = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })
19	2, 18	eqtrid 2816	. 2 ⊢ (𝐾 ∈ V → 𝐻 = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })
20	1, 19	syl 18	1 ⊢ (𝐾 ∈ 𝐴 → 𝐻 = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  {crab 3423  Vcvv 3463   class class class wbr 5110  ‘cfv 6534  Basecbs 17265  1.cp1 18474   ⋖ ccvr 39921  LHypclh 40643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6490  df-fun 6536  df-fv 6542  df-lhyp 40647

This theorem is referenced by:  islhp  40655