Theorem lhpset 39377

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 3487	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		lhpset.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
3		fveq2 6884	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		lhpset.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2784	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6		eqidd 2727	. . . . . 6 ⊢ (𝑘 = 𝐾 → 𝑤 = 𝑤)
7		fveq2 6884	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
8		lhpset.c	. . . . . . 7 ⊢ 𝐶 = ( ⋖ ‘𝐾)
9	7, 8	eqtr4di 2784	. . . . . 6 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
10		fveq2 6884	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
11		lhpset.u	. . . . . . 7 ⊢ 1 = (1.‘𝐾)
12	10, 11	eqtr4di 2784	. . . . . 6 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
13	6, 9, 12	breq123d 5155	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑤( ⋖ ‘𝑘)(1.‘𝑘) ↔ 𝑤𝐶 1 ))
14	5, 13	rabeqbidv 3443	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)} = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })
15		df-lhyp 39370	. . . 4 ⊢ LHyp = (𝑘 ∈ V ↦ {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)})
16	4	fvexi 6898	. . . . 5 ⊢ 𝐵 ∈ V
17	16	rabex 5325	. . . 4 ⊢ {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 } ∈ V
18	14, 15, 17	fvmpt 6991	. . 3 ⊢ (𝐾 ∈ V → (LHyp‘𝐾) = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })
19	2, 18	eqtrid 2778	. 2 ⊢ (𝐾 ∈ V → 𝐻 = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })
20	1, 19	syl 17	1 ⊢ (𝐾 ∈ 𝐴 → 𝐻 = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468   class class class wbr 5141  ‘cfv 6536  Basecbs 17151  1.cp1 18387   ⋖ ccvr 38643  LHypclh 39366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-lhyp 39370

This theorem is referenced by:  islhp  39378