Theorem lhpset 39981

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 3476	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		lhpset.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
3		fveq2 6865	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		lhpset.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2783	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6		eqidd 2731	. . . . . 6 ⊢ (𝑘 = 𝐾 → 𝑤 = 𝑤)
7		fveq2 6865	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
8		lhpset.c	. . . . . . 7 ⊢ 𝐶 = ( ⋖ ‘𝐾)
9	7, 8	eqtr4di 2783	. . . . . 6 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
10		fveq2 6865	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
11		lhpset.u	. . . . . . 7 ⊢ 1 = (1.‘𝐾)
12	10, 11	eqtr4di 2783	. . . . . 6 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
13	6, 9, 12	breq123d 5129	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑤( ⋖ ‘𝑘)(1.‘𝑘) ↔ 𝑤𝐶 1 ))
14	5, 13	rabeqbidv 3430	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)} = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })
15		df-lhyp 39974	. . . 4 ⊢ LHyp = (𝑘 ∈ V ↦ {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)})
16	4	fvexi 6879	. . . . 5 ⊢ 𝐵 ∈ V
17	16	rabex 5302	. . . 4 ⊢ {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 } ∈ V
18	14, 15, 17	fvmpt 6975	. . 3 ⊢ (𝐾 ∈ V → (LHyp‘𝐾) = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })
19	2, 18	eqtrid 2777	. 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 1540   ∈ wcel 2109  {crab 3411  Vcvv 3455   class class class wbr 5115  ‘cfv 6519  Basecbs 17185  1.cp1 18389   ⋖ ccvr 39247  LHypclh 39970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-iota 6472  df-fun 6521  df-fv 6527  df-lhyp 39974

This theorem is referenced by:  islhp  39982