Theorem lplnset 36217

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

Hypotheses

Ref	Expression
lplnset.b	⊢ 𝐵 = (Base‘𝐾)
lplnset.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
lplnset.n	⊢ 𝑁 = (LLines‘𝐾)
lplnset.p	⊢ 𝑃 = (LPlanes‘𝐾)

Assertion

Ref	Expression
lplnset	⊢ (𝐾 ∈ 𝐴 → 𝑃 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})

Distinct variable groups:   𝑦,𝑁   𝑥,𝐵   𝑥,𝑦,𝐾

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝐶(𝑥,𝑦)   𝑃(𝑥,𝑦)   𝑁(𝑥)

Proof of Theorem lplnset

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3458	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		lplnset.p	. . 3 ⊢ 𝑃 = (LPlanes‘𝐾)
3		fveq2 6545	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		lplnset.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	syl6eqr 2851	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6		fveq2 6545	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LLines‘𝑘) = (LLines‘𝐾))
7		lplnset.n	. . . . . . 7 ⊢ 𝑁 = (LLines‘𝐾)
8	6, 7	syl6eqr 2851	. . . . . 6 ⊢ (𝑘 = 𝐾 → (LLines‘𝑘) = 𝑁)
9		fveq2 6545	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10		lplnset.c	. . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾)
11	9, 10	syl6eqr 2851	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
12	11	breqd 4979	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥 ↔ 𝑦𝐶𝑥))
13	8, 12	rexeqbidv 3364	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥))
14	5, 13	rabeqbidv 3433	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})
15		df-lplanes 36187	. . . 4 ⊢ LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
16	4	fvexi 6559	. . . . 5 ⊢ 𝐵 ∈ V
17	16	rabex 5133	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥} ∈ V
18	14, 15, 17	fvmpt 6642	. . 3 ⊢ (𝐾 ∈ V → (LPlanes‘𝐾) = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})
19	2, 18	syl5eq 2845	. 2 ⊢ (𝐾 ∈ V → 𝑃 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})
20	1, 19	syl 17	1 ⊢ (𝐾 ∈ 𝐴 → 𝑃 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})

Syntax hints:   → wi 4   = wceq 1525   ∈ wcel 2083  ∃wrex 3108  {crab 3111  Vcvv 3440   class class class wbr 4968  ‘cfv 6232  Basecbs 16316   ⋖ ccvr 35950  LLinesclln 36179  LPlanesclpl 36180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-iota 6196  df-fun 6234  df-fv 6240  df-lplanes 36187

This theorem is referenced by:  islpln  36218