Theorem atpsubN 39747

Description: The set of all atoms is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
atpsub.a	⊢ 𝐴 = (Atoms‘𝐾)
atpsub.s	⊢ 𝑆 = (PSubSp‘𝐾)

Assertion

Ref	Expression
atpsubN	⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆)

Proof of Theorem atpsubN

Dummy variables 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3969	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		ax-1 6	. . . . 5 ⊢ (𝑟 ∈ 𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴))
3	2	rgen 3046	. . . 4 ⊢ ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴)
4	3	rgen2w 3049	. . 3 ⊢ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴)
5	1, 4	pm3.2i 470	. 2 ⊢ (𝐴 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴))
6		eqid 2729	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
7		eqid 2729	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
8		atpsub.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
9		atpsub.s	. . 3 ⊢ 𝑆 = (PSubSp‘𝐾)
10	6, 7, 8, 9	ispsubsp 39739	. 2 ⊢ (𝐾 ∈ 𝑉 → (𝐴 ∈ 𝑆 ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴))))
11	5, 10	mpbiri 258	1 ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3914   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  lecple 17227  joincjn 18272  Atomscatm 39256  PSubSpcpsubsp 39490

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 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-psubsp 39497

This theorem is referenced by:  pclvalN  39884  pclclN  39885