Theorem atpsubN 40245

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 3937	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		ax-1 6	. . . . 5 ⊢ (𝑟 ∈ 𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴))
3	2	rgen 3055	. . . 4 ⊢ ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴)
4	3	rgen2w 3058	. . 3 ⊢ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴)
5	1, 4	pm3.2i 471	. 2 ⊢ (𝐴 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴))
6		eqid 2739	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
7		eqid 2739	. . 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 40237	. 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 259	1 ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ⊆ wss 3883   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  lecple 17218  joincjn 18268  Atomscatm 39755  PSubSpcpsubsp 39988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-psubsp 39995

This theorem is referenced by:  pclvalN  40382  pclclN  40383