Theorem atpsubN 36891

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 3991	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		ax-1 6	. . . . 5 ⊢ (𝑟 ∈ 𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴))
3	2	rgen 3150	. . . 4 ⊢ ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴)
4	3	rgen2w 3153	. . 3 ⊢ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴)
5	1, 4	pm3.2i 473	. 2 ⊢ (𝐴 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴))
6		eqid 2823	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
7		eqid 2823	. . 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 36883	. 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 260	1 ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140   ⊆ wss 3938   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  lecple 16574  joincjn 17556  Atomscatm 36401  PSubSpcpsubsp 36634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-psubsp 36641

This theorem is referenced by:  pclvalN  37028  pclclN  37029