Theorem ispsubsp2 39728

Description: The predicate "is a projective subspace". (Contributed by NM, 13-Jan-2012.)

Hypotheses

Ref	Expression
psubspset.l	⊢ ≤ = (le‘𝐾)
psubspset.j	⊢ ∨ = (join‘𝐾)
psubspset.a	⊢ 𝐴 = (Atoms‘𝐾)
psubspset.s	⊢ 𝑆 = (PSubSp‘𝐾)

Assertion

Ref	Expression
ispsubsp2	⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))))

Distinct variable groups:   𝐴,𝑟   𝑞,𝑝,𝑟,𝐾   𝑋,𝑝,𝑞,𝑟   𝐴,𝑝,𝑞

Allowed substitution hints:   𝐷(𝑟,𝑞,𝑝)   𝑆(𝑟,𝑞,𝑝)   ∨ (𝑟,𝑞,𝑝)   ≤ (𝑟,𝑞,𝑝)

Proof of Theorem ispsubsp2

Step	Hyp	Ref	Expression
1		psubspset.l	. . 3 ⊢ ≤ = (le‘𝐾)
2		psubspset.j	. . 3 ⊢ ∨ = (join‘𝐾)
3		psubspset.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
4		psubspset.s	. . 3 ⊢ 𝑆 = (PSubSp‘𝐾)
5	1, 2, 3, 4	ispsubsp 39727	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))))
6		ralcom 3257	. . . . . . 7 ⊢ (∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑝 ∈ 𝐴 ∀𝑟 ∈ 𝑋 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
7		r19.23v 3156	. . . . . . . 8 ⊢ (∀𝑟 ∈ 𝑋 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
8	7	ralbii 3075	. . . . . . 7 ⊢ (∀𝑝 ∈ 𝐴 ∀𝑟 ∈ 𝑋 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑝 ∈ 𝐴 (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
9	6, 8	bitri 275	. . . . . 6 ⊢ (∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑝 ∈ 𝐴 (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
10	9	ralbii 3075	. . . . 5 ⊢ (∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑞 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
11		ralcom 3257	. . . . . 6 ⊢ (∀𝑞 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝑋 (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
12		r19.23v 3156	. . . . . . 7 ⊢ (∀𝑞 ∈ 𝑋 (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
13	12	ralbii 3075	. . . . . 6 ⊢ (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝑋 (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
14	11, 13	bitri 275	. . . . 5 ⊢ (∀𝑞 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
15	10, 14	bitri 275	. . . 4 ⊢ (∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
16	15	a1i 11	. . 3 ⊢ (𝐾 ∈ 𝐷 → (∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋)))
17	16	anbi2d 630	. 2 ⊢ (𝐾 ∈ 𝐷 → ((𝑋 ⊆ 𝐴 ∧ ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋)) ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))))
18	5, 17	bitrd 279	1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3905   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353  lecple 17186  joincjn 18235  Atomscatm 39244  PSubSpcpsubsp 39478

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 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

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 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-psubsp 39485

This theorem is referenced by:  psubspi  39729  paddclN  39824