Theorem ispsubsp2 39703

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 39702	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))))
6		ralcom 3295	. . . . . . 7 ⊢ (∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑝 ∈ 𝐴 ∀𝑟 ∈ 𝑋 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
7		r19.23v 3189	. . . . . . . 8 ⊢ (∀𝑟 ∈ 𝑋 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
8	7	ralbii 3099	. . . . . . 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 3099	. . . . 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 3295	. . . . . 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 3189	. . . . . . 7 ⊢ (∀𝑞 ∈ 𝑋 (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
13	12	ralbii 3099	. . . . . 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 629	. 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 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  lecple 17318  joincjn 18381  Atomscatm 39219  PSubSpcpsubsp 39453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-psubsp 39460

This theorem is referenced by:  psubspi  39704  paddclN  39799