Theorem ispsubsp2 37042

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 37041	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))))
6		ralcom 3307	. . . . . . 7 ⊢ (∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑝 ∈ 𝐴 ∀𝑟 ∈ 𝑋 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
7		r19.23v 3238	. . . . . . . 8 ⊢ (∀𝑟 ∈ 𝑋 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
8	7	ralbii 3133	. . . . . . 7 ⊢ (∀𝑝 ∈ 𝐴 ∀𝑟 ∈ 𝑋 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑝 ∈ 𝐴 (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
9	6, 8	bitri 278	. . . . . 6 ⊢ (∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑝 ∈ 𝐴 (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
10	9	ralbii 3133	. . . . 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 3307	. . . . . 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 3238	. . . . . . 7 ⊢ (∀𝑞 ∈ 𝑋 (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
13	12	ralbii 3133	. . . . . 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 278	. . . . 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 278	. . . 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 631	. 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 282	1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  lecple 16564  joincjn 17546  Atomscatm 36559  PSubSpcpsubsp 36792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-psubsp 36799

This theorem is referenced by:  psubspi  37043  paddclN  37138