Theorem ispsubsp2 41069

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 41068	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))))
6		ralcom 3294	. . . . . . 7 ⊢ (∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑝 ∈ 𝐴 ∀𝑟 ∈ 𝑋 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
7		r19.23v 3193	. . . . . . . 8 ⊢ (∀𝑟 ∈ 𝑋 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
8	7	ralbii 3112	. . . . . . 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 3112	. . . . 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 3294	. . . . . 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 3193	. . . . . . 7 ⊢ (∀𝑞 ∈ 𝑋 (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
13	12	ralbii 3112	. . . . . 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 642	. 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))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451  lecple 17479  joincjn 18535  Atomscatm 40586  PSubSpcpsubsp 40819

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-psubsp 40826

This theorem is used by:  psubspi  41070  paddclN  41165