Theorem ispsubsp2 40209

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 40208	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))))
6		ralcom 3266	. . . . . . 7 ⊢ (∀𝑟 ∈ 𝑋 ∀𝑝 ∈ 𝐴 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ ∀𝑝 ∈ 𝐴 ∀𝑟 ∈ 𝑋 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
7		r19.23v 3165	. . . . . . . 8 ⊢ (∀𝑟 ∈ 𝑋 (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
8	7	ralbii 3084	. . . . . . 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 3084	. . . . 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 3266	. . . . . 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 3165	. . . . . . 7 ⊢ (∀𝑞 ∈ 𝑋 (∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
13	12	ralbii 3084	. . . . . 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 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 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 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890   class class class wbr 5086  ‘cfv 6493  (class class class)co 7361  lecple 17221  joincjn 18271  Atomscatm 39726  PSubSpcpsubsp 39959

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7364  df-psubsp 39966

This theorem is referenced by:  psubspi  40210  paddclN  40305