Theorem psubspi 39765

Description: Property of 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
psubspi	⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟)) → 𝑃 ∈ 𝑋)

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

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

Proof of Theorem psubspi

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psubspset.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
2		psubspset.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
3		psubspset.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
4		psubspset.s	. . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾)
5	1, 2, 3, 4	ispsubsp2 39764	. . . . 5 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))))
6	5	simplbda 499	. . . 4 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
7	6	ex 412	. . 3 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 → ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋)))
8		breq1 5092	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ (𝑞 ∨ 𝑟) ↔ 𝑃 ≤ (𝑞 ∨ 𝑟)))
9	8	2rexbidv 3195	. . . . 5 ⊢ (𝑝 = 𝑃 → (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) ↔ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟)))
10		eleq1 2817	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ 𝑋 ↔ 𝑃 ∈ 𝑋))
11	9, 10	imbi12d 344	. . . 4 ⊢ (𝑝 = 𝑃 → ((∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟) → 𝑃 ∈ 𝑋)))
12	11	rspccv 3572	. . 3 ⊢ (∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) → (𝑃 ∈ 𝐴 → (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟) → 𝑃 ∈ 𝑋)))
13	7, 12	syl6 35	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 → (𝑃 ∈ 𝐴 → (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟) → 𝑃 ∈ 𝑋))))
14	13	3imp1 1348	1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟)) → 𝑃 ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110  ∀wral 3045  ∃wrex 3054   ⊆ wss 3900   class class class wbr 5089  ‘cfv 6477  (class class class)co 7341  lecple 17160  joincjn 18209  Atomscatm 39281  PSubSpcpsubsp 39514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6433  df-fun 6479  df-fv 6485  df-ov 7344  df-psubsp 39521

This theorem is referenced by:  psubspi2N  39766  paddidm  39859