Theorem psubspi 38556

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 38555	. . . . 5 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))))
6	5	simplbda 501	. . . 4 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
7	6	ex 414	. . 3 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 → ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋)))
8		breq1 5150	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ (𝑞 ∨ 𝑟) ↔ 𝑃 ≤ (𝑞 ∨ 𝑟)))
9	8	2rexbidv 3220	. . . . 5 ⊢ (𝑝 = 𝑃 → (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) ↔ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟)))
10		eleq1 2822	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ 𝑋 ↔ 𝑃 ∈ 𝑋))
11	9, 10	imbi12d 345	. . . 4 ⊢ (𝑝 = 𝑃 → ((∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟) → 𝑃 ∈ 𝑋)))
12	11	rspccv 3609	. . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   ⊆ wss 3947   class class class wbr 5147  ‘cfv 6540  (class class class)co 7404  lecple 17200  joincjn 18260  Atomscatm 38071  PSubSpcpsubsp 38305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7407  df-psubsp 38312

This theorem is referenced by:  psubspi2N  38557  paddidm  38650