Theorem psubspi 40319

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 40318	. . . . 5 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))))
6	5	simplbda 502	. . . 4 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
7	6	ex 415	. . 3 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 → ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋)))
8		breq1 5097	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ (𝑞 ∨ 𝑟) ↔ 𝑃 ≤ (𝑞 ∨ 𝑟)))
9	8	2rexbidv 3221	. . . . 5 ⊢ (𝑝 = 𝑃 → (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) ↔ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟)))
10		eleq1 2844	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ 𝑋 ↔ 𝑃 ∈ 𝑋))
11	9, 10	imbi12d 346	. . . 4 ⊢ (𝑝 = 𝑃 → ((∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟) → 𝑃 ∈ 𝑋)))
12	11	rspccv 3573	. . 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 1357	1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟)) → 𝑃 ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136  ∀wral 3070  ∃wrex 3080   ⊆ wss 3899   class class class wbr 5094  ‘cfv 6510  (class class class)co 7385  lecple 17269  joincjn 18319  Atomscatm 39835  PSubSpcpsubsp 40068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6512  df-fv 6518  df-ov 7388  df-psubsp 40075

This theorem is referenced by:  psubspi2N  40320  paddidm  40413