Theorem psubspi2N 39705

Description: Property of a projective subspace. (Contributed by NM, 13-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
psubspset.l	⊢ ≤ = (le‘𝐾)
psubspset.j	⊢ ∨ = (join‘𝐾)
psubspset.a	⊢ 𝐴 = (Atoms‘𝐾)
psubspset.s	⊢ 𝑆 = (PSubSp‘𝐾)

Assertion

Ref	Expression
psubspi2N	⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ (𝑄 ∈ 𝑋 ∧ 𝑅 ∈ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃 ∈ 𝑋)

Proof of Theorem psubspi2N

Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7455	. . . 4 ⊢ (𝑞 = 𝑄 → (𝑞 ∨ 𝑟) = (𝑄 ∨ 𝑟))
2	1	breq2d 5178	. . 3 ⊢ (𝑞 = 𝑄 → (𝑃 ≤ (𝑞 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ 𝑟)))
3		oveq2 7456	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑅))
4	3	breq2d 5178	. . 3 ⊢ (𝑟 = 𝑅 → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ 𝑅)))
5	2, 4	rspc2ev 3648	. 2 ⊢ ((𝑄 ∈ 𝑋 ∧ 𝑅 ∈ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) → ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟))
6		psubspset.l	. . 3 ⊢ ≤ = (le‘𝐾)
7		psubspset.j	. . 3 ⊢ ∨ = (join‘𝐾)
8		psubspset.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
9		psubspset.s	. . 3 ⊢ 𝑆 = (PSubSp‘𝐾)
10	6, 7, 8, 9	psubspi 39704	. 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟)) → 𝑃 ∈ 𝑋)
11	5, 10	sylan2 592	1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ (𝑄 ∈ 𝑋 ∧ 𝑅 ∈ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃 ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∃wrex 3076   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  lecple 17318  joincjn 18381  Atomscatm 39219  PSubSpcpsubsp 39453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-psubsp 39460

This theorem is referenced by:  pclclN  39848  pclfinN  39857  pclfinclN  39907