Theorem psubspi 40996

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 40995	. . . . 5 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))))
6	5	simplbda 505	. . . 4 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))
7	6	ex 418	. . 3 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 → ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋)))
8		breq1 5109	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ (𝑞 ∨ 𝑟) ↔ 𝑃 ≤ (𝑞 ∨ 𝑟)))
9	8	2rexbidv 3231	. . . . 5 ⊢ (𝑝 = 𝑃 → (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) ↔ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟)))
10		eleq1 2852	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ 𝑋 ↔ 𝑃 ∈ 𝑋))
11	9, 10	imbi12d 347	. . . 4 ⊢ (𝑝 = 𝑃 → ((∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋) ↔ (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟) → 𝑃 ∈ 𝑋)))
12	11	rspccv 3577	. . 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 36	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 → (𝑃 ∈ 𝐴 → (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟) → 𝑃 ∈ 𝑋))))
14	13	3imp1 1366	1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟)) → 𝑃 ∈ 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   class class class wbr 5106  ‘cfv 6558  (class class class)co 7443  lecple 17464  joincjn 18514  Atomscatm 40512  PSubSpcpsubsp 40745

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-psubsp 40752

This theorem is used by:  psubspi2N  40997  paddidm  41090