Theorem ispsubsp 40547

Description: The predicate "is a projective subspace". (Contributed by NM, 2-Oct-2011.)

Hypotheses

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

Assertion

Ref	Expression
ispsubsp	⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋))))

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

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

Proof of Theorem ispsubsp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psubspset.l	. . . 4 ⊢ ≤ = (le‘𝐾)
2		psubspset.j	. . . 4 ⊢ ∨ = (join‘𝐾)
3		psubspset.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
4		psubspset.s	. . . 4 ⊢ 𝑆 = (PSubSp‘𝐾)
5	1, 2, 3, 4	psubspset 40546	. . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑆 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥))})
6	5	eleq2d 2848	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥))}))
7	3	fvexi 6895	. . . . 5 ⊢ 𝐴 ∈ V
8	7	ssex 5290	. . . 4 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ∈ V)
9	8	adantr 485	. . 3 ⊢ ((𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)) → 𝑋 ∈ V)
10		sseq1 3961	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
11		eleq2 2851	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑟 ∈ 𝑥 ↔ 𝑟 ∈ 𝑋))
12	11	imbi2d 343	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥) ↔ (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)))
13	12	ralbidv 3187	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥) ↔ ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)))
14	13	raleqbi1dv 3332	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥) ↔ ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)))
15	14	raleqbi1dv 3332	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥) ↔ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)))
16	10, 15	anbi12d 643	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥)) ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋))))
17	9, 16	elab3 3644	. 2 ⊢ (𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥))} ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)))
18	6, 17	bitrdi 290	1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {cab 2740  ∀wral 3078  Vcvv 3454   ⊆ wss 3904   class class class wbr 5108  ‘cfv 6536  (class class class)co 7412  lecple 17323  joincjn 18373  Atomscatm 40065  PSubSpcpsubsp 40298

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415  df-psubsp 40305

This theorem is used by:  ispsubsp2  40548  0psubN  40551  snatpsubN  40552  linepsubN  40554  atpsubN  40555  psubssat  40556  pmapsub  40570  pclclN  40693  pclfinN  40702