Theorem ispsubsp 40895

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 40894	. . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑆 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥))})
6	5	eleq2d 2846	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥))}))
7	3	fvexi 6899	. . . . 5 ⊢ 𝐴 ∈ V
8	7	ssex 5281	. . . 4 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ∈ V)
9	8	adantr 486	. . 3 ⊢ ((𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)) → 𝑋 ∈ V)
10		sseq1 3933	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
11		eleq2 2849	. . . . . . . 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 3185	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥) ↔ ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)))
14	13	raleqbi1dv 3329	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥) ↔ ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)))
15	14	raleqbi1dv 3329	. . . 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 644	. . 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 3639	. 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 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  lecple 17440  joincjn 18490  Atomscatm 40413  PSubSpcpsubsp 40646

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-psubsp 40653

This theorem is used by:  ispsubsp2  40896  0psubN  40899  snatpsubN  40900  linepsubN  40902  atpsubN  40903  psubssat  40904  pmapsub  40918  pclclN  41041  pclfinN  41050