Theorem ispsubsp 36883

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 36882	. . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑆 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥))})
6	5	eleq2d 2900	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥))}))
7	3	fvexi 6686	. . . . 5 ⊢ 𝐴 ∈ V
8	7	ssex 5227	. . . 4 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ∈ V)
9	8	adantr 483	. . 3 ⊢ ((𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)) → 𝑋 ∈ V)
10		sseq1 3994	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
11		eleq2 2903	. . . . . . . 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 3199	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥) ↔ ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)))
14	13	raleqbi1dv 3405	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑞 ∈ 𝑥 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑥) ↔ ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋)))
15	14	raleqbi1dv 3405	. . . 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 632	. . 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 3676	. 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	syl6bb 289	1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2801  ∀wral 3140  Vcvv 3496   ⊆ wss 3938   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  lecple 16574  joincjn 17556  Atomscatm 36401  PSubSpcpsubsp 36634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-psubsp 36641

This theorem is referenced by:  ispsubsp2  36884  0psubN  36887  snatpsubN  36888  linepsubN  36890  atpsubN  36891  psubssat  36892  pmapsub  36906  pclclN  37029  pclfinN  37038