Theorem pclcmpatN 40926

Description: The set of projective subspaces is compactly atomistic: if an atom is in the projective subspace closure of a set of atoms, it also belongs to the projective subspace closure of a finite subset of that set. Analogous to Lemma 3.3.10 of [PtakPulmannova] p. 74. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pclfin.a	⊢ 𝐴 = (Atoms‘𝐾)
pclfin.c	⊢ 𝑈 = (PCl‘𝐾)

Assertion

Ref	Expression
pclcmpatN	⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑃 ∈ (𝑈‘𝑋)) → ∃𝑦 ∈ Fin (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦)))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑈   𝑦,𝐾   𝑦,𝑋   𝑦,𝑃

Proof of Theorem pclcmpatN

Step	Hyp	Ref	Expression
1		pclfin.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
2		pclfin.c	. . . . . 6 ⊢ 𝑈 = (PCl‘𝐾)
3	1, 2	pclfinN 40925	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦))
4	3	eleq2d 2847	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑃 ∈ (𝑈‘𝑋) ↔ 𝑃 ∈ ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦)))
5		eliun 4955	. . . 4 ⊢ (𝑃 ∈ ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦) ↔ ∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈‘𝑦))
6	4, 5	bitrdi 290	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑃 ∈ (𝑈‘𝑋) ↔ ∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈‘𝑦)))
7		elin 3915	. . . . . . 7 ⊢ (𝑦 ∈ (Fin ∩ 𝒫 𝑋) ↔ (𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋))
8		elpwi 4564	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
9	8	anim2i 629	. . . . . . 7 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋))
10	7, 9	sylbi 220	. . . . . 6 ⊢ (𝑦 ∈ (Fin ∩ 𝒫 𝑋) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋))
11	10	anim1i 627	. . . . 5 ⊢ ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (𝑈‘𝑦)) → ((𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋) ∧ 𝑃 ∈ (𝑈‘𝑦)))
12		anass 474	. . . . 5 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋) ∧ 𝑃 ∈ (𝑈‘𝑦)) ↔ (𝑦 ∈ Fin ∧ (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦))))
13	11, 12	sylib 221	. . . 4 ⊢ ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (𝑈‘𝑦)) → (𝑦 ∈ Fin ∧ (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦))))
14	13	reximi2 3096	. . 3 ⊢ (∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈‘𝑦) → ∃𝑦 ∈ Fin (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦)))
15	6, 14	biimtrdi 256	. 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑃 ∈ (𝑈‘𝑋) → ∃𝑦 ∈ Fin (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦))))
16	15	3impia 1135	1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑃 ∈ (𝑈‘𝑋)) → ∃𝑦 ∈ Fin (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ ciun 4951  ‘cfv 6531  Fincfn 8957  Atomscatm 40288  AtLatcal 40289  PClcpclN 40912

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-om 7867  df-1o 8460  df-en 8958  df-fin 8961  df-proset 18448  df-poset 18467  df-plt 18482  df-lub 18498  df-glb 18499  df-join 18500  df-meet 18501  df-p0 18577  df-lat 18586  df-covers 40291  df-ats 40292  df-atl 40323  df-psubsp 40528  df-pclN 40913

This theorem is used by: (None)