Theorem pclcmpatN 39903

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 39902	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦))
4	3	eleq2d 2827	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑃 ∈ (𝑈‘𝑋) ↔ 𝑃 ∈ ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦)))
5		eliun 4995	. . . 4 ⊢ (𝑃 ∈ ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦) ↔ ∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈‘𝑦))
6	4, 5	bitrdi 287	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑃 ∈ (𝑈‘𝑋) ↔ ∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈‘𝑦)))
7		elin 3967	. . . . . . 7 ⊢ (𝑦 ∈ (Fin ∩ 𝒫 𝑋) ↔ (𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋))
8		elpwi 4607	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
9	8	anim2i 617	. . . . . . 7 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋))
10	7, 9	sylbi 217	. . . . . 6 ⊢ (𝑦 ∈ (Fin ∩ 𝒫 𝑋) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋))
11	10	anim1i 615	. . . . 5 ⊢ ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (𝑈‘𝑦)) → ((𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋) ∧ 𝑃 ∈ (𝑈‘𝑦)))
12		anass 468	. . . . 5 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋) ∧ 𝑃 ∈ (𝑈‘𝑦)) ↔ (𝑦 ∈ Fin ∧ (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦))))
13	11, 12	sylib 218	. . . 4 ⊢ ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (𝑈‘𝑦)) → (𝑦 ∈ Fin ∧ (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦))))
14	13	reximi2 3079	. . 3 ⊢ (∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈‘𝑦) → ∃𝑦 ∈ Fin (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦)))
15	6, 14	biimtrdi 253	. 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑃 ∈ (𝑈‘𝑋) → ∃𝑦 ∈ Fin (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦))))
16	15	3impia 1118	1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑃 ∈ (𝑈‘𝑋)) → ∃𝑦 ∈ Fin (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∃wrex 3070   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600  ∪ ciun 4991  ‘cfv 6561  Fincfn 8985  Atomscatm 39264  AtLatcal 39265  PClcpclN 39889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-om 7888  df-1o 8506  df-en 8986  df-fin 8989  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-covers 39267  df-ats 39268  df-atl 39299  df-psubsp 39505  df-pclN 39890

This theorem is referenced by: (None)