Theorem pclcmpatN 38760

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 38759	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦))
4	3	eleq2d 2819	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑃 ∈ (𝑈‘𝑋) ↔ 𝑃 ∈ ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦)))
5		eliun 5000	. . . 4 ⊢ (𝑃 ∈ ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦) ↔ ∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈‘𝑦))
6	4, 5	bitrdi 286	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑃 ∈ (𝑈‘𝑋) ↔ ∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈‘𝑦)))
7		elin 3963	. . . . . . 7 ⊢ (𝑦 ∈ (Fin ∩ 𝒫 𝑋) ↔ (𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋))
8		elpwi 4608	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
9	8	anim2i 617	. . . . . . 7 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋))
10	7, 9	sylbi 216	. . . . . 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 469	. . . . 5 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋) ∧ 𝑃 ∈ (𝑈‘𝑦)) ↔ (𝑦 ∈ Fin ∧ (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦))))
13	11, 12	sylib 217	. . . 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	syl6bi 252	. 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑃 ∈ (𝑈‘𝑋) → ∃𝑦 ∈ Fin (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦))))
16	15	3impia 1117	1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑃 ∈ (𝑈‘𝑋)) → ∃𝑦 ∈ Fin (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3070   ∩ cin 3946   ⊆ wss 3947  𝒫 cpw 4601  ∪ ciun 4996  ‘cfv 6540  Fincfn 8935  Atomscatm 38121  AtLatcal 38122  PClcpclN 38746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-om 7852  df-1o 8462  df-en 8936  df-fin 8939  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-covers 38124  df-ats 38125  df-atl 38156  df-psubsp 38362  df-pclN 38747

This theorem is referenced by: (None)