Theorem pclcmpatN 40271

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 40270	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦))
4	3	eleq2d 2823	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑃 ∈ (𝑈‘𝑋) ↔ 𝑃 ∈ ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦)))
5		eliun 4952	. . . 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 3919	. . . . . . 7 ⊢ (𝑦 ∈ (Fin ∩ 𝒫 𝑋) ↔ (𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋))
8		elpwi 4563	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
9	8	anim2i 618	. . . . . . 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 616	. . . . 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 3071	. . 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 1542   ∈ wcel 2114  ∃wrex 3062   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  ∪ ciun 4948  ‘cfv 6500  Fincfn 8895  Atomscatm 39633  AtLatcal 39634  PClcpclN 40257

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-om 7819  df-1o 8407  df-en 8896  df-fin 8899  df-proset 18229  df-poset 18248  df-plt 18263  df-lub 18279  df-glb 18280  df-join 18281  df-meet 18282  df-p0 18358  df-lat 18367  df-covers 39636  df-ats 39637  df-atl 39668  df-psubsp 39873  df-pclN 40258

This theorem is referenced by: (None)