![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > pclcmpatN | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
pclfin.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pclfin.c | ⊢ 𝑈 = (PCl‘𝐾) |
Ref | Expression |
---|---|
pclcmpatN | ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑃 ∈ (𝑈‘𝑋)) → ∃𝑦 ∈ Fin (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pclfin.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | pclfin.c | . . . . . 6 ⊢ 𝑈 = (PCl‘𝐾) | |
3 | 1, 2 | pclfinN 35689 | . . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦)) |
4 | 3 | eleq2d 2825 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑃 ∈ (𝑈‘𝑋) ↔ 𝑃 ∈ ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦))) |
5 | eliun 4676 | . . . 4 ⊢ (𝑃 ∈ ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦) ↔ ∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈‘𝑦)) | |
6 | 4, 5 | syl6bb 276 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑃 ∈ (𝑈‘𝑋) ↔ ∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈‘𝑦))) |
7 | elin 3939 | . . . . . . 7 ⊢ (𝑦 ∈ (Fin ∩ 𝒫 𝑋) ↔ (𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋)) | |
8 | elpwi 4312 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋) | |
9 | 8 | anim2i 594 | . . . . . . 7 ⊢ ((𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋)) |
10 | 7, 9 | sylbi 207 | . . . . . 6 ⊢ (𝑦 ∈ (Fin ∩ 𝒫 𝑋) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋)) |
11 | 10 | anim1i 593 | . . . . 5 ⊢ ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (𝑈‘𝑦)) → ((𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋) ∧ 𝑃 ∈ (𝑈‘𝑦))) |
12 | anass 684 | . . . . 5 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋) ∧ 𝑃 ∈ (𝑈‘𝑦)) ↔ (𝑦 ∈ Fin ∧ (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦)))) | |
13 | 11, 12 | sylib 208 | . . . 4 ⊢ ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (𝑈‘𝑦)) → (𝑦 ∈ Fin ∧ (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦)))) |
14 | 13 | reximi2 3148 | . . 3 ⊢ (∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈‘𝑦) → ∃𝑦 ∈ Fin (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦))) |
15 | 6, 14 | syl6bi 243 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴) → (𝑃 ∈ (𝑈‘𝑋) → ∃𝑦 ∈ Fin (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦)))) |
16 | 15 | 3impia 1110 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑃 ∈ (𝑈‘𝑋)) → ∃𝑦 ∈ Fin (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ∃wrex 3051 ∩ cin 3714 ⊆ wss 3715 𝒫 cpw 4302 ∪ ciun 4672 ‘cfv 6049 Fincfn 8121 Atomscatm 35053 AtLatcal 35054 PClcpclN 35676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-en 8122 df-fin 8125 df-preset 17129 df-poset 17147 df-plt 17159 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-p0 17240 df-lat 17247 df-covers 35056 df-ats 35057 df-atl 35088 df-psubsp 35292 df-pclN 35677 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |