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Mathbox for Norm Megill |
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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 | β’ π΄ = (AtomsβπΎ) |
pclfin.c | β’ π = (PClβπΎ) |
Ref | Expression |
---|---|
pclcmpatN | β’ ((πΎ β AtLat β§ π β π΄ β§ π β (πβπ)) β βπ¦ β Fin (π¦ β π β§ π β (πβπ¦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pclfin.a | . . . . . 6 β’ π΄ = (AtomsβπΎ) | |
2 | pclfin.c | . . . . . 6 β’ π = (PClβπΎ) | |
3 | 1, 2 | pclfinN 39429 | . . . . 5 β’ ((πΎ β AtLat β§ π β π΄) β (πβπ) = βͺ π¦ β (Fin β© π« π)(πβπ¦)) |
4 | 3 | eleq2d 2811 | . . . 4 β’ ((πΎ β AtLat β§ π β π΄) β (π β (πβπ) β π β βͺ π¦ β (Fin β© π« π)(πβπ¦))) |
5 | eliun 4995 | . . . 4 β’ (π β βͺ π¦ β (Fin β© π« π)(πβπ¦) β βπ¦ β (Fin β© π« π)π β (πβπ¦)) | |
6 | 4, 5 | bitrdi 286 | . . 3 β’ ((πΎ β AtLat β§ π β π΄) β (π β (πβπ) β βπ¦ β (Fin β© π« π)π β (πβπ¦))) |
7 | elin 3955 | . . . . . . 7 β’ (π¦ β (Fin β© π« π) β (π¦ β Fin β§ π¦ β π« π)) | |
8 | elpwi 4605 | . . . . . . . 8 β’ (π¦ β π« π β π¦ β π) | |
9 | 8 | anim2i 615 | . . . . . . 7 β’ ((π¦ β Fin β§ π¦ β π« π) β (π¦ β Fin β§ π¦ β π)) |
10 | 7, 9 | sylbi 216 | . . . . . 6 β’ (π¦ β (Fin β© π« π) β (π¦ β Fin β§ π¦ β π)) |
11 | 10 | anim1i 613 | . . . . 5 β’ ((π¦ β (Fin β© π« π) β§ π β (πβπ¦)) β ((π¦ β Fin β§ π¦ β π) β§ π β (πβπ¦))) |
12 | anass 467 | . . . . 5 β’ (((π¦ β Fin β§ π¦ β π) β§ π β (πβπ¦)) β (π¦ β Fin β§ (π¦ β π β§ π β (πβπ¦)))) | |
13 | 11, 12 | sylib 217 | . . . 4 β’ ((π¦ β (Fin β© π« π) β§ π β (πβπ¦)) β (π¦ β Fin β§ (π¦ β π β§ π β (πβπ¦)))) |
14 | 13 | reximi2 3069 | . . 3 β’ (βπ¦ β (Fin β© π« π)π β (πβπ¦) β βπ¦ β Fin (π¦ β π β§ π β (πβπ¦))) |
15 | 6, 14 | biimtrdi 252 | . 2 β’ ((πΎ β AtLat β§ π β π΄) β (π β (πβπ) β βπ¦ β Fin (π¦ β π β§ π β (πβπ¦)))) |
16 | 15 | 3impia 1114 | 1 β’ ((πΎ β AtLat β§ π β π΄ β§ π β (πβπ)) β βπ¦ β Fin (π¦ β π β§ π β (πβπ¦))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwrex 3060 β© cin 3938 β wss 3939 π« cpw 4598 βͺ ciun 4991 βcfv 6543 Fincfn 8962 Atomscatm 38791 AtLatcal 38792 PClcpclN 39416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-om 7869 df-1o 8485 df-en 8963 df-fin 8966 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-lat 18423 df-covers 38794 df-ats 38795 df-atl 38826 df-psubsp 39032 df-pclN 39417 |
This theorem is referenced by: (None) |
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