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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 39284 | . . . . 5 β’ ((πΎ β AtLat β§ π β π΄) β (πβπ) = βͺ π¦ β (Fin β© π« π)(πβπ¦)) |
4 | 3 | eleq2d 2813 | . . . 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 4994 | . . . 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 287 | . . 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 3959 | . . . . . . 7 β’ (π¦ β (Fin β© π« π) β (π¦ β Fin β§ π¦ β π« π)) | |
8 | elpwi 4604 | . . . . . . . 8 β’ (π¦ β π« π β π¦ β π) | |
9 | 8 | anim2i 616 | . . . . . . 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 614 | . . . . 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 468 | . . . . 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 3073 | . . 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 | syl6bi 253 | . 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwrex 3064 β© cin 3942 β wss 3943 π« cpw 4597 βͺ ciun 4990 βcfv 6537 Fincfn 8941 Atomscatm 38646 AtLatcal 38647 PClcpclN 39271 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-om 7853 df-1o 8467 df-en 8942 df-fin 8945 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-lat 18397 df-covers 38649 df-ats 38650 df-atl 38681 df-psubsp 38887 df-pclN 39272 |
This theorem is referenced by: (None) |
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