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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 38366 | . . . . 5 β’ ((πΎ β AtLat β§ π β π΄) β (πβπ) = βͺ π¦ β (Fin β© π« π)(πβπ¦)) |
4 | 3 | eleq2d 2824 | . . . 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 4959 | . . . 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 3927 | . . . . . . 7 β’ (π¦ β (Fin β© π« π) β (π¦ β Fin β§ π¦ β π« π)) | |
8 | elpwi 4568 | . . . . . . . 8 β’ (π¦ β π« π β π¦ β π) | |
9 | 8 | anim2i 618 | . . . . . . 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 616 | . . . . 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 470 | . . . . 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 3083 | . . 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 1118 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwrex 3074 β© cin 3910 β wss 3911 π« cpw 4561 βͺ ciun 4955 βcfv 6497 Fincfn 8884 Atomscatm 37728 AtLatcal 37729 PClcpclN 38353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-om 7804 df-1o 8413 df-en 8885 df-fin 8888 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-lat 18322 df-covers 37731 df-ats 37732 df-atl 37763 df-psubsp 37969 df-pclN 38354 |
This theorem is referenced by: (None) |
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