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Mirrors > Home > MPE Home > Th. List > Mathboxes > psubssat | Structured version Visualization version GIF version |
Description: A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011.) |
Ref | Expression |
---|---|
atpsub.a | β’ π΄ = (AtomsβπΎ) |
atpsub.s | β’ π = (PSubSpβπΎ) |
Ref | Expression |
---|---|
psubssat | β’ ((πΎ β π΅ β§ π β π) β π β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 β’ (leβπΎ) = (leβπΎ) | |
2 | eqid 2728 | . . 3 β’ (joinβπΎ) = (joinβπΎ) | |
3 | atpsub.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | atpsub.s | . . 3 β’ π = (PSubSpβπΎ) | |
5 | 1, 2, 3, 4 | ispsubsp 39250 | . 2 β’ (πΎ β π΅ β (π β π β (π β π΄ β§ βπ β π βπ β π βπ β π΄ (π(leβπΎ)(π(joinβπΎ)π) β π β π)))) |
6 | 5 | simprbda 497 | 1 β’ ((πΎ β π΅ β§ π β π) β π β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3058 β wss 3949 class class class wbr 5152 βcfv 6553 (class class class)co 7426 lecple 17247 joincjn 18310 Atomscatm 38767 PSubSpcpsubsp 39001 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-psubsp 39008 |
This theorem is referenced by: psubatN 39260 paddidm 39346 paddclN 39347 paddss 39350 pmodlem1 39351 pmod1i 39353 pmodl42N 39356 elpcliN 39398 pclidN 39401 pclbtwnN 39402 pclunN 39403 pclun2N 39404 pclfinN 39405 polssatN 39413 psubclsubN 39445 |
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