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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 2726 | . . 3 β’ (leβπΎ) = (leβπΎ) | |
2 | eqid 2726 | . . 3 β’ (joinβπΎ) = (joinβπΎ) | |
3 | atpsub.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | atpsub.s | . . 3 β’ π = (PSubSpβπΎ) | |
5 | 1, 2, 3, 4 | ispsubsp 39129 | . 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 498 | 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 395 = wceq 1533 β wcel 2098 βwral 3055 β wss 3943 class class class wbr 5141 βcfv 6537 (class class class)co 7405 lecple 17213 joincjn 18276 Atomscatm 38646 PSubSpcpsubsp 38880 |
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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 df-ov 7408 df-psubsp 38887 |
This theorem is referenced by: psubatN 39139 paddidm 39225 paddclN 39226 paddss 39229 pmodlem1 39230 pmod1i 39232 pmodl42N 39235 elpcliN 39277 pclidN 39280 pclbtwnN 39281 pclunN 39282 pclun2N 39283 pclfinN 39284 polssatN 39292 psubclsubN 39324 |
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