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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0psubN | Structured version Visualization version GIF version |
Description: The empty set is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0psub.s | β’ π = (PSubSpβπΎ) |
Ref | Expression |
---|---|
0psubN | β’ (πΎ β π β β β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4397 | . . 3 β’ β β (AtomsβπΎ) | |
2 | ral0 4513 | . . 3 β’ βπ β β βπ β β βπ β (AtomsβπΎ)(π(leβπΎ)(π(joinβπΎ)π) β π β β ) | |
3 | 1, 2 | pm3.2i 472 | . 2 β’ (β β (AtomsβπΎ) β§ βπ β β βπ β β βπ β (AtomsβπΎ)(π(leβπΎ)(π(joinβπΎ)π) β π β β )) |
4 | eqid 2733 | . . 3 β’ (leβπΎ) = (leβπΎ) | |
5 | eqid 2733 | . . 3 β’ (joinβπΎ) = (joinβπΎ) | |
6 | eqid 2733 | . . 3 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
7 | 0psub.s | . . 3 β’ π = (PSubSpβπΎ) | |
8 | 4, 5, 6, 7 | ispsubsp 38664 | . 2 β’ (πΎ β π β (β β π β (β β (AtomsβπΎ) β§ βπ β β βπ β β βπ β (AtomsβπΎ)(π(leβπΎ)(π(joinβπΎ)π) β π β β )))) |
9 | 3, 8 | mpbiri 258 | 1 β’ (πΎ β π β β β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3062 β wss 3949 β c0 4323 class class class wbr 5149 βcfv 6544 (class class class)co 7409 lecple 17204 joincjn 18264 Atomscatm 38181 PSubSpcpsubsp 38415 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-psubsp 38422 |
This theorem is referenced by: (None) |
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