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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > psubcli2N | Structured version Visualization version GIF version |
Description: Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psubcli2.p | β’ β₯ = (β₯πβπΎ) |
psubcli2.c | β’ πΆ = (PSubClβπΎ) |
Ref | Expression |
---|---|
psubcli2N | β’ ((πΎ β π· β§ π β πΆ) β ( β₯ β( β₯ βπ)) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . 3 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
2 | psubcli2.p | . . 3 β’ β₯ = (β₯πβπΎ) | |
3 | psubcli2.c | . . 3 β’ πΆ = (PSubClβπΎ) | |
4 | 1, 2, 3 | ispsubclN 39112 | . 2 β’ (πΎ β π· β (π β πΆ β (π β (AtomsβπΎ) β§ ( β₯ β( β₯ βπ)) = π))) |
5 | 4 | simplbda 499 | 1 β’ ((πΎ β π· β§ π β πΆ) β ( β₯ β( β₯ βπ)) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 β wss 3948 βcfv 6543 Atomscatm 38437 β₯πcpolN 39077 PSubClcpscN 39109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-psubclN 39110 |
This theorem is referenced by: psubclsubN 39115 pmapidclN 39117 poml6N 39130 osumcllem3N 39133 osumclN 39142 pmapojoinN 39143 pexmidN 39144 pexmidlem6N 39150 |
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