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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > psubspi2N | Structured version Visualization version GIF version |
Description: Property of a projective subspace. (Contributed by NM, 13-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psubspset.l | β’ β€ = (leβπΎ) |
psubspset.j | β’ β¨ = (joinβπΎ) |
psubspset.a | β’ π΄ = (AtomsβπΎ) |
psubspset.s | β’ π = (PSubSpβπΎ) |
Ref | Expression |
---|---|
psubspi2N | β’ (((πΎ β π· β§ π β π β§ π β π΄) β§ (π β π β§ π β π β§ π β€ (π β¨ π ))) β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7423 | . . . 4 β’ (π = π β (π β¨ π) = (π β¨ π)) | |
2 | 1 | breq2d 5155 | . . 3 β’ (π = π β (π β€ (π β¨ π) β π β€ (π β¨ π))) |
3 | oveq2 7424 | . . . 4 β’ (π = π β (π β¨ π) = (π β¨ π )) | |
4 | 3 | breq2d 5155 | . . 3 β’ (π = π β (π β€ (π β¨ π) β π β€ (π β¨ π ))) |
5 | 2, 4 | rspc2ev 3614 | . 2 β’ ((π β π β§ π β π β§ π β€ (π β¨ π )) β βπ β π βπ β π π β€ (π β¨ π)) |
6 | psubspset.l | . . 3 β’ β€ = (leβπΎ) | |
7 | psubspset.j | . . 3 β’ β¨ = (joinβπΎ) | |
8 | psubspset.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
9 | psubspset.s | . . 3 β’ π = (PSubSpβπΎ) | |
10 | 6, 7, 8, 9 | psubspi 39276 | . 2 β’ (((πΎ β π· β§ π β π β§ π β π΄) β§ βπ β π βπ β π π β€ (π β¨ π)) β π β π) |
11 | 5, 10 | sylan2 591 | 1 β’ (((πΎ β π· β§ π β π β§ π β π΄) β§ (π β π β§ π β π β§ π β€ (π β¨ π ))) β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwrex 3060 class class class wbr 5143 βcfv 6543 (class class class)co 7416 lecple 17239 joincjn 18302 Atomscatm 38791 PSubSpcpsubsp 39025 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7419 df-psubsp 39032 |
This theorem is referenced by: pclclN 39420 pclfinN 39429 pclfinclN 39479 |
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