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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 7412 | . . . 4 β’ (π = π β (π β¨ π) = (π β¨ π)) | |
2 | 1 | breq2d 5153 | . . 3 β’ (π = π β (π β€ (π β¨ π) β π β€ (π β¨ π))) |
3 | oveq2 7413 | . . . 4 β’ (π = π β (π β¨ π) = (π β¨ π )) | |
4 | 3 | breq2d 5153 | . . 3 β’ (π = π β (π β€ (π β¨ π) β π β€ (π β¨ π ))) |
5 | 2, 4 | rspc2ev 3619 | . 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 39131 | . 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 592 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwrex 3064 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: pclclN 39275 pclfinN 39284 pclfinclN 39334 |
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