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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lshplss | Structured version Visualization version GIF version |
Description: A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.) |
Ref | Expression |
---|---|
lshplss.s | β’ π = (LSubSpβπ) |
lshplss.h | β’ π» = (LSHypβπ) |
lshplss.w | β’ (π β π β LMod) |
lshplss.u | β’ (π β π β π») |
Ref | Expression |
---|---|
lshplss | β’ (π β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lshplss.u | . . 3 β’ (π β π β π») | |
2 | lshplss.w | . . . 4 β’ (π β π β LMod) | |
3 | eqid 2726 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
4 | eqid 2726 | . . . . 5 β’ (LSpanβπ) = (LSpanβπ) | |
5 | lshplss.s | . . . . 5 β’ π = (LSubSpβπ) | |
6 | lshplss.h | . . . . 5 β’ π» = (LSHypβπ) | |
7 | 3, 4, 5, 6 | islshp 38361 | . . . 4 β’ (π β LMod β (π β π» β (π β π β§ π β (Baseβπ) β§ βπ£ β (Baseβπ)((LSpanβπ)β(π βͺ {π£})) = (Baseβπ)))) |
8 | 2, 7 | syl 17 | . . 3 β’ (π β (π β π» β (π β π β§ π β (Baseβπ) β§ βπ£ β (Baseβπ)((LSpanβπ)β(π βͺ {π£})) = (Baseβπ)))) |
9 | 1, 8 | mpbid 231 | . 2 β’ (π β (π β π β§ π β (Baseβπ) β§ βπ£ β (Baseβπ)((LSpanβπ)β(π βͺ {π£})) = (Baseβπ))) |
10 | 9 | simp1d 1139 | 1 β’ (π β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 βwrex 3064 βͺ cun 3941 {csn 4623 βcfv 6536 Basecbs 17150 LModclmod 20703 LSubSpclss 20775 LSpanclspn 20815 LSHypclsh 38357 |
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-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-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 6488 df-fun 6538 df-fv 6544 df-lshyp 38359 |
This theorem is referenced by: lshpnel 38365 lshpnelb 38366 lshpne0 38368 lshpdisj 38369 lshpcmp 38370 lshpsmreu 38491 lshpkrlem1 38492 lshpkrlem5 38496 lshpkr 38499 dochshpncl 40767 dochshpsat 40837 lclkrlem2f 40895 |
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