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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lshpne | Structured version Visualization version GIF version |
Description: A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.) |
Ref | Expression |
---|---|
lshpne.v | β’ π = (Baseβπ) |
lshpne.h | β’ π» = (LSHypβπ) |
lshpne.w | β’ (π β π β LMod) |
lshpne.u | β’ (π β π β π») |
Ref | Expression |
---|---|
lshpne | β’ (π β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lshpne.u | . . 3 β’ (π β π β π») | |
2 | lshpne.w | . . . 4 β’ (π β π β LMod) | |
3 | lshpne.v | . . . . 5 β’ π = (Baseβπ) | |
4 | eqid 2737 | . . . . 5 β’ (LSpanβπ) = (LSpanβπ) | |
5 | eqid 2737 | . . . . 5 β’ (LSubSpβπ) = (LSubSpβπ) | |
6 | lshpne.h | . . . . 5 β’ π» = (LSHypβπ) | |
7 | 3, 4, 5, 6 | islshp 37470 | . . . 4 β’ (π β LMod β (π β π» β (π β (LSubSpβπ) β§ π β π β§ βπ£ β π ((LSpanβπ)β(π βͺ {π£})) = π))) |
8 | 2, 7 | syl 17 | . . 3 β’ (π β (π β π» β (π β (LSubSpβπ) β§ π β π β§ βπ£ β π ((LSpanβπ)β(π βͺ {π£})) = π))) |
9 | 1, 8 | mpbid 231 | . 2 β’ (π β (π β (LSubSpβπ) β§ π β π β§ βπ£ β π ((LSpanβπ)β(π βͺ {π£})) = π)) |
10 | 9 | simp2d 1144 | 1 β’ (π β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2944 βwrex 3074 βͺ cun 3913 {csn 4591 βcfv 6501 Basecbs 17090 LModclmod 20338 LSubSpclss 20408 LSpanclspn 20448 LSHypclsh 37466 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-lshyp 37468 |
This theorem is referenced by: lshpnel 37474 lshpcmp 37479 lkrshp3 37597 lkrshp4 37599 dochshpncl 39876 dochlkr 39877 dochkrshp 39878 dochsatshpb 39944 |
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