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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 2727 | . . . . 5 β’ (LSpanβπ) = (LSpanβπ) | |
5 | eqid 2727 | . . . . 5 β’ (LSubSpβπ) = (LSubSpβπ) | |
6 | lshpne.h | . . . . 5 β’ π» = (LSHypβπ) | |
7 | 3, 4, 5, 6 | islshp 38455 | . . . 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 1140 | 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 2936 βwrex 3066 βͺ cun 3945 {csn 4630 βcfv 6551 Basecbs 17185 LModclmod 20748 LSubSpclss 20820 LSpanclspn 20860 LSHypclsh 38451 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-iota 6503 df-fun 6553 df-fv 6559 df-lshyp 38453 |
This theorem is referenced by: lshpnel 38459 lshpcmp 38464 lkrshp3 38582 lkrshp4 38584 dochshpncl 40861 dochlkr 40862 dochkrshp 40863 dochsatshpb 40929 |
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