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Mirrors > Home > MPE Home > Th. List > obsipid | Structured version Visualization version GIF version |
Description: A basis element has length one. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
obsipid.h | β’ , = (Β·πβπ) |
obsipid.f | β’ πΉ = (Scalarβπ) |
obsipid.u | β’ 1 = (1rβπΉ) |
Ref | Expression |
---|---|
obsipid | β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β (π΄ , π΄) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | obsipid.h | . . . 4 β’ , = (Β·πβπ) | |
3 | obsipid.f | . . . 4 β’ πΉ = (Scalarβπ) | |
4 | obsipid.u | . . . 4 β’ 1 = (1rβπΉ) | |
5 | eqid 2727 | . . . 4 β’ (0gβπΉ) = (0gβπΉ) | |
6 | 1, 2, 3, 4, 5 | obsip 21660 | . . 3 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅ β§ π΄ β π΅) β (π΄ , π΄) = if(π΄ = π΄, 1 , (0gβπΉ))) |
7 | 6 | 3anidm23 1418 | . 2 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β (π΄ , π΄) = if(π΄ = π΄, 1 , (0gβπΉ))) |
8 | eqid 2727 | . . 3 β’ π΄ = π΄ | |
9 | 8 | iftruei 4537 | . 2 β’ if(π΄ = π΄, 1 , (0gβπΉ)) = 1 |
10 | 7, 9 | eqtrdi 2783 | 1 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β (π΄ , π΄) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 ifcif 4530 βcfv 6551 (class class class)co 7424 Basecbs 17185 Scalarcsca 17241 Β·πcip 17243 0gc0g 17426 1rcur 20126 OBasiscobs 21641 |
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-pow 5367 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-pw 4606 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-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fv 6559 df-ov 7427 df-obs 21644 |
This theorem is referenced by: obsne0 21664 |
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