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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 2726 | . . . 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 2726 | . . . 4 β’ (0gβπΉ) = (0gβπΉ) | |
6 | 1, 2, 3, 4, 5 | obsip 21612 | . . 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 2726 | . . 3 β’ π΄ = π΄ | |
9 | 8 | iftruei 4530 | . 2 β’ if(π΄ = π΄, 1 , (0gβπΉ)) = 1 |
10 | 7, 9 | eqtrdi 2782 | 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 395 = wceq 1533 β wcel 2098 ifcif 4523 βcfv 6536 (class class class)co 7404 Basecbs 17151 Scalarcsca 17207 Β·πcip 17209 0gc0g 17392 1rcur 20084 OBasiscobs 21593 |
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-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-obs 21596 |
This theorem is referenced by: obsne0 21616 |
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