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Mirrors > Home > MPE Home > Th. List > obsss | Structured version Visualization version GIF version |
Description: An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
obsss.v | β’ π = (Baseβπ) |
Ref | Expression |
---|---|
obsss | β’ (π΅ β (OBasisβπ) β π΅ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | obsss.v | . . 3 β’ π = (Baseβπ) | |
2 | eqid 2728 | . . 3 β’ (Β·πβπ) = (Β·πβπ) | |
3 | eqid 2728 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
4 | eqid 2728 | . . 3 β’ (1rβ(Scalarβπ)) = (1rβ(Scalarβπ)) | |
5 | eqid 2728 | . . 3 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
6 | eqid 2728 | . . 3 β’ (ocvβπ) = (ocvβπ) | |
7 | eqid 2728 | . . 3 β’ (0gβπ) = (0gβπ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isobs 21647 | . 2 β’ (π΅ β (OBasisβπ) β (π β PreHil β§ π΅ β π β§ (βπ₯ β π΅ βπ¦ β π΅ (π₯(Β·πβπ)π¦) = if(π₯ = π¦, (1rβ(Scalarβπ)), (0gβ(Scalarβπ))) β§ ((ocvβπ)βπ΅) = {(0gβπ)}))) |
9 | 8 | simp2bi 1144 | 1 β’ (π΅ β (OBasisβπ) β π΅ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 βwral 3057 β wss 3945 ifcif 4524 {csn 4624 βcfv 6542 (class class class)co 7414 Basecbs 17173 Scalarcsca 17229 Β·πcip 17231 0gc0g 17414 1rcur 20114 PreHilcphl 21549 ocvcocv 21585 OBasiscobs 21629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 df-obs 21632 |
This theorem is referenced by: obsne0 21652 obselocv 21655 obslbs 21657 |
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